Numerical analysis of projectile penetration and perforation of …anil-lab/others/lectures/CMCE/33... · 2016. 8. 6. · NUMERICAL ANALYSIS OF CONCRETE PENETRATION AND PERFORATION

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. (2016)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2555

Numerical analysis of projectile penetration and perforation ofplain and fiber reinforced concrete slabs

Jovanca Smith1,*,† and Gianluca Cusatis2

1Lecturer, Department of Civil and Environmental Engineering, The University of the West Indies,St. Augustine, Trinidad and Tobago

2Associate Professor, Department of Civil and Environmental Engineering, Northwestern University, Tech BuildingRoom A125, Evanston, IL 60208

SUMMARY

The research presented in this paper deals with the numerical analysis of projectile impact on regular strengthconcrete (RSC), high-strength concrete (HSC), and engineered cementitious composites (ECC) using theLattice Discrete Particle Model (LDPM). The LDPM is chosen in this study as it naturally captures the fail-ure mechanisms at the length scale of coarse aggregate of concrete, and its capabilities include the accuratedepiction of both intrinsic and apparent rate effects in concrete, as well as fiber reinforcement effects. Themodel is used to predict the experimental impact response performed by four independent testing laborato-ries, and for each data set the model parameters are calibrated and validated using a combination of uniaxialcompression, triaxial compression, uniaxial strain compression, and dogbone tests. In the first study, per-foration experiments on RSC and HSC for varied impact velocities are carried out, and the exit velocity iscompared with the available experimental data. The second study focuses on ECC, where multiple impactof steel and plastic fiber reinforced concrete panels are explored. A third investigation is performed on fourRSC panels with varied thicknesses and subjected to the same impact velocity. In this instance, the model isused to predict the penetration depths for the different cases. Finally, in the last study, the response of large-thickness infinite panels of sizes ranging from 300 mm to 700 mm under projectile impact is considered.Copyright © 2016 John Wiley & Sons, Ltd.

Received 16 December 2015; Revised 25 May 2016; Accepted 6 June 2016

KEY WORDS: engineered cementitious composites; high performance concrete; lattice discrete particlemodel; multiple impact; perforation; penetration

1. INTRODUCTION

Safety of protective structural applications can be improved through penetration resistance andreduced fragmentation from a dynamic impact. Traditionally, engineers have used classical struc-tural design and standard steel reinforced concrete to improve impact resistance to civilian andmilitary structures, but the creation of ultra-high performance fiber-reinforced concrete (UHPFRC)has offered improved micro-cracking, strain-hardening and energy absorption capabilities fordesigning impact resistant structures. Engineered cement composites (ECC) or UHPFRCs are clas-sified as concretes with high compressive strength and increased ductility obtained from fiberreinforcement, a modern method to straw reinforced clay bricks used in ancient times. These ECCshave already been used worldwide, providing additional crack control for bridges, buildings, anddams. Their effectiveness against impact is determined through penetration depth, ballistic limit,crater size, and failure pattern, which also gives essential information to determine the local failureprocesses taking place. Important parameters to consider when analyzing these responses are panel

*Correspondence to: Jovanca Smith, Lecturer, Department of Civil and Environmental Engineering, The University ofthe West Indies, St. Augustine, Trinidad and Tobago.

†E-mail: [email protected]

Copyright © 2016 John Wiley & Sons, Ltd.

J. SMITH AND G. CUSATIS

properties (eg. strength, size, microstructure, reinforcement, and boundary conditions), projectileproperties (eg. material properties, size, and shape), and impact conditions (eg. velocity/strain-rate,ideal impact, and non-ideal impact) [1].

Many authors [2–20] have investigated the response of concrete under ideal dynamic impactconditions, and demonstrated that the penetration and perforation resistance of the material is pro-portional to material strength. This phenomenon is typically explained using wave propagationconcepts. Simply speaking, when a compressive wave is generated upon impact, it propagatesthrough the material, and upon reaching the back end of the panel-air interface, it is reflected as atensile wave. When this tensile wave becomes greater than the tensile capacity of the specimen atthe rear face, scabbing occurs. The tensile wave travels through the specimen until it again meets aless dense medium and it is reflected again as a compressive wave. This back and forth continuesdamping the amplitude of the stress waves, and damage of the panel ceases once the stress amplitudeis dampened below the tensile capacity of the material [21]. Failure comparison for similar appliedstresses in regular strength concrete (RSC) and UHPFRC materials shows, in general, reduced fail-ure for the UHPFRC because the maximum applied stresses in UHPFRC must overcome highermaterial strength.

This elastic stress wave phenomenon also reveals why fiber addition to the panels has little effecton penetration depth, but significantly affects crater size. The discontinuous nature of fibers resultsin little contribution to the tensile strength of concrete panels, but they can boost energy dissipationand arrest crack propagation. A study by Riedel [12] found the fracture energy of UHPFRC understatic loading conditions to be two orders of magnitude higher than the RSC. In conventional con-crete, interlocking of aggregates transfer forces across the crack, but in the case of fiber-reinforcedconcrete, the fiber provides an additional avenue for force transfer across crack. Moreover, ECCswith high modulus fibers, like steel, have lower strains with higher tensile strength and cause higherstrength and toughness of the material to improve impact resistance [22]. Low modulus fibers, likepolyvinyl alcohol or PE, have higher fiber-bridging capability and better energy absorption capa-bility, provide high strain capability, and reduce the width of micro-cracking to improve impactresistance. These observations have led to studies on the use of hybrid fiber reinforcement contain-ing mixtures of various types of fibers. Almusallam [21] investigated the response of hybrid-fiber(steel and polyolyphene mix) reinforced concrete slabs to hard projectile and concluded the optimaldesign ratio for the steel:plastic hybrid fibers is 2:1.

Concrete with steel reinforcement grids, also called Ferrocement, is often used with plain concretepanels and ECCs. Kamal [16] determined that one steel layer and three steel layers of Ferroce-ment on each face of plain panels reduces the penetration depth by 27.5% and 30%, respectively,but increasing the quantity has little effect on the penetration depth. However, similar to the fiberphenomenon, the increased Ferrocement has more significant effect on the front/rear face damage.Likewise, Vossoughi [23] used polypropelene and zylon fabric to reinforce concrete externally, andobserved that the materials held the rear crater together more efficiently than unreinforced spec-imens. When investigating impact, it is also important to consider the strain rate effects on thereinforcement in addition to the concrete. An investigation by Riedel et al. [12] studied fiber pull-outunder both quasi-static and dynamic loading. The obtained exponential softening law of UHPFRCshowed that, although the fracture energy — the area under the softening curve — is quite similarunder static and dynamic conditions, differences are observed in the peak ultimate strength and theshape of the softening curve.

Projectile size and shape also have tremendous effect on the observed failure characteristics, andby extension the penetration depth and ballistic limit. Some common projectile types are blunt,conical, and ogive nose. Blunt projectile failure cause plugging, while ogive and conical shapesresult in enlargement of hole [1]. Shiu [24] explored how impact force varies for different noseshaped projectiles, and concluded that for the same target thickness and missile mass, flat shapes canhave greater penetration than the sharper nose if the energy converted from the pressure wave duringimpact is greater for the flat nose. For a ratio of target thickness to missile diameter less than orequal to 1, the amplitude of the transmitted wave is reflected on the opposite side, causing scabbingand creating internal friction — this makes it easier for the flat nose missile to penetrate due to theinternal damage of the specimen, while the conical nose rebounds for the given velocity. However,

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2016)DOI: 10.1002/nag

NUMERICAL ANALYSIS OF CONCRETE PENETRATION AND PERFORATION

when the ratio of target thickness to missile diameter is greater than 1, the conical projectile has agreater penetration.

Smith et al. [13] performed a parametric study of ideal penetration impact with an ogive nosepenetrator on concrete comparing RSC, ultra high performance concrete (UHPC) with no reinforce-ment, a 25% stronger and weaker UHPC called UHPC-U and UHPC-L respectively, and UHPFRCwith 3% and 5% steel fiber reinforcement. This investigation showed that the UHPC-U and UHPC-L had a very similar penetration depth and exit velocity as UHPC for various impact velocities, andthe results deviated only around the ballistic limit. The same patterned behavior was observed forthe addition of fiber reinforcement. Adding the reinforcement, or increasing the quantity of rein-forcement did not affect the penetration depth, but the results were scattered around the ballisticlimit. Discrete jump in energy state around the ballistic limit proves that small deviations in materialand panel properties lead to significant change in the results and this is the reason why this zone iscalled ‘zone of mixed results’ [12].

Although many investigations focus solely on symmetric impact of penetrators into targets,penetrator-target interaction is complex and asymmetric, and it is important to also consider impactunder imperfect or non-ideal conditions to obtain realistic assessment of impact resistance in prac-tical applications. Investigation of some non-ideal conditions include multiple hits, various anglesof attack and obliquity [25, 26], rotating projectiles, and fiber angle. Under extreme strain rates, forvelocities above 1000 m/s, the projectile can no longer be considered rigid, as it may erode or meltduring the penetration. Forrestal [27] reported a mass decrease of about 3.5% for striking velocitiesof 1000 m/s on steel projectiles.

The various methods used to investigate concrete behavior under impact include: experimental,empirical, analytical, and numerical analysis and combinations of any type with the objective tounderstand the penetration phenomenon and strengthen the material to withstand the toughest loadsat minimal cost. The drawbacks with experimental testing is cost effectiveness, the quantity of teststhat can be performed, and repeatability of exact testing conditions, to obtain precise results. This isespecially important because around the ballistic limit, there is a discrete jump in energy and hencea wider range of results. Moreover, tests are often scaled when performed in an experimental setting,and size effect [28–31] may not be accurately accounted for in the analysis. When using empiricalmethods to analyze the penetration/perforation phenomena, the results are limited to the boundaryconditions that were used for the test, with little or no insight of the failure mechanisms involved.Because the actual mechanics are not explicitly captured, it can be dangerous to use these methodsto perform predictions for a wide range of protective structures. Two popular analytical methodsused to analyze impact are: 1. the two stage model [7], and 2. the (spherical) cavity expansion model[27, 32]. These methods are rooted in mechanics as they are based on the use of conservation lawsfor continuum solids, but do not capture all the necessary features of the penetration/perforationevent as they are restricted to one mode of failure to simplify the analysis. A numerical model(discrete or continuous) provides more detailed analysis of stresses and deformations occurringduring the impact and gives in depth information about the mechanics involved in penetration. Forquasi-brittle materials, like concrete, it is necessary to use numerical models that provide accuraterepresentation of the cracking phenomenon observed during the experiments to ensure accuratepredictions. Additionally, it is necessary to accurately depict the various components for the entiresystem like the projectile and target dimension with updated strains and deformations during impact,strain-rate effects, frictional effects, hardening, softening, and shear loading effects, crack initiationand propagation, and reinforcement contributions.

The objective of this study is to evaluate the capabilities of the numerical Lattice Discrete ParticleModel (LDPM) under ideal impact conditions through:

1. Calibration, validation, and prediction of various concrete mixes under both quasi-static anddynamic loading conditions with accurate failure responses

2. Evidence of accurate intrinsic and apparent rate effects in the dynamic loading regime3. Demonstration of accurate energy dissipation during penetration and perforation as well as

energy transfer between concrete and fiber inclusions utilizing the LDPM fiber law, and togain insight on damage suppression techniques for various types of concrete.



2. REVIEW OF THE LATTICE DISCRETE PARTICLE MODEL

The LDPM [33, 34] originated from the Confinement Shear Lattice (CSL) Model by Cusatis et al.[35, 36] and it depicts accurate material behavior under numerous loading regimes with the addedbenefit of explicitly modeling fracture and fragmentation processes and replicating the randomnessof the crack surface of concrete. Description of the model here is conveniently broken down into 1)particle mesh generation, 2) constitutive law, 3) fiber interaction law, and 4) rate dependence ofthe model.

2.1. Definition of the discrete particle generation

Through the adoption of a granulometric distribution of particles and a Fuller curve, a random sizedistribution of aggregate particles is generated for a given volume, cement content .c/, aggregateto cement ratio .a=c/, water to cement ratio .w=c/, maximum .da/ and minimum aggregate size.d0/. Aggregates are assumed to be spheres, which are randomly introduced in the volume of inter-est in a way that prevents particle overlapping and ensures all particles are included in the givenvolume. Subsequent to aggregate generation, a triangulation of particle centers is executed form-ing tetrahedra. Following this process is a tessellation of the tetrahedra with exterior sides to forma polyhedral cell. These polyhedral cells contain an aggregate particle with concrete matrix. Thetriangular matrix interface, named facets, represent the location where neighboring cells interactand stresses and strains are defined. The geometric idealization of the concrete mesostructure withreference to two adjacent particles is displayed in Figure 1(a) and further explained in [34].

2.2. Kinematics and constitutive behavior

The LDPM describes the meso-structure deformation of concrete through the rigid body kinematicsof each individual cell. For certain displacements and rotation rates of the polyhedral cells, the

Figure 1. (a) Discrete adjacent polyhedral cell with internal aggregate sphere and exterior facet,(b) Deformed configuration for one LDPM tetrahedron edge, (c) Fiber and facet interaction, (d) Effec-tive stress versus effective strain curve with strain-rate effect; (e) Elastic domain with strain-rate effect,

(f) Dynamic Increase Factor for Hanchak RSC with and without intrinsic rate effect.



relative displacement rates at the facet centroids are used to obtain the facet strain rates in the currentdeformed configuration (Figure 1(b)):

P"N DnT� PuC �

`I P"M D

mT� PuC �

`I P"L D

lT� PuC �

`(1)

where n represents the vector component normal to the projected facet (parallel to the tetrahedronedge), m and l represent the shear components on the facets (mutually orthogonal to n), �u� isthe displacement jump between facets associated with two neighboring aggregate particles `, is theinitial length of the tetrahedron edge associated with the facet. In the elastic regime as well as duringelastic unloading/reloading the stress rates are assumed to be proportional to the correspondingstrain rates: P�N D EN P"N ; P�M D ET P"M ; P�L D ET P"L where EN=E0, the effective normalmodulus, ET =˛EN , and ˛ D shear-normal coupling parameter.

For stresses and strains beyond the elastic limit, LDPM meso-scale failure is characterized bythree mechanisms: 1) Fracture and cohesion due to tension and tension-shear, 2) Compaction andpore collapse from compression, and 3) Friction due to compression-shear.

For tensile loading ("N > 0), the fracturing behavior is formulated through an effective stress � ,

and effective strain ", given by � Dq�2N C .�

2M C �

2L/=˛ and " D

q"2N C ˛."

2M C "

2L/ which

define the normal and shear stresses as �N D "N�="I �M D ˛"M�="I �L D ˛"L�=".The effective stress � , must satisfy the inequality 0 � � � �bt ."; !/, where ! characterizes the

interaction between normal and total shear strains and stresses: tan.!/ D "N =p˛"T D �N

p˛=�T

The strain dependent stress boundary �bt , is an exponential function of the maximum effectivestrain

�bt D �0.!/ exp

��H0.!/

h" � "0.!/i

�0.!/

�(2)

where �0.!/ D �tr2st .� sin.!/ C

qsin2.!/C 4˛ cos2.!/=r2st /=Œ2˛ cos2.!/�, hxi D max¹x; 0º,

H0.!/ D Hs=˛C.Ht�.Hs=˛//.2!=�/nt ,Hs D rsE0,Ht D 2E0= .`t=` � 1/, `t D 2E0Gt=�2t ,

rst D �s=�t , rs D shear softening parameter, and �t , �s , Gt , and `t are material parameters.For compressive loading ("N < 0), the normal stress is computed through the inequality��bc."D; "V / � �N � 0, where �bc is a strain-dependent stress boundary function of the volumet-ric strain, "V D .V � V0/=3V0, and the deviatoric strain, "D D "N � "V , where V and V0 are thecurrent and initial LDPM tetrahedron volumes, respectively. The upper elastic limit is defined by avolumetric strain of "c0 D �c0=E0, where �c0 is the mesoscale compressive yield stress. Beyondthe elastic limit, pore collapse occurs with a linear evolution of the compressive boundary, �bc D�c0Ch�"V � "c0iHc.rDV / until the volumetric strain limit "c1 D �c0"c0, beyond which reharden-ing is modeled with an exponential evolution �bc D �c1.rDV / exp Œ.�"V � "c1/Hc.rDV /�c1.rDV /�where Hc.rDV / D Hc0=.1 C �c2/, rDV D "D="V , �c1.rDV / D �c0 C ."c1 � "c0/Hc.rDV / and�c0, Hc0, �c0, �c1, and �c2 are material parameters.

Incremental plasticity is used to simulate the increase in shear strength from friction due to normalcompressive stresses. Plastic strain increments, P"pM D

P�@'=@�M , and P"pL DP�@'=@�L, where � is

the plastic multiplier, are used to calculate incremental shear stresses P�M D ET .P"M � P"pM / and

P�L D ET .P"L� P"pL/. The plastic potential is given by ' D �T ��bs.�N /, where �bs is the nonlinear

frictional law for shear stress �bs.�N / D �s C .�0 � �1/�N0Œ1 � exp.�N =�N0/� � �1�N . Thenormal stress value �N0, defines the internal friction coefficient transition from its initial value �0to its final value �1.

2.3. Fiber interaction law in LDPM

This section describes the fiber interaction law in LDPM as formulated in [37, 38] and adoptedhere to simulate fiber-reinforced UHPC. The number of fibers, Nf in the concrete is first fashionedthrough the fiber volume fraction Vf , as Nf D INT Œ4Vf V=�d2fAf � where V is the concrete



volume, Af is the cross-sectional area of the fiber, and df is an equivalent diameter for the fiber.The interaction between the fibers and the matrix is defined at the intersection of the fiber and thefacet. Figure 1(c) depicts this interaction between a projected facet and a fiber. In the figure, Ls andLl depict the shorter and longer embedment lengths of the fiber respectively and the unit vectors forthe fiber, nf and facet, n is also shown.

At the meso-level, equilibrium allows the assumption of parallel coupling of the fibers and facetsgiving the total stress, � on each facet as � D � c C .1=Ac/

P.f�Ac/

Pf .w/, where � c is the facetstress vector, Ac is the facet area, Pf D Pf nf is the crack bridging force, and uf is the directionof the fiber (Figure 1(c)). For an elastic material response or compressive and compressive-shearloads, the fiber force, Pf , can be assumed to be negligible. For tensile loading, Pf is assumed tobe a function of the crack opening, which can be computed as w D wNnCwMmCwLl when thedeformation of the tensile facet exceeds the elastic limit, and

wN D `.�N � "N =EN /I wM D `.�M � "M=ET /I wL D `.�L � "L=ET / (3)

In turn, the crack opening can be related to the relative displacement between concrete and thefiber (slippage) on either sides of the crack. The bridging force versus slippage law is based on thework of Li et al. [10]. For a fiber orthogonal to the crack plane and subject to axial load only, one canwrite P.v/ D Œ.�2Ef d

3f.�0v C Gd /=2�

1=2 during the debonding phase and P.v/ D P0Œ1 � .v �

vd=Le/�Œ1C .ˇ.v � vd /=df /� during the frictional pull-out phase. The frictional pull-out starts atfull debonding characterized by the critical slippage vd D .2�0L2e=Ef df /C .8GdL

2e=Ef df /

1=2.In the previous equations, P0 D �Ledf �0, Le is a generic embedment length, Gd is the bondfracture energy, and Ef is the fiber elastic modulus. When the interfacial friction is independent,increases, or decreases with slippage, ˇ D 0;> 0, or < 0, respectively.

For more general situations and loading conditions, when the orientation of the embedded seg-ment and the crack bridging segment of fiber are different, localized fracture and crushing can occurreducing the embedded fiber length by a spalling length, sf D PfN sin.=2/=ksp�tdf cos2.=2/,where PfN is the crack-bridging force component orthogonal to the crack, is the angle betweenthe normals to the embedded segment and the crack, �t is the meso-scale tensile strength, '0

fis the

change in fiber orientation, and ksp is a material parameter.The snubbing model adopted in LDPM-F assumes that the fiber in a perfectly flexible manner,

wraps around the exit of the tunnel crack that was shortened by spalling. The change in the ten-sile load for the flexible tendon is expressed as Pf D exp.ksn'0f /P.v/ where P is the summationof all slip-friction and debonding forces parallel to the embedment length, and ksn is the snubbingparameter. For a Pf leading to fiber rupture, there will be no contribution from the fibers to struc-tural strength and ductility. Single fiber pullout tests on polyvinyl alcohol (PVA) revealed lowerrupture loads for increasing '0

f. This strength reduction (due to bending) is adopted in LDPM-F,

and expressed as �f � �uf e�krup'

0

f where �f is the axial stress in the crack-bridging segment,krup is a material parameter and �uf is the ultimate tensile strength of the fiber.

The micro-mechanics of fiber pull-out are well documented in the literature and are furtherdiscussed in detail by [37, 38].

2.4. Strain rate dependence of concrete behavior in LDPM

The clear change in mechanical properties that exists for a change of loading rate of concrete cate-gorizes it as a rate-dependent material [39–48]. Depending on the strain rate, this behavioral changecan be caused by intrinsic effects in the material composition, or by apparent effects of the materialand external boundaries.

At the meso-scale, the intrinsic concrete effects are due to 1) creep, 2) rate dependent cohe-sive crack opening, and 3) load carrying capacity of capillary and adsorbed water. Concrete creep,an increase in deformation rate for a constantly applied load, is believed to be a result of shearslips among the Calcium Silicate Hydrates platelets [49, 50], and should be considered for strainrates below 10�7 s�1. This visco-elastic process does not play a major role in high rate penetrationsimulations, and was not included in this study.



The second effect follows the activation energy theory for ruptured bonds governed by theclassical Maxwell-Boltzmann equation. The LDPM adopts the theory formulated by Bažant [51],in which the static cohesive law is scaled vertically by a function of the crack opening rate�ch.w; Pw/ D Œ1C c1arcsinh. Pw=c0/�f .w/

In the earlier equation, �ch is the cohesive stress, w is the crack opening, Pw is the crack openingrate, f .w/ is the cohesive law under static conditions, and c0 and c1 are two material parameters.The elastic rate can be considered negligible compared with the opening rate during late stage soft-ening, hence, the LDPM effective strain can be written as P" D Pw=`CP�=E0 � Pw=`, and the cohesivelaw can be transformed in a strain dependent stress boundary extending the one given in Equation 2to include the rate effect as given below

�bt ."; P"; !/ D F.P"/�0.!/ exp

��H0.!/

h" � "0.P"; !/i

�0.!/

�(4)

where "0.P"; !/ D F.P"/�0.!/=E0, and F.P"/ D 1C c1arcsinh.P"=.c0=`//Figure 1(d) shows the effective stress versus effective strain curve with strain rate dependence,

and Figure 1(e) shows the rate effect in terms of shear stress versus normal stress failure envelopeunder the assumption of rate-independent frictional behavior.

The third effect, influence of pore water on concrete strength, is negligible under quasi-staticloading (loading rates around 10�7s�1 to 10�3s�1) as the water is more compressible than theconcrete matrix. However, for triaxial compressive loading under high strain rates, pore water hassignificant effect on material properties as pressure builds up during pore collapse with insufficienttime for the water to drain out. Forquin and coworkers [52] report an increase of 55% of the hydro-static stress measured in saturated concrete samples from a strain of 0.08 to a strain-rate of 120 s�1.Moreover, there is also a reported further increase in saturated concrete strength, compared with dryconcrete, under tensile loading for higher rates. One hypothesis for such behavior, is the proportion-ality between the force and velocity of rigid plates containing Newtonian fluid being pulled apart.This concept is called the Stefan effect, and while admired by some authors [53, 54], it is questionedby others who doubt its applicability to absorbed and hindered water in concrete, especially underpartial saturation.

An exploration of the concrete curing techniques relevant to the experimental data labeled asJinzhu, WES, and ECC, revealed that tests were not performed immediately following moist curing.Jinzhu [17] obtained a dry condition for the samples that were tested 60 to 90 days after casting;Cargile [55] reported also dry conditions for the WES samples; and Almusallam [21] reported thatECC specimens were allowed to dry for an additional 2 weeks prior to testing. Hanchak [18] curingtechniques were not expounded, but a limiting w/c ratio of 0.22 was adopted for the high strengthconcretes (HSC), which naturally reduces the pore water capacity of the material [56]. As a firstapproximation, the effect of pore water was not considered in the current work for all the performednumerical simulations.

The discrete nature of LDPM naturally captures many apparent rate effects within the spatial andtemporal resolution of the model, that must be accounted for separately with continuum models.During a dynamic event, stress waves travel through concrete and depend on 1) effect of the externalboundaries, or 2) inertia effects. A common misconception is for the associated apparent strengthincrease to be seen as an inherent or natural material property as opposed to the effect of boundaryconditions and inertia. The presence of external boundaries cause fictitious confinement and arti-ficial strength increase [57]. Internal effects of inertia result in multiple crack initiation and crackbranching instead of large localized crack formation. Under high rates, many cracks initiate in theinitial stage, propagate throughout the material and branch into additional crack paths [58]. Internalheterogeneity provides additional effects, and strength increase [57], while microscopic crack pat-tern changes [53, 59] as a result of the crack passing through the aggregate particles instead of thetypical and weaker interfacial transitional zone of the aggregate-binder mixture.

Another effect observed under extreme strain rates is comminution or pulverization [60]. Com-minution is the process where particles disintegrate due to enhanced energy dissipation caused by



the higher strain rates. Bažant and Caner [61, 62] have explored this hypothesis to explain addi-tional energy dissipation needed to match exit velocities of perforating projectiles in the Microplanemodel. Numerical modeling of the phenomenon depends on the spatial resolution of the model. Inthe LDPM, sub meso-scale particle generation, smaller than the LDPM resolution, would need to beaccounted for in the constitutive law. However, results on penetration and perforation simulationsdiscussed in the paper demonstrate the capability of LDPM to capture accurate energy dissipationduring these higher rates without the effect of comminution.

2.5. Parameter calibration

The calibration of the LDPM parameters used in the simulations was pursued by fitting basicmechanical properties available for the concrete used in the penetration/perforation tests. Rarelythese data were complete to the point of fully calibrating all LDPM parameters. Hence, someparameters were estimated on the basis of the parameter values reported in Cusatis at al. [33] forsimilar concretes.

3. NUMERICAL SIMULATION OF HANCHAK DATA

An experimental campaign of steel projectiles impacting reinforced concrete slabs at different veloc-ities was performed by Hanchack et al. [18]. The slabs had dimensions 610 mm � 610 mm �178 mm, and they were reinforced by three layers of square pattern rebars, having a diameter of5.69 mm, with a spacing of 76.2 mm. The ogival nose projectiles had a length of 101.6 mm (4 in), adiameter of 25.4 mm (1 in), and a mass of 0.5 kg. The test setup used in the experimental tests andthe numerical investigation is given in Figure 2(a), and the meshing and measurement of the projec-tile is shown in Figure 2(b). Impact tests were performed on both RSC and HSC having 20 MPa and50 MPa triaxial test compressive strengths of 48 MPa and 140 MPa, respectively. Velocities rangedfrom 310 m/s to 1058 m/s for the RSC and 376 m/s to 998 m/s for the HSC, and in no case theyimpacted the rebars. Hydrostatic (HC), triaxial (TXC), uniaxial unconfined compression (UC), anduniaxial tension (UT) tests were performed on cylinders with radius 31.75 mm and height 127 mm tocharacterize the material. The value of the unconfined compressive strength was estimated throughextrapolation of a quadratic curve fitting from the triaxial compression test results, and the resultsare given in Figure 3(b). The mix design parameters of the RSC and HSC are given in Table I.

The LDPM meso parameters were calibrated by simulating the estimated uniaxial UC test results,the HC results, and the 155 MPa and 620 MPa triaxial tests (TXC) for RSC and 90 MPa and

Figure 2. Experimental setup for Hanchak penetration tests showing (a) slab dimension (b) projectiledimension and numerical meshing.



Figure 3. Calibration and validation results for Hanchak RSC and HSC (a) Hydrostatic compression test, and(b) Triaxial compression test; Perforation results for (c) RSC without intrinsic rate effect (d) HSC without

intrinsic rate effect, (e) RSC with intrinsic rate effect, and (f) HSC with intrinsic rate effect.

Table I. Mix designs for concrete.

Hanchak WES

Symbol RSC HSC ECC 5000 Jinzhu

c (kg/m3) 359.35 621.09 446 319.8 469.9w=c 0.41 0.22 0.4 0.46 0.44a=c 5.38 2.84 3.93 5.87 3.71da (mm) 9.5 9.5 10 9.5 15d0 (mm) 4 4 5 4 5nF 0.56 0.56 0.5 0.56 0.58

643 MPa for HSC. Without further change to the model parameters, all other triaxial compressiontests for RSC (20 MPa, 270 MPa, 400 MPa, 615 MPa, 620 MPa) and HSC (50 MPa, 170 MPa,305 MPa, 420 MPa, 477 MPa, 542 MPa) were used to validate the calibrated parameters. The LDPMparameters calibrated for each test is shown is Table II. Each test was performed with three differ-ent concrete meso-structure arrangements to provide numerical experimental scatter in the results,and the respective average value was used for comparison with the experimental data. Detaileddescriptions of the various quasi static tests are given below.

The uniaxial compression test was modeled through the application of velocity to the top of thecylinder through a rigid steel disc in contact with the cylinder under a high sliding friction constraint.This friction constraint is designed to force a set of nodes to move over a surface. The constraintperpendicular to the surface is treated using a master-slave formulation, and resistance to slidingwithin the plane is treated with a stick-slip friction model. Once the tangential in plane force, Ft isless than the maximum friction force, Fmax D �.s/Fn, where Fn is the force normal to the surface,the LDPM surface nodes stick to the steel surface nodes. When Ft exceeds Fmax , the nodes startsslipping and the friction force is gradually reduced. The high friction static, �s , dynamic, �d , andcharacteristic length parameters, s0, define�.s/ D �d C .�s ��d /s0=.s0Cs/, and values of 0.18,



Table II. Meso-scale parameters for concrete.

Hanchak WES

Static parameter RSC HSC ECC 5000 Jinzhu

Compressive yielding stress (MPa) 140 200 215 150 100Elastic normal modulus (MPa) 27000 115000 58642 31250 46480Initial hardening modulus ratio 0.7 0.7 7 1.6 0.4Densification ratio 1 1 0.3 1 1Transitional strain ratio 1.5 1 1 7 4Shear strength ratio 1.5 10 4.75 2.8 4.1Initial friction 0.4 0.2 0.15 0.5 0.2Softening exponent 0.2 0.2 0.01 0.2 0.2Deviatoric strain threshold ratio 0.5 0.5 1 1 1Deviatoric damage parameter 5 5 5 5 5Asymptotic friction 0 0 0 0 0Transitional stress (MPa) 600 1 600 600 600Tensile strength (MPa) 4 8 4.4 3.4 2.65Tensile characteristic length (mm) 60 110 128 100 100Shear softening modulus ratio 0 0 0 0.01 0

0.015, and 1.5 mm were used, respectively. The parameter, s is the cumulative slip computed duringthe simulation. Figure 3(b) shows the UC calibration results for the RSC and HSC. The HC test wasperformed by applying a hydrostatic load to the specimen. The load was applied in pressure controlin axial and radial directions, and the comparison between numerical and experimental results areshown in Figure 3(a) for RSC and HSC, respectively. Triaxial compression tests were performedin two stages, a hydrostatic compression stage after which the radial pressure was held constant,and the axial force was increased by applying an axial load in displacement control. Figure 3(b)shows the result of the calibration (lighter points) as well as the validation (darker points) for theRSC and HSC, respectively. Finally, the uniaxial tensile test was modeled through contact with arigid hexahedral steel block, interacting with the specimen through a master and slave algorithm,for an applied load in displacement control. An average peak stress of 3.6 MPa was obtained for thenumerical simulation, and compared well with the experimental value of 4 MPa. Prior to complet-ing the numerical penetration investigation, the intrinsic rate dependent parameters were estimatedbased on the RSC Hanchak parameters and a typical dynamic increase factor (DIF) versus strain ratecurve. The dynamic increase factor is a ratio of the quasi-static concrete strength to the strength ofthe material under various strain rates. The numerical response of this test was compared with exper-imental data [57] for compression and the plot comparison is given in Figure 1(f) for a c0 D 10��61/s and c1 D 10 � �2. These rate effect parameters were used for all the simulations presented inthis paper to capture the rate dependence associated with intrinsic effects.

The computational model used to simulate the penetration is based on different sub-models: theconcrete slab is modeled using an LDPM network, the steel reinforcement using elasto-plastic beamelements, and the projectile using rigid solid hexahedral finite elements. The numerical simulationin this section as well as the ones in the rest of the paper were performed by using Modeling andAnalysis of the Response of Structures (MARS), which is a computational framework very effectiveat handling complex systems characterized by the interaction of different components [63]. The pro-jectile was simulated as a rigid body, akin to experimental observation, which showed only minornose erosion and did not fracture during penetration. Both the concrete-rebar interaction and theprojectile-concrete contact were modeled through a penalty algorithm. The concrete-rebar constraintwas modeled by penalty between the node list of the rebars and the LDPM tetrahedra overlappingwith the rebars, which allowed for the required non-linear bond slippage although an initial per-fect bond is assumed. In making a comparison from the HSC to the RSC, identical reinforcementplacement and bond were used generating a fair opportunity to visualize the effect of the variedconcrete types on overall behavior and failure. Contact forces between projectile and concrete weremodeled through a node-node penalty law. Furthermore, a stick-slip friction model was used forthe force contact list for the concrete-projectile interaction. This friction algorithm computes the



incremental tangential force between the two objects perpendicular to the normal direction at thecontact point. The tangential force is controlled by a static friction force parameter if the projectileis sticking, or a kinetic friction force parameter if it is slipping. For a tangential force exceeding thestatic force, the contact switches from stick to slip and vice-versa when the tangential force is lessthan the kinetic friction force [63]. Typical static and kinetic friction parameters of 0.35 and 0.25for the steel-concrete interaction were used.

The concrete slab undergoes an initial compressive phase followed by the tunneling phase forsufficiently thick panels, and finally exterior cracking upon exit. For velocities below the ballisticlimit (the minimum velocity which allows the projectile to perforate the slab) of the panel, the pro-jectile velocity terminates at 0 m/s and perforation of the panel is prevented. Results for projectileexit velocity versus striking velocity curves (Figure 3(c–f)) have an excellent agreement with theexperimental results. They capture the ballistic limit very well, in addition to the discrete jump in theenergy state around the ballistic limit. An added benefit of numerical simulations is the easy abil-ity to perform additional impact tests at low costs. In addition to the experimental tests carried out,additional numerical tests were performed around the ballistic limit to highlight the transition of thediscrete energy jump. In addition to the average of three seeds, the results show the maximum andminimum of the tests performed as the top and bottom points of the error bars, respectively. More-over, the importance of the apparent rate effect is clearly revealed when comparing the experimentsperformed neglecting the intrinsic rate for RSC and HSC (Figure 3(c) and (d) respectively), to thosewith the intrinsic rate effects for RSC and HSC (Figure 3(e) and (f) respectively). When the intrinsicrate dependent phenomena are omitted, there is about 50 m/s difference in the ballistic limit.

In addition to the numerical plots, another essential part of investigating the penetra-tion/perforation of concrete is the failure modes. Figure 4(a) and (b) gives the experimental failureobserved during the experiments for the impact and exit faces for RSC respectively, and Figure 4(c)

Figure 4. Perforation failure pattern comparison for Hanchak RSC at 749 m/s for experimental (a) impactface (b) rear face; and numerical (c)impact face (d) rear face.



and (d) gives the respective numerical front and rear face. When the failure pattern for HSC, Figure5, is compared to the failure pattern observed for RSC, Figure 4, one can observe an increase inthe damage zone in the HSC. Although there is a higher ballistic limit for the HSC, this result isattributed to the lower w/c ratio in the HSC mix, leading to a stronger but more brittle material. Thecolor gives the crack opening, and extent of damage of the panel for a range of 0.05 to 0.3 mm forblue to red. The adopted post-processor allowed for the removal of the outer skin in the numericalresults to visualize damage throughout the specimen. The actual spalling observed in the outer facewas outlined in the figures with a magenta line. The percent differences in the failure zone areas ofthe RSC and HSC between the experimental and numerical results were 40.6% and 28.3% at thepanel front and 15.7% and 8.3% at the panel rear, respectively. The overestimation of the resultscan be attributed to the fact that the maximum cracks, while viewed as red, may not have resultedin the concrete visually spalling. Careful observation of the experimental results displays concretesections within a fine crack boundary still intact. The larger inclusions and lower density on the RSCwould contribute to less surface damage than the HSC, and also explain the reason for an increasein the damage zone with a higher ballistic limit for the HSC. Additional variation of the areas canbe attributed to the scatter expected for concrete with only one experimental failure result provided.

4. NUMERICAL SIMULATION OF ENGINEERED CEMENTITIOUS COMPOSITE

An experimental investigation on regular strength cementitious composite (RSCC) and high strengthcementitious composite (HSCC) with a combination of steel and polyolyphene plastic fibers wereperformed by Almusallam [21] to obtain the optimal fiber combination to reduce concrete damageunder projectile loading. The square hybrid fiber reinforced slabs with side dimensions 600 mmand thickness of 90 mm were mounted onto steel plates (Figure 6a). The HSCC was modeled usingLDPM for various volume fractions of fiber reinforcement: 0% (HM0), 0.6% steel fiber (HM1), and

Figure 5. Perforation failure pattern comparison for Hanchak HSC at 743 m/s for experimental (a) impactface (b) rear face; and numerical (c) impact face (d) rear face.



Figure 6. Experimental setup for Almusallam penetration tests showing (a) slab dimension (b) projectiledimension and numerical meshing.

Table III. Concrete fiber interaction material parameters.

Parameter Steel fiber Plastic fiber

Elastic modulus (GPa) 200 4Spalling (MPa) 150 150Fiber strength (MPa) 2600 30Fiber strength decay 0.4 0.3Snubbing parameter 0.05 0.5Bond strength (MPa) 11.5 3Debonding fracture energy (N/m) 0 6Pull out hardening 0 0.005Plastic parameter 1 1Cook Gordon parameter 2 2

0.6% plastic fiber (HM4). Steel fibers had a length of 13 mm, diameter of 0.2 mm, with 2600 MPatensile strength and 200 GPa elastic modulus, whereas, the plastic polyolyphene fibers had lengthequal to 50 mm, a diameter of 0.015 mm, tensile strength of 30 MPa and elastic modulus of 4 GPa.The concrete fiber interaction parameters, obtained from previous LDPM numerical simulation [37,38], are given in Table III. In addition to the fiber reinforcement, the composites were reinforcedwith Grade 420 steel rebars having diameter 8 mm, spaced at 100 mm on center across the panel intwo perpendicular oriented layers, 15 mm from the rear face of the panel as shown in Figure 6(a).The mesh was oriented to ensure there was no obstruction to the projectile penetration at the centerof the panels. The projectile used for the impact experiments was a 0.8 kg hardened steel bi-conicprojectile with a length of 95 mm and diameter 40 mm (Figure 6(a)). Modeling of the projectile wasperformed in the Cubit platform [64], and the mesh was imported into the Modeling and Analysisof the Response of Structures code.

The nominal unconfined compressive strength of 60 MPa for HSCC was obtained from cylin-ders with 152.4 mm diameter and 304.8 mm length. Because of the lack of experimental data,apart from the compressive strength, the parametric identification of the LDPM concrete parameterswas performed on data relevant to a concrete having similar composition and similar compressivestrength tested at Northwestern University [65]. Three tests were used for the calibration of concretemeso parameters (Table I): UC cube, a 3-point bending (3pbt) notched, and lastly a uniaxial strain(UX) test.

The cube had a length dimension of 40 mm and was sandwiched between two steel cubic blocksmodeled as solid hexahedral rigid body elements having side length 80 mm. The velocity wasapplied to the top of the specimen via compressive displacement of the top block, while the bottomblock was fixed in space. Contact between the LDPM specimen and the rigid blocks were modeled



using static, dynamic, and characteristic length parameters of 0.08, 0.015, and 0.5 mm, respectively,to achieve medium frictional resistance. Results of the calibration are shown in Figure 7(a), whichcompares the average of numerical and experimental specimens.

The 3-point bending test (3pbt) was modeled with beam height D D 92 mm, width B D 40 mm,notch height a D 30 mm, clear span L D 202 mm, and a specimen length of 226 mm. A veloc-ity was applied at the midspan of specimen, and the support boundaries were modeled by havingfixed nodes along the locations of support. To provide normalized experimental versus numericalcomparison, the nominal bending stress was defined according as �n D 3PL=BD2, where P is theapplied load, and the nominal strain was taken to be u=D where u is the crack mouth opening dis-placement (CMOD). Cognizant a localized crack will develop from the notch tip, only 100 mm inthe middle of the beam was modeled with LDPM, the edges were modeled as elastic hexahedral FEblocks as they would not be stressed beyond the elastic regime. The blocks were tied to the LDPMmesh using a master slave constraint. This process was performed to reduce the amount of LDPMmesh particles and the simulation time. Figure 7(b) gives the average result of experimental andnumerical specimens.

A cylinder comprising a height of 40 mm and a diameter of 50 mm was used for the UX tests.The experiments were performed with a steel jacket of 14.15 mm thickness encasing the cylindricalspecimen. The test setup was modeled in LDPM by constraining the nodes on the circumference ofthe specimen to prevent any deformation in the radial direction while the velocity is applied to nodesat the top of the specimen, and the nodes are the bottom of the specimen are held fixed. Figure 7(c)shows the calibration of the paramaters for the UX tests.

Validation of calibrated parameters were performed on an ASTM C39 uniaxial compression testand ASTM modulus of rupture test with no further change to calibrated parameters. The com-pression cylindrical specimen had diameter 76.2 mm and height 152.4 mm, and was modeled bythe application of loads through cubic steel blocks of length 80 mm. The compressive strengthwas obtained by dividing the maximum load force by the cross sectional area of the cylinder. Thenumerical average compressive strength of 59 MPa compares well to the experimental average of55.6 MPa having coefficient of variation (CoV) 3.7%. Modulus of rupture tests had a nominal cross

Figure 7. Calibration, validation and perforation results for Almusallam test results (a) cube compression(b) 3-point bending test (c) confined compression (d) LDPM-FE block comparison; Multiple impact results

showing (e) projectile velocity, and (f) penetration depth.



Table IV. Comparison of experimental and numerical results for ECC.

Mix Specimen name Impact velocity (m/s) Penetration depth (%) Exit velocity (m/s)

Experimental Numerical Numerical

HM0S1 124.9 100 100 2.4S2 107.6 38.9 21.2 n/aS3 91 33.3 17.6 n/a

HM1S1 124.8 100 100 2S2 131.9 100 100 5.3S3 107.7 100 20.7 n/a

HM4S1 125.4 46.7 25.6 n/aS2 128.5 100 100 4S3 107.8 36.8 20.2 n/a

sectional dimension,D of 153 mm, a length of 559 mm and a support span of 457 mm. The nominalbending stress was defined according to beam theory, � D 3PL=2D3, where P is the applied load.The average maximum experimental stress was 8.3 MPa with CoV 3.6%, and the average maximumnumerical stress was 7.7 MPa.

Subsequent to obtaining material parameters, the LDPM was used to investigate the responseof HPCC to projectile impact. Each numerical simulation was repeated for three seeds to changethe random configuration of the LDPM particles, and for the cases with fiber, the fibers were alsoregenerated for each seed to provide optimal scatter in results. The LDPM panel was first optimizedto obtain the smallest area around the projectile that can be modeled with the fine LDPM meshwithout affecting the results. This optimization was necessary as the number of particles obtained bymodeling the entire concrete block with LDPM particles proved to be too large for the computationcontaining fiber reinforcement. Figure 7(d) shows the velocity versus time curve for the scatter ofresults for plain unreinforced HM0 specimen modeled with the entire LDPM block, compared withmodeling smaller sizes of LDPM blocks with finite element (FE) combination. The results portraythat a range of LDPM-FE block sizes from 100 mm to 500 mm are within reason of expectedexperimental scatter. Akin to the Hanchak experiments, an initial perfect bond is assumed betweenrebar and concrete, but the smaller percentage of locally un-bonded rebar to concrete is highly undershadowed by fiber addition in arresting the crack growth and failure zone making the limitation ofthe model tolerant. Table IV gives the comparison of results for the experimental and numericalpenetration depth for the specimens. The HM0 and HM1 specimens were modeled with the full 600mm-sided LDPM panel, but the HM4 specimen was modeled with a LDPM block of 140 mm.

Analysis of results presented in Table IV shows the ballistic limit to be within the range of theselected impact velocities. Additionally, the results demonstrate a phenomenon observed by Smithet al. [13], where the penetration depth is not highly affected by the presence of fibers. However,the true benefit of the fibers can be seen in Figure 8(a) and (b) showing comparison of panelswithout fibers and panels with fiber reinforcement. The panels with fiber reinforcement have a muchsmaller crater size damage than the panel without fibers. Figure 7(e) and (f), showing the resultsof multiple impact on the concrete panels, also reinforce this effect. Following the first impact,one simulation impacted the panel in the same location as impact one, whereas another simulationimpacted the panel 50 mm away from impact 1. Figure 7(e) shows the second impact to causecomplete perforation of the projectile, whereas the projectile did not perforate in the first impactfor both multiple impact scenarios. However, in the simulation with the impact two, a distance of50 mm away from impact one, the exit velocity was lower, as there was less damage from the initialimpact point. Moreover, although the projectile penetrated only 27% in the initial impact, this wasenough to cause complete perforation in a second impact within a distance of 50 mm from the initialimpact location. This is confirmed by the failure patterns observed in Figure 9(a) and (b) for multipleimpact at a distance of 0 mm and 50 mm, respectively, from the initial impact. The projectile hasmore concrete resistance to penetration a distance of 50 mm away from the initial impact. However,



Figure 8. Comparison of (a) Plain concrete, and (b) Fiber reinforced concrete for Almusallam penetrationtests.

although a second impact in the same location provides less resistance, a second impact 50 mmaway extends the final damage zone of the simulation.

5. NUMERICAL SIMULATION OF WES-5000 CONCRETE TEST DATA

Perforation experiments were performed by the Geotechnical and Structures Laboratory of the U.S.Army Engineer Research and Development Center (ERDC) on 45 MPa compressive strength con-crete named WES-5000 [55, 66]. A 2.3 kg semi-armor piercing (SAP) ogive nosed projectile witha diameter of 50.8 mm, and length-to-diameter ratio of 7 was launched at conventional concrete ata nominal velocity of 310 m/s. The unreinforced concrete targets were cast in corrugated steel cul-verts with nominal diameter of 1520 mm having thicknesses of 127 mm, 216 mm, 254 mm, and284 mm. Edge effects in the experiments were ignored by ensuring the diameter of the targets weregreater than 25 calibers (projectile diameters).

Calibration and validation of the LDPM parameters were performed on cylinders of length 101.6mm and diameter 50.8 mm on the following quasi-static experiments: UX, UC, TXC, and UT. TheUC, TXC, and UT tests were executed as described in Hanchak experiments earlier, and the UX asdescribed in the ECC tests. Calibration was performed on the 14 MPa, 100 MPa and 400 MPa TXCtests, while other TXC tests (20 MPa, 7 MPa, and 200 MPa) were used for parametric validation. Thecompressive strength of the material was given as 45 MPa, which matched the average experimentalof 48 MPa. Table II gives the concrete meso-scale parameters and Table I gives the concrete mixdesign. Figure 11(a) and (b) shows the final numerical and experimental plot comparison of thedata TXC and UX data respectively, where the lighter grey results are those used for the parametric



Figure 9. Multiple impact plot comparison for Almusallam penetration tests on HM0 at 125 m/s (a) 0 mmfrom initial impact, and (b) 50 mm from initial impact.

Figure 10. Panel and projectile setup for numerical simulations with meshing scheme used.

calibration. The numerical results for the UT test was an exact match of the experimental results of4 MPa.

The numerical penetration simulations were modeled following the same procedure as the exper-iments earlier, with rigid projectile. Contrary to the experimental test that used a circular concretepanel with steel culvert, the numerical experiments were modeled with square LDPM blocks andexterior nodes were fixed in place to simulate external boundary conditions. Similarly to the simu-lations in the ECC, only a 600 mm central portion of the block was simulated with LDPM particlesand elastic FE was used to simulate the exterior block. Figure 10 gives the setup of the block, alongwith the dimensions and meshing of the projectile. The FE block mesh was selected based on thescale of the aggregates. A penalty algorithm was used for the contact between the projectile and theconcrete, without friction parameters.

Figure 11(c) shows the comparison for exit velocity with thickness for numerical and experi-mental results. The error bar gives the maximum and minimum values of the scatter obtained forthe numerical results. The results underestimate the exit velocity of the two thinner panels, but the



Figure 11. Calibration and validation of WES-5000 concrete for (a) Triaxial compression tests (b) Uniaxialstrain tests; Penetration prediction results for WES-5000 concrete for (c) Exit velocity for various paneldepths, (d) Change in velocity for various panel depths; Contact force between concrete and projectile for

(e) 127-mm specimen, (f) 284-mm specimen.

Figure 12. Detailed crack pattern for projectile penetrating WES-5000 127-mm concrete panel.

larger panels matched the experimental results quite accurately. More experimental tests should beperformed to determine if this could be the result of experimental scatter. The large scatter observedfor the 254 mm specimen is most likely due to the fact that the impact velocity used is the ballis-tic limit for that specimen depth, resulting in a discrete energy jump. Figure 11(d) gives the changein velocity for the four specimen depths, and the evolution of the penetrator through the panel isshown in Figure 12 for the 127 mm specimen. The first slope after impact in Figure 11(d) is dueto the initial compression of the specimen, the change in slope is due to the tunneling effect of theprojectile and occurs around 0.4 ms, and the final change is around 0.1 ms where the panel is com-pletely perforated, and the projectile freely exits. Figure 11(e) and f gives the contact force between



Figure 13. Comparison of impact and exit crater profiles for numerical simulations and experimental tests.

the panel and the projectile for the smallest and largest specimens, respectively. The direction ofapplied velocity to the projectile is in the Z-direction. Moreover, the front section gives the sum offorces acting in contact with the projectile nose, and the rear shows the forces in contact with thetail. Results show the forces are present only in the projectile nose and mostly in the direction ofmovement. A similar response is observed for the larger panel depth, but the forces do not go to 0until a time of about 2 ms, which corresponds to the time the penetrator come to stop in the panel.

The ability of LDPM to accurately represent the failure profiles of the specimen is demonstratedin Figure 13. The final profile for the LDPM block of the perforated specimens are shown overlayedwith the final profile observed in the experimental investigation. The blue outline shows the LDPMparticles left behind after separation from the scabbing and spalling surfaces. The numerical profileand the experimental profile provide an excellent match, confirming the predictive capability of themodel in obtaining accurate failure during perforation.

6. NUMERICAL SIMULATION OF JINZHU CONCRETE TEST DATA

Perforation experiments on semi-infinite concrete targets of thicknesses (300, 400, 500, 600, and700 mm) by an ogive nose steel projectile having nominal striking velocity 400 m/s were car-ried out by Jinzhu et al. [17]. Projectiles had a mass of 4.8 kg, and were rigid being machinedfrom low alloyed, high-strength 35CrMnSiA steel and having showed little deformation duringthe experimental process. A 152 mm gas gun was used to launch the projectiles, and the strik-ing velocity and altitudes were recorded using a high-speed camera system. The panels, cast in1200 mm diameter cylindrical steel culverts 3 mm thick, were made of C30 concrete with an aver-age experimental compressive strength of 34.26 MPa. The cement content, c = 470 kg/m3, was



calculated from the given normalized concrete mix proportions and dry density, D 2420 kg/m3

using D c. 1 C a=c C w=c/, where w=c is the water-to-cement ratio D 0:44, and a=c is theaggregate-to-cement ratioD 3:71.

The numerical meso-parameters (Table II) were obtained from a numerical concrete of similaraverage compressive strength, 34.3 MPa, and mix design (Table I) calibrated and validated in pre-vious LDPM research [33]. Although the experimental campaign used 1200 mm diameter cylindersof varied thickness for the the targets, as they were semi-infinite, they were modeled as 1200 mm �1200 mm blocks with varied thickness (300, 400, 500, 600, and 700) mm. This allowed the use ofa smaller inner LDPM block of 600 mm � 600 mm, with an outer elastic block as done in previousexperiments to reduce the computational cost of the simulations. The experimental setup showingthe projectile dimensions and concrete block is given in Figure 14. To replicate the fixed boundaryconditions of the 3 mm-thick steel culvert used in the experiments, the outer nodes of the elasticblock were fixed to prevent displacement and rotation.

Figure 15(a) demonstrates how the projectile velocity changes with time for the various paneldepths. As expected, the projectile velocity decreases for all panel sizes as it meets the resistance ofthe panel, and the panels with greater depth provide additional resistance to the projectile and hencefurther lowers its velocity. This phenomenon is more visible in Figure 15(b), with both numericaland experimental residual velocity comparison. The residual velocity is taken as the final velocityof the projectile during penetration or the exit velocity during complete perforation. In the previouspenetration simulations, the wide variation of results around the ballistic limit due to the discreteenergy jump can also be observed in these simulations. This energy jump coupled with the scatterof concrete results due to its meso-structure can account for the difference of of residual velocityfor the numerical and experimental results. However, despite these discrepancies, the numericalresults are still a good prediction of the experimental results. The total nominal KE applied to thesystem is 384000 J. Figure 15(c) shows the final kinetic energy (KE) consumed for the panels of

Figure 14. Panel and projectile setup for numerical simulations of C30 concrete.

Figure 15. (a) Velocity-time plot results for varied panel sizes, (b) Comparison of numerical and experimen-tal residual velocity for varied panel sizes, (c) Consumed Kinetic Energy for varied panel sizes and along

700 mm panel.



varying thickness as well as the KE consumed through the depth of 700 mm panel. Figure 15(c)clearly shows for increase in both panel thickness, t , and penetration depth, d , there is an increasein consumed KE as more resistance is provided. Additionally, there is only slightly more energyconsumed for t < .1=2/p, and there is otherwise no significant variation with KE of projectiletunneling for t > .1=2/p. The experimental results did not show any general trend in the front andback crater depth for the given velocity on the specimen, and due to the smaller size block usedfor the LDPM block, an assessment on the numerical front and back crater depths would not becompletely accurate.

7. CONCLUSIONS

1. The LDPM model has shown superb capability in the calibration and prediction of a varietyof concrete mixes with great level of accuracy under quasi-static and dynamic loading.

2. Within the dynamic regime of penetration, the apparent rate effects play a much greater rolein comparison to the intrinsic rate effect, and the discrete nature of the model naturally allowsit to capture concrete behavior in this loading regime.

3. Numerous sizes of concrete blocks were modeled, and LDPM successfully predicted the fail-ure response and intricate mechanisms like scabbing, tunneling, and spalling associated witheach specimen. In-depth analysis of the failure mechanisms are possible with the numericalmodel.

4. The fiber-law in the LDPM can predict accurate energy transfer between concrete and rein-forcements under both static and dynamic loading, proves capable of confining the craterregion of concrete during impact, and was able to predict the response for penetrationresistance in ECC with both steel and plastic fibers.

5. The LDPM demonstrates accurate energy dissipation during penetration and perforation. Theconsumed kinetic energy increases with an increase in panel size, and for a specific panel sizethere is an increase in consumed kinetic energy at a decreasing rate along the panel depth.

6. The discrete nature of LDPM allows an accurate depiction of crater area, tunneling, and over-all failure patterns when the damage zone is modeled within the actual size of the LDPMmeso-structure. As computational tools continue to evolve, LDPM will be able to model largerdegrees of freedom, and can be an essential breakthrough in futuristic modeling of concrete.

ACKNOWLEDGEMENTS

This material is based upon work supported by the National Science Foundation under grant CMMI-1237920. The work of the first author was also supported by the National Science Foundation under grantDGE-0948017 and the National GEM Consortium.

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Numerical analysis of projectile penetration and perforation of …anil-lab/others/lectures/CMCE/33... · 2016. 8. 6. · NUMERICAL ANALYSIS OF CONCRETE PENETRATION AND PERFORATION