c p

d a

d De

, No

1 February 2013Accepted 5 February 2013Available online 14 February 2013

Keywords:Small-arms bullets

on material strength but also on local ductility [16,17]. Since high-velocity perforation is an extremely localised process, the

inium alloys an

hot-rolling at the Hydro Research Laboratory in Bonn, Germany,before they were heat treated to tempers O (annealed), T4 (natu-rally aged), T6 (peak strength) and T7 (overaged). This manipula-tion of the wrought alloy signicantly changed the mechanicalproperties of the material, while retaining the grain structure. Thestressestrain behaviour of the different tempers was characterisedby conducting quasi-static tension tests at room temperature. Theanisotropy in ow stress and fracture strain of the materials wasobtained from tests in three different in-plane directions. Based on

* Corresponding author. Structural Impact Laboratory (SIMLab), Centre forResearch-based Innovation (CRI) and Department of Structural Engineering, Nor-wegian University of Science and Technology, NO-7491 Trondheim, Norway.Tel.: 47 73 59 46 47; fax: 47 73 59 47 01.

Contents lists available at

International Journal o

journal homepage: www.else

International Journal of Impact Engineering 57 (2013) 119e133E-mail address: tore.borvik@ntnu.no (T. Brvik).good or even better perforation resistance than steel when arealmass is taken into account [14,15], while other studies have sug-gested that the ballistic properties of plates are not only dependent

In this study, the ballistic properties of the high-strengthaluminium alloy AA6070 in different tempers have been investi-gated. Plates of 20mm thickness were rst produced by casting andstrength-to-density ratio (see e.g. [2e13]). Recent studies haveindicated that high-strength aluminium alloys may have equally

tion resistance [18], possibly making aluminteresting alternative for protective structures.Mobile protective structures are often made of thin plates ofhigh-strength steel because of its excellent combination ofstrength, ductility, price and formability [1]. Aluminium alloys arealso considered for armour applications owing to their high

nation of strength and ductility may under certain conditions besuperior energy absorbers for ballistic impact applicationscompared to special alloys having higher strength at the expense ofductility. Aluminium alloys can through alloying and heat treat-ments gain a number of useful properties with respect to perfora-Aluminium armourBallistic impact tests3D numerical simulationsCylindrical cavity expansion theory

1. Introduction0734-743X/$ e see front matter 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.ijimpeng.2013.02.002(overaged). The stressestrain behaviour of the different tempers was characterised by quasi-static ten-sion tests and was found to vary considerably with temper in regards to strength, strain hardening andductility. Ballistic impact tests using 7.62 mm APM2 bullets were then carried out, and the ballistic limitvelocities were obtained for all tempers. In the material tests it was shown that the O-temper was mostductile and almost no fragmentation took place during the ballistic impact tests. The T6-temper provedto be least ductile, and fragmentation was commonly seen. The experiments show that despite frag-mentation, strength is a more important feature than ductility in ballistic impact for this alloy, at least forthe given projectile and within the velocity range investigated. A thermoelasticethermoviscoplasticconstitutive relation and a ductile fracture criterion were determined for each temper, and niteelement analyses were performed using the IMPETUS Afea Solver with fully integrated 3rd-order 64-node hexahedrons. The numerical simulations predicted the same variation in ballistic limit velocitywith respect to temper condition as found in the experiments, but the results were consistently to theconservative side. In addition, analytical calculations using the cylindrical cavity expansion theory (CCET)were carried out. The ballistic limit velocities resulting from these calculations were found to be in goodagreement with the experimental data.

2013 Elsevier Ltd. All rights reserved.

consequence is that structural materials with a balanced combi-Received 2 October 2012Received in revised formplates of 20 mm thickness in tempers O (annealed), T4 (naturally aged), T6 (peak strength) and T7

Article history: The ballistic properties of the aluminium alloy AA6070 in different tempers are studied, using targetEffects of heat treatment on the ballisti

J.K. Holmen a, J. Johnsen a, S. Jupp b, O.S. Hopperstaa Structural Impact Laboratory (SIMLab), Centre for Research-based Innovation (CRI) anNO-7491 Trondheim, NorwaybHydro Rolled Products GmbH, Research & Development, D-53117 Bonn, GermanycNorwegian Defence Estates Agency, Research and Development Section, NO-0015 Oslo

a r t i c l e i n f o a b s t r a c tAll rights reserved.roperties of AA6070 aluminium alloy

, T. Brvik a,c,*

partment of Structural Engineering, Norwegian University of Science and Technology,

rway

SciVerse ScienceDirect

f Impact Engineering

vier.com/locate/ i j impeng

the material tests, a thermoelasticethermoviscoplastic constitutiverelation and a ductile fracture criterion were determined for eachtemper. Ballistic impact tests using 7.62 mm APM2 bullets werethen carried out, and the ballistic limit velocities were obtained. Inthe material tests it was shown that the O-temper was the most

550 C for 4 h before it was hot-rolled to a thickness of 20mm. After

Table 1Chemical components (in weight%) of aluminium alloy AA6070.

Al Si Fe Cu Mn Mg Others

Balance 1.38 0.22 0.26 0.54 1.23 0.15

J.K. Holmen et al. / International Journal of120production of the plates different heat treatments were applied toobtain tempers O (annealed), T4 (naturally aged), T6 (peakstrength) and T7 (overaged), which gave a range of differentstrength and hardening properties to the alloy. The various steps inthe heat treatment processes are described in Table 2. These heattreatments are not expected to change the grain structure of thematerial [18], as also conrmed by the tri-planar optical micro-graphs in Fig. 1 showing the grain structure after heat treatments totemper O and T6.

Quasi-static tensile tests were carried out using smoothaxisymmetric specimens with a gauge length of 40mm and a cross-section diameter of 6 mm (Fig. 2). The tensile axes of the specimenswere oriented at 0, 45 and 90 with respect to the rolling direc-tion (RD) of the plate, and three repeat tests were carried out ineach direction at room temperature (giving a total of 36 tests). The

Table 2Heat treatment processes of AA6070 to obtain the different tempers.

Temper Solutionizing Cooling Annealing/Articial Coolingductile and almost no fragmentation took place in the ballistic tests.The T6-temper proved to be less ductile, and fragmentation duringimpact was commonly seen. It was found also for this alloy thatstrength is a more important feature than ductility in ballisticimpact, at least within the limitations of this study. Finite elementanalyses were performed using the IMPETUS Afea Solver [19] withfully integrated 3rd-order 64-node hexahedrons. Ballistic limitvelocities were calculated on the basis of the numerical results andcompared to the experimental values. The numerical simulationspredicted the same variation in ballistic limit velocity with respectto temper condition as observed in the experiments, but the resultswere consistently to the conservative side. Finally, analytical cal-culations using the cylindrical cavity expansion theory (CCET) wereperformed. These results were found to be in good agreement withthe experimental data.

2. Experimental study

2.1. Target and bullet materials

Plates of the aluminium alloy AA6070 from Hydro are consid-ered in this study. The chemical composition of the alloy can befound in Table 1. The melt was rst DC-cast (direct chilled) to athickness of 126 mm, then machined to 103 mm, homogenised ataging

O 90 min at560 C (5 C)

Waterquench

24 h at350 C (5 C)

Slowcooling

T4 90 min at560 C (5 C)

Waterquench

e e

T6 90 min at560 C (5 C)

Waterquench

64 h at160 C (5 C)

Slowcooling

T7 90 min at560 C (5 C)

Waterquench

8 h at200 C (5 C)

Slowcoolingcross-head velocity of the test machine was 1.2 mm/min, corre-sponding to an average strain rate before necking of 5 104 s1.The force and diameter at minimum cross-section of the specimenwere continuously measured until fracture. The latter was madepossible using a meter with two perpendicular lasers that accu-rately measured the specimen diameter. The lasers were installedon a mobile frame to ensure that the diameter during straining wasalways measured at minimum cross-section. The specimen diam-eter wasmeasured in the thickness direction (DZ) of the plate and inthe transverse direction (Dt) of the specimen. The Cauchy (true)stress and the logarithmic (true) strain were calculated as

s FA; 3 ln A0

A(1)

where F is the force, A0 pD20=4 is the initial cross-section areaand D0 is the initial diameter of the gauge section. The current areaof the cross-section is given as

A p4DZDt (2)

The plastic strain is then obtained as 3p 3 s/E, where E isYoungs modulus. Note that plastic incompressibility and negligibleelastic strain have been assumed in Eq. (1). Since there might bevariations in stress and strain over the cross-section, s and 3shouldbe considered as average values.

Typical true stressestrain curves to fracture are shown in Fig. 3(since the spread in experimental results between repeat tests wasfound to be negligible), while some material data are given inTable 3. Here s0 is the 0.2% offset yield stress, su is the ultimatetensile strength, spt is the true peak stress, 3f is the true failure strainandWc is the plastic work to failure given as the area under the truestressestrain curve (i.e. the CockcrofteLatham parameter that willbe further discussed in Section 3). The effects of the heat treatmentare found to be signicant, and considerable variation in yieldstress, strain hardening and strain to failure with temper isobserved. The general trends are that the anisotropy in ow stressof AA6070 is negligible, while the anisotropy in ductility is signif-icant. The annealed O-temper is considerably more ductile than theother tempers. The fracture strain in the rolling direction is morethan 3 times as high as for temper T6, while in the 90 direction it ismore than 16 times as high. The opposite trends are seen for thestrength. Temper T6 has a yield stress almost 8 times the O-temper.The yield stress is about 10% lower for temper T7, while temper T4has only half the yield stress of temper T6. However, the strainhardening in temper T4 is much stronger than for the other tem-pers. Fig. 3 and Table 3 further conrm that increased strengthtakes place at the expense of ductility. Temper T4 exhibits thehighest Wc value independent of specimen direction due to thebalanced combination of strength and ductility.

The plates were impacted by APM2 bullets red at various ve-locities. The 7.62mmdiameter,10.5 0.25 gmass, bullet consists ofa brass jacket, a lead tip and a 5 0.25 g, ogival-nose, hardened steelcorewith a calibre-radius-head of 3 and a Rockwell C hardness of 63.The purpose of the lead tip is to stabilize the bullet during ight andin the initial stage of penetration. Fig. 4 shows the dimensions andthe various parts that make up the APM2 bullet. More informationregarding the bullet materials can be found in Brvik et al. [1].

2.2. Experimental set-up

Theballistic impact testswere carriedout in a compressedgas-gunfacility described in detail by Brvik et al. [20,21]. A 7.62 63 mmspecially designed smooth-bored Mauser rie with a barrel length of

Impact Engineering 57 (2013) 119e133about 1 mwas in this study used to re the APM2 bullets. The stock

Fig. 1. Tri-planar optical micrographs showing the grain structure after heat treatments todirection, TD gives the transverse direction and ND gives the normal direction of the plate

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133 121was removed and the rie was mounted in a rigid rack inside theimpact chamber of the gas-gun. This xture guaranteed a well-dened impact point in all tests, and the rie could be red by amagnetic trigger from a safe distance. Square target plates of AA6070with in-plane dimensions of 300 mm and thickness of 20 mm werermly clamped to a rigid frame by two beams. This provided a xedboundary for the horizontal sides of the targets, while the vertical

Fig. 2. Geometry of the tensile specimen (dimensions in mm).

Fig. 3. Typical true stressestrain curves of all tempers and test orientations (where theblack s indicate the fracture strain).sides remained free. The in-plane distance between each impact pointand the target boundary was roughly 100 mm. Striking and residualvelocities were measured with various laser optical devices that havebeen found to be accurate to within 1e2% [1]. In addition, the overallperforationprocesswasphotographedwith aPhotron FASTCAMSA1.1high-speed video camera. To tune the impact velocity, the ammuni-tion was adjusted so that the bullet impacted the target at velocitieswell above and just below the ballistic limit. Even though this wasdonewith great care, some spread in initial impact velocitywas found.Taking this uncertainty into account, six to nine tests had to be con-ducted for each temper to get an accurate prediction of the ballisticlimit velocity.

2.3. Experimental results

Based on a number of experimental tests, the ballistic limitcurve and the ballistic limit velocity for each temper of the 20 mmthick AA6070 target plates have been determined. Measured initial(vi) versus residual (vr) velocities of the bullet from each experi-ment are plotted in Fig. 5. The solid lines through the data points

temper (a) O and (b) T6 for the aluminium alloy AA6070, where RD gives the rolling.are tted using a generalized version of the RechteIpson model[22] given as

vr avpi v

pbl

1=p(3)

where a and p may be taken as empirical constants, and vbl is theballistic limit velocity. Since the bullets mainly pierced the plates byductile hole growth, a was chosen equal to unity (although some

Table 3Material data for typical AA6070 material tests.

Temper Orientation () s0 (MPa) su (MPa) spt (MPa) 3f Wc (MPa)

O 0 51 139 243 0.79 15145 51 136 231 0.76 14290 50 138 225 0.66 118

T4 0 187 320 487 0.52 21145 186 328 474 0.46 18090 187 328 457 0.39 150

T6 0 373 393 475 0.26 11545 379 396 423 0.07 2990 387 399 416 0.04 19

T7 0 341 354 419 0.32 12845 346 360 396 0.14 5590 340 356 391 0.13 49

fragmentswere ejected fromoneor both surfaces of the target at thehighest impact velocities), while p and vblweretted to the test datausing themethod of least squares. Obtained values of a, p and vbl aregiven in Table 4. Even though some spread is seen (Fig. 5), theagreement between the experimental data points and the Rechte

5%. Larger yaw angles may, on the other hand, signicantly reduce

Fig. 4. Schematic drawings and geometry of the 7.62 mm bullet (dimensions in mm).

J.K. Holmen et al. / International Journal of122Ipson model is good. The data further indicates a strong andrather linear increase in perforation resistance with target yieldstress, and the results are in this sense in close agreement withsimilar data reported in Ref. [1] for steel plates. This conrms thatthematerial strength is amore important feature than themeasuredstrain to failure in the design of protective structures against small-arms bullets (as will be further discussed in Section 5).

Fig. 6a shows high-speed video images from the perforationprocess of a 20 mm thick plate in temper O at maximum impactvelocity. Some fragmentation fromthe rear-side of the target plate isclearly seen. Note that the brass jacket enclosing the steel core of thebullet survived the test (see also Fig. 15a). Similar high-speed videoimages at medium impact velocity are given in Fig. 6b. The impactvelocity is considerably lower in this test, and no fragmentation ofsignicance is seen during perforation. Inspection of the bullet holesafter each test conrmed that no fragmentation took place in the O-temper at the lowest impact velocities, but small petalswere formedon both sides of the cavity. This behaviour was as expected for arelatively ductile material as the annealed aluminium alloy. High-speed video images of the perforation process of a 20 mm thickplate in temper T6 at maximum and medium impact velocities areshown in Fig. 7. The T6-temper exhibited thehighest yield stress, butalso the lowest fracture strain (see Fig. 3), and this behaviourrevealed itself in the ballistic tests through a higher degree of frag-mentation during perforation. A similar behaviour was observedduring impact of the high-strength aluminium alloy AA7075-T651as described in Refs. [16,17]. It is alsoworth noting that as the impactFig. 5. Ballistic limit curves for the four different tempers of AA6070.the penetration capacity of the projectile.Cross-sections of sliced plates in tempers O and T6 after impact

are shown in Figs. 8 and 9, respectively. The transition from rear-side fragmentation to front-side fragmentation with a decrease inimpact velocity is clearly illustrated in Fig. 9. In the centre part ofthe plates, the cavities are mostly smooth and of circular shape,indicating penetration by ductile hole growth. The more ductilefailure mode of the O-temper can clearly be seen in Fig. 8 whencomparing it to the cross-sections of the T6-temper in Fig. 9. Notethat even though the impact in general was orthogonal (see e.g.Fig. 6b), a distinct obliquity developed during the perforation pro-cess (Fig. 8b) in some of the tests. Tempers T4 and T7 have strengthand hardening properties that lie between tempers O and T6, andtheir behaviour during impact was a mixture of these two ex-tremes. Temper T4 exhibited more petalling than the other tem-pers. Fragmentation was found to be important at the highestimpact velocities for the least ductile tempers (T6 and T7), whileductile hole growthwas dominating in themore ductile tempers (Oand T4) and at the lowest impact velocities. Due to this, the frag-mentation process is believed to have a relatively small effect on theballistic limit velocity in this study. It should however bementioned that for other impact conditions than those studied here(see e.g. [16]) the effect of target fragmentation may be essential.

3. Finite element simulations

3.1. Constitutive relation and fracture criterion

A modied version of the JohnsoneCook (MJC) constitutiverelation was used to model the target materials (see [1,24,25]). Theconstitutive behaviour is assumed to be isotropic and modelledwith the von Mises yield criterion even though the materialsvelocity decreased, the front-side fragmentation increased and thefragmentation from the exit side decreased (albeit some fragmen-tation on the exit sidewas still present). Only small pitch/yawangles(below 1e2%) were registered in these tests (see e.g. the high-speedcamera images in Figs. 6 and 7). Such small angles are assumed notto affect the ballistic limit of the target material. From the experi-mental results reported by Goldsmith [23], it appears that thepenetration process is hardly affected by total yaw angles up to 3e

Table 4Ballistic limit velocities and RechteIpson constants from experimental tests andnite element simulations.

Temper Experimental data Numerical simulations Deviation (%)

a p vbl (m/s) a p vbl (m/s)

O 1 1.94 348.0 1 1.94 296.0 14.9T4 1 2.05 506.2 1 2.05 438.1 13.5T6 1 2.21 562.5 1 2.21 480.6 14.6T7 1 2.20 529.1 1 2.20 438.3 17.2

Impact Engineering 57 (2013) 119e133exhibit some anisotropy (but mainly in the strain to failure, seeFig. 3). The equivalent stress is then expressed as

seq A

X2i1

Qi1 exp Ci 3eq

!1 _3*eq

c1 T*m

(4)

where 3eq is the equivalent plastic strain and (A,Q1,C1,Q 2,C2,c,m) aremodel parameters. Note that the usual power law strain hardeningin Eq. (4) has been replaced by a two-term Voce strain hardeninglaw (see e.g. [26]) for the target material in this study, since thelatter generally gives a better t of the stressestrain curve foraluminium alloys. The dimensionless plastic strain rate is given by

k AA

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133 123_3*eq _3eq=_30, where _30 is a user-dened reference strain rate. Thehomologous temperature is dened as T* (T Tr)/(Tm Tr), whereT is the absolute temperature, Tr is the room temperature and Tm isthe melting temperature of the material. The temperature changedue to adiabatic heating is calculated as

DT Z3eq0

cseqd 3eqrCp

(5)

where r is the material density, Cp is the specic heat and c is the

Fig. 6. High-speed video images showing the perforation process of a 20 mm thicTayloreQuinney coefcient that represents the proportion ofplastic work converted into heat.

Fig. 7. High-speed video images showing the perforation process of a 20 mm thick AAFailure is modelled using a criterion proposed by Cockcroft andLatham (CL) [27]

W Z3eq0

hs1id 3eq Wc (6)

where s1 is the major principal stress, hs1i s1 when s1 0 andhs1i 0 when s1 < 0. From Eq. (6) it is seen that failure cannotoccur when there are no tensile stresses operating. The modelconstantWc is the value ofW at failure, and can be determined from

6070 plate in temper O impacted by the APM2 bullet at two different velocities.a simple uniaxial tensile test. It was shown by e.g. Dey et al. [28] andKane et al. [29] that the one-parameter CL criterion gives equally

6070 plate in temper T6 impacted by the APM2 bullet at two different velocities.

good results as more advanced failure criteria in simulations ofperforation of steel plates under various stress states and projectilenose shapes. It should however be noted that owing to the aniso-tropic behaviour of the material and the uncertainty in the cali-bration of the CL criterion,Wc should not be regarded as a materialcharacteristic. In this study, the deviatoric stresses in the elementare set to zero when W reaches its critical value Wc in a speciednumber of integration points. This is dened as material failure.However, the element continues to take compressive hydrostaticstresses until the time step size drops below a critical level. This isdened as element erosion. The constitutive relation and the failurecriterion have been implemented in the non-linear explicit niteelement codes IMPETUS Afea Solver [19] and LS-DYNA [30].

The bullet materials weremodelledwith themodied JohnsoneCook constitutive relation using the usual JohnsoneCook strainhardening law to describe the work hardening, while failure wasdescribed using the CL criterion. For more details regarding themodelling of the various bullet parts, see [1].

3.2. Identication of model parameters

The yield stress (A) and the hardening parameters (Q1,C1,Q2,C2)were rst tted by a direct calibration to Bridgman-corrected truestress versus true plastic strain curves from the material tests in therolling direction (Fig. 3). These parameters were then used as initialvalues in LS-OPT [30], which is an optimization tool that interacts

Fig. 8. Pictures of cross-sections of 20 mm thick target plates of AA6070 in temper O perforated by APM2 bullets at various impact velocities.

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133124Fig. 9. Pictures of cross-sections of 20 mm thick target plates of AA6070 in temper T6 perforated by APM2 bullets at various impact velocities.

with LS-DYNA. An axisymmetric nite element model of the tensilespecimen was created and several successive analyses were run insequential order using a hybrid optimization algorithm withdefault values. Measured forceediameter reduction curves fromthe tensile tests in the rolling direction of the plates were used asresponse curves. Models with both coarse and ne element mesheswere used in the optimization, but the difference in results wasfound to be minor. Optimized model parameters from the tensile

temperature (as assumed for aluminium alloys in Ref. [32]). Thefracture parameterWc was calibrated by use of the uniaxial tensiletest in the rolling direction. In this case we simply have thatWc

R 3f0 sd 3

p. An alternative approach would have been to extractWc from the numerical simulations at the same diameter reductionas failure occurred in the experiments. This would give a somewhathigher value of Wc compared to the value from the direct calibra-tion. Note that the difference in Wc with material direction is

Fig. 10. 3D solid element meshes used in the numerical simulations: (a) aluminium plate with 8-node linear elements in the periphery and 64-node cubic elements in the impactarea (two times the radius of the bullet), and (b) APM2 bullet with 64-node cubic elements.

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133 125tests in the rolling direction, together with mean square errors(MSE) from LS-OPT, are given in Table 5. The agreement betweenmeasured and tted curves after optimization was excellent. Thiswas also validated by running full 3D numerical simulations of thetensile tests using the IMPETUS Afea Solver and thematerial data inTable 5.

The rate sensitivity of high-strength aluminium alloys is usuallyfound to be small at room temperature, and several experimentalstudies indicate an increase less than 10% from quasi-static loadingconditions to a strain rate in the order of 103 s1 (see e.g. [16,31]).Owing to the lack of tensile test data at elevated strain rates andtemperatures, the strain rate sensitivity constant c was given asmall positive value [31], while _30 was taken equal to the strain ratein the quasi-static tensile tests. It is thus assumed that the strainrate sensitivity of the material is low and independent of temper.Further, the temperature sensitivity parameter m was set to unity,implying a linear decrease in the ow stress with increasingFig. 11. Perforation process (where t provides the time after impact) of a 20 mm thick AA6residual velocity vr 784 m/s.substantial (see Table 3). Since the stress level was not directionallydependent, the variation inWc reects the anisotropy of the failurestrain in uniaxial tension. This variation has not been taken intoaccount in the simulations in this study, but it was discussed insome detail in Ref. [16] for the anisotropic high-strength aluminiumalloy AA7075-T651. Thus, only data from the tests in the rollingdirection have been used in the calibration. Physical constants andmodel parameters common to all materials are provided in Table 6.The physical constants were given nominal values for aluminiumalloys provided in the literature.

Constitutive relation, fracture criterion and model parametersfor the different bullet materials were taken from Brvik et al. [1].

3.3. Finite element models

All numerical simulations presented in the following were car-ried out using the explicit nite element code IMPETUS Afea Solver070 plate in temper T6 by the full APM2 bullet with impact velocity vi 903 m/s and

[19]. The projectile and the region in the target plate that undergolarge plastic deformations were modelled using fully integrated3rd-order 64-node hexahedrons. The target plate was modelledsomewhat smaller (80 80 mm2) than in the test to savecomputational time. Some analyses were also carried out with afull-size plate of 300 300 mm2, and the discrepancy in results

The symmetry in the problem was exploited by modelling onlyhalf the bullet and the plate. Outside the impact region 20 fullyintegrated linear elements with a node spacing of 1 mm in all di-rections were used, while 20 cubic 3rd-order 64-node elementswere used over the thickness in the impact region. This gave a nodespacing of 0.33 mm in all directions in the critical region. Plots of

Fig. 12. Perforation process (where t provides the time after impact) of 20 mm thick AA6070 plate in temper T6 by only the hard core of the APM2 bullet with impact velocityvi 903 m/s and residual velocity vr 754 m/s.

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133126compared to the reduced model was found to be negligible. Thebulk of the simulations were done with only the rigid, hard steelcore instead of the full bullet. This signicantly reduced the CPUtime without compromising the results too much, since it wasshown experimentally in Refs. [14,15] that the brass jacket and thelead tip have a relatively small effect on the perforation process ofmonolithic 20 mm thick aluminium plates.Fig. 13. Plots of the perforation process of 20 mm thick AA6070 plate at two different impvi 377 m/s and vr 231 m/s, (c) T4-temper with vi 894 m/s and vi 769 m/s, (d) T4-temthe 3D nite element meshes used in the bulk of the simulationsare shown in Fig. 10. Earlier studies have shown that perforationproblems involving blunt projectiles causing shear localization aremesh-size sensitive, while the mesh-size dependency is lessdistinct for pointed nose projectiles causing failure by ductile holegrowth (see e.g. [33,34]). To check themesh-size dependency in thecurrent problem, simulations were run on AA6070-O plates usingact velocities: (a) O-temper with vi 900 m/s and vr 834 m/s, (b) O-temper withper with vi 509 m/s and vi 266 m/s. The fringes show the equivalent plastic strain.

Fig. 14. Plots of the perforation process of 20 mm thick AA6070 plate at two different impact velocities (a) T6-temper with vi 903 m/s and vr 754 m/s, (b) T6-temper withvi 615 m/s and vr 395 m/s, (c) T7-temper with vi 901 m/s and vi 764 m/s, (d) T7-temper with vi 538 m/s and vi 374 m/s. The fringes show the equivalent plastic strain.

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133 127respectively 7, 10, 20, 30, 40 and 50 cubic elements over the targetthickness in the impact region. A constant impact velocity ofvi 550 m/s was applied in these simulations. If less than 20 ele-ments were used over the target thickness the mesh-size sensi-

tivity was found to be strong, and the residual velocity increased by

Fig. 15. Comparison of APM2 bullets after perforation in tests and simulations. All20% when going from 7 to 20 elements. However, a slight reductionin residual velocity of 4% was found when going from 20 to 50 el-ements over the target thickness. As a compromise between ac-curacy and CPU time, 20 cubic elements were used over the target

thickness in this study.

pictures and predictions are from the maximum velocity tests (vi z900 m/s).

Regardless of the increased computational time of simulationsapplying the full APM2 bullet, a study was conducted to see if wecould predict the deformation of the bullet and the stripping of the

(Figs. 6 and 7). Thus, the cylindrical cavity expansion theory (CCET)

Table 5Optimised model parameters for different tempers of AA6070 (valid for the rollingdirection).

Temper A Q C Q C W Mean

J.K. Holmen et al. / International Journal of Impact Engineering 57 (2013) 119e133128The constitutive behaviour of the target materials was modelledusing the MJC relation with Voce hardening given by Eq. (4), whilematerial failure was modelled using the CL failure criterion denedin Eq. (6). The bullet materials were modelled in a similar way as inBrvik et al. [1]. All simulationswere runwith themodel parametersfor the target plate based on the tensile tests in the rolling direction.When 16 of the 64 integration points in the higher-order elementsreached failure, all deviatoric stresseswere set to zero. However, thematerial was still allowed to take compressive hydrostatic stressesand failed elements were not eroded until their time step droppedbelow a user-dened critical level. In addition to failure caused bydamage, the elements were also allowed to fail if the temperaturereached the melting temperature of the material. Element erosionwas used to prevent overly distorted elements which reduced thetime step towards zero and could cause error termination. Thecritical time step for element erosion was set to 3 ns in this study.

Contact between the various parts was established using apenalty-based node-to-surface contact algorithm available in theIMPETUS Afea Solver. Friction between parts in contact wasneglected since this will give a conservative estimate. In the currentwork, all exterior nodes and element faces were active in thecontact. Free nodes of failed elements kept their mass and mo-mentum, and remained active in the contact. Being given a physicalradius to correctly represent the volume of eroded elements, freenodes were also in contact with each other.

3.4. Numerical results

Typical simulations of the perforation process of a 20 mm thickAA6070 target plate in temper T6 impacted by the full bullet and byonly the hard core of the bullet are shown in Figs. 11 and 12,respectively. Even though the mass of the full bullet is about twiceas high as the core, the obtained difference in residual velocity islow. Thus, the experimental observation that the brass jacket andthe lead tip only have a small effect on the perforation process wasalso captured in the numerical simulations.

Fringe and deformation plots of the perforation process fortemper O and temper T4 at two different initial velocities are givenin Fig. 13, while similar plots for temper T6 and temper T7 areshown in Fig. 14. The fringes represent the equivalent plastic strain.Also the numerically obtained front-side and rear-side surfaces ofthe target plates after perforation are shown in the gures. Frag-mentation has been found difcult to capture in numerical simu-lations [16], so only slight differences in physical behaviour

(MPa)1

(MPa)1 2

(MPa)2 c

(MPa) square error

O 38.8 79.5 56.9 88.2 4.0 151 2.4 $ 105

T4 172.7 35.6 80.6 247.7 6.5 211 6.2 $ 105

T6 350.0 30.1 185.9 72.8 7.7 115 9.4 $ 104

T7 292.5 55.3 317.2 31.1 10.0 128 6.0 $ 104between the various tempers are seen. These plots also reveal thatonly a small part of the target plate is affected by the highlylocalised perforation process, and hardly any global deformation ispresent. The area of the target plate with severe plastic strains islimited to a region with diameter just over two times the diameterof the projectile.

Table 6Physical constants and model parameters common to all tempers of AA6070.

Temper E (MPa) n () r (kg/m3) a (K1) Cp (J/kgO, T4, T6, T7 70,000 0.3 2700 2.3 $ 105 910seems to be applicable for analytical modelling of the penetrationand perforation process.

The CCET was rst proposed by Bishop et al. [35] and furtherdeveloped by Forrestal and co-authors (see e.g. [2e6]). Only a briefsummary of the main equations as given by Forrestal and Warren[36] and Brvik et al. [14] will be presented in this study. Toapproximate ductile hole growth, the theory idealises the target asthin, independent layers that are compressed normal to thepenetration direction [4]. Thus, the analysis is simplied to one-dimensional motion in the radial plate dimension for an elasticeplastic material. The cavity is then expanded from an initialradius of zero at a constant velocity V. This expansion producesboth elastic and plastic responses. The elastic region has Youngsmodulus E and Poissons ratio n, while the plastic region is taken asan incompressible, power law hardening material. Thus, the one-dimensional material response may be simplied as

s

8>:

E 3; s < s0

s0

E 3s0

n; s s0

(7)

K) _30 (s1) c () c () Tr (K) Tm (K) m ()brass jacket during perforation. Analyses were conducted for alltempers at vi 900 m/s, and the numerical results were comparedto post-perforation pictures taken from the high-speed videos. Theresults are presented in Fig. 15, and it is seen that the deformationand stripping process are well captured. The brass jacket iscompletely removed from the steel core after perforation of platesin tempers T4, T6 and T7, while it is still attached to the core afterperforation of plates in temper O. This is consistent with the high-speed video images from the experiments.

Finally, the main results from a large number of simulations arepresented as ballistic limit curves (i.e. curves of the residual versusinitial velocity emanating from the ballistic limit of the target) inFig. 16, while predicted ballistic limit velocities are given in Table 4.The numerical predictions are found to be conservative, i.e. theballistic limit velocities are under-estimated for all tempers. Thenumerical results exposed a maximum deviation in ballistic limitvelocity from the experimental data of 17.2% for temper T7, and aminimum deviation of 13.5% for temper T4. Except for the frag-mentation, the main trends observed in the tests were wellcaptured in the simulations. All ballistic limit curves in Fig. 16 havebeen obtained using the RechteIpson model in Eq. (3) with a 1and p from the respective experimental data (Table 4).

4. Analytical method

4.1. CCET model

Post-experimental inspections of the target plates indicated thatthe predominant failure mode during perforation is ductile holegrowth (Figs. 8 and 9), although some fragmentation was observedespecially at the highest impact velocities. The high-speed videos ofthe ballistic experiments further suggested that the hard core of theAPM2 bullet remained rigid throughout the perforation process5 $ 104 0.001 0.9 293 893 1.0

J.K. Holmen et al. / International Journal ofwhere s is the true stress, 3is the true strain, s0 is the yield stressand n is the strain hardening exponent. The model further requiresthe relation between the radial stress sr at the cavity surface versusthe cavity expansion velocity V. It was shown in Ref. [4] that

sr ss rtBV2 (8)

where ss is the quasi-static radial stress required to open the cy-lindrical cavity, rt is the target density and B is given as

B 12

(1

1 n1a2

p ln"1

1a2

p

a

#g22ln g1

)(9)

a2 3

p1 2n

21 nrtV

2

s0

and g2 21 ns0

3p

E(10)

Assuming the von Mises yield criterion, ss becomes

Fig. 16. Predicted ballistic limit curve for (a) AA6070-O, (b) AA6070-T4, (c) AA6070-T6 and

Table 7Results from CCET calculations.

Temper s0 (MPa) n B0 () C vbl (m/s) Deviation (%)O 51 0.213 3.63 0.0 332.4 4.5

0.142 357.3 2.7T4 187 0.166 2.99 0.0 498.0 1.6

0.117 528.4 4.4T6 373 0.050 2.64 0.0 562.9 0.1

0.103 593.1 5.4T7 341 0.036 2.69 0.0 534.7 1.1

0.105 563.9 6.6Impact Engineering 57 (2013) 119e133 129ss s03

p8