John F.MOXNES*，Anne K.PRYTZ，?yvind FR?YLAND1，Stian SKRIUDALEN1，Steinar B?RVE1，Gard?DEG?RDSTUEN

aLand Systems Division，Norwegian Defence Research Establishment（FFI），P.O.Box 25，NO-2027 Kjeller，Norway

bNammo Raufoss AS，P.O.Box 162，NO-2831 Raufoss，Norway

Received 3 July 2014；revised 26 August 2014；accepted 27 August 2014 Available online 22 December 2014

Strain rate dependency and fragmentation pattern of expanding warheads

John F.MOXNESa，*，Anne K.PRYTZb，2，?yvind FR?YLANDa，1，Stian SKRIUDALENa，1，Steinar B?RVEa，1，Gard?DEG?RDSTUENb，2

aLand Systems Division，Norwegian Defence Research Establishment（FFI），P.O.Box 25，NO-2027 Kjeller，Norway

bNammo Raufoss AS，P.O.Box 162，NO-2831 Raufoss，Norway

Received 3 July 2014；revised 26 August 2014；accepted 27 August 2014 Available online 22 December 2014

For the characterization of the behaviors of a metalmaterial in events like expanding w arheads，it is necessary to know its strength and ductility at high strain rates，around 104-105/s.The flyer plate impact testing produces the uniform stress and strain rates but the testing is expensive.The Taylor test is relatively inexpensivebutproducesnon-uniform stressand strain fields，and the resultsarenotso easily inferred for materialmodeling.In the split-Hopkinson bar（SHB），whichmay be used in compression，tension and torsion testing，the strain rates never exceeds 103/s.In the presentwork，we use the expanding ring testwhere the strain rate is 104-105/s.A streak camera is used to examine the expanding ring velocity，and aw ater tank isused to collect the fragments.The experimental resultsare compared w ith the numerical simulations using the hydrocodes AUTODYN，IMPETUSA fea and a regularized smooth particle（RSPH）softw are.The number of fragments increasesw ith the increase in the expansion velocity of the rings.The number of fragments is similar to the experimental results.The RSPH software shows much the same results as the AUTODYN where the Lagrangian solver is used for the ring.The IMPETUSA fea solver shows a somewhat different fragmentation characteristic due to the node splitting algorithm that induces pronounced tensile splitting.

Warhead；Fragmentation；Simulation；Fracturemodel；Expanding ring

1.Introduction

Plasticity-based analytical modeling and finite element methods（FEM）may be used to predict the fragmentation pattern of warheads.However，the viability of the predictions relies on the material constitutive models describing the plastic flow stress and fracture.For an expanding thin wall casing，the tangential strain rates are typically in the range of 104-105/sand thequasistatic establishedmaterialmodelmay not be viable.Main research issues are the dependency of fracture strain on triaxiality（thatmeans on the proportion of invariant I1to J2），the influence of the third invariant，i.e.，strain rate，on ductility，element size and the connection to adiabatic shear bands at high strain rate，and whether statistical failure predicts the size distribution of fragments better than a homogeneous failure model［1-6］.

Failure process of ductile materials is caused by the nucleation，growth and coalescence of voids to fracture.The fracture coalescence depends on pressure or triaxiality（that means on the proportion of invariant I1to J2）［7］.In general，the larger the triaxiality is，the smaller the fracture strain at failure becomes.This is in agreementw ith theoreticalmodels for void grow th［8，9］.Recently，Bao and W ierzbicki［10，11］compared the different models to cover the influence of triaxiality.They concluded that none of themodelswere able to capture the fracture behavior in the entire range of triaxiality.The void grow th was the dom inated ductile failuremodeat large triaxialities（say above 0.4），while the shear of voids dominates at low triaxialities.Themain conclusion was that therewas indeed a possible slope discontinuity in the fracture locus corresponding to the pointof fracture transition［11］.A dependency of the third invariant has been forecasted.

Both yield strength and ultimate tensile strength usually increase w ith strain rate for steelmaterials.The ductility of quenched and tempered steelmay increase w ith strain rate，while the ductility of the material which high strength is achieved by precipitation hardening process may decrease w ith strain rate.Body-centered cubic（bcc）materials can also behave different from face-centered cubic（fcc）materials. Thermal softening decreases strength and increases ductility. Thus the ductility ofmaterials could increasew ith smallstrain ratesbutcould decreasew ith higher strain rates due to thermal softening.Decreased ductility at higher strain rates may be explained by shear localization due to adiabatic heating［12］. Unstable adiabatic shear transfers theentireburden of strain to a finite number of these shear planes（adiabatic shear bands）. Due to restriction on computational time，the elementsizesare traditionally too coarse to resolve the shear bands by direct sim ulation.

W ilkinsetal.［13］concludedmany yearsago that theorderof the applied loads，i.e.hydrostatic pressure followed by shear or viceversa，should be importantin failuremodeling.To account for the order of the applied loads，the cumulative damage criterion has been applied［13］.Fracture occursat a pointof the materialwhere aweightedmeasure of the accumulated plastic strain reachesa critical value.Theweighing function depends on the triaxiality and/or the third invariant I3.Finding an appropriateweighting function isstillan active field of research［14，15］.In the Johnson-Cook（J-C）model［16］，an uncoupled（passive）damage evolution formulationw ith no third invariant dependency isadopted，which entails that there isno coupling between the stress-strain behavior and the damage evolution until fracture occursat the critical damage.

The split-Hopkinson bar（SHB），which may be used in compression，tension and torsion testing，is the most w idespread method formaterial high strain rate characterization. However the strain rate never exceeds 103/s and is thusmuch lower than that achieved under explosive loadings.Many ductilematerials display an increase in yield stress for strain rates above 103/s［17，18］.It is challenging to conductmaterial tests at the strain rates of larger than 103/s.The flyer plate im pact testing produce uniform stress and strain rates but the testing is expensive.The Taylor testing is relatively inexpensive and data could be obtained from simple post-testmeasurements.However，the Taylor test produces non-uniform stress and strain fields and the results are not so easily interpreted formaterialmodeling.

In this article，the fracture behavior of steel rings，taken from a 25mm warhead，is studied.To reach the strain ratesof more than 103/s，an expanding ring test isperformed.A streak camerawasused to exam ine the ring velocity，and awater tank was used to collect the fragments［19］.

A quasistatic strengthmodelof the steelwasestablished by using a smooth uniaxial tensile test to find the von M ises flow plastic function in a J-C strengthm odel.The parameters of a J-C damage developmentmodel are found using the results from quasi static tensile tests in which three different sample geometries are used［20］.

The Lagrangian processor iscomputationally fastand gives good definition ofmaterial interfaces.However，the ability of the Lagrangian processor to simulate explosive events can only be enhanced by use of an erosion algorithm which removes the zones that have reached a user-specific strain，typically in the order of 75%-150%.The Eulerian processor，which uses a fixed grid through whichmaterial flow s，ismuch more expensive in calculation than the Lagrangian processor，but is well suited form odeling larger deformations and fluid flow.See Refs.［21，22］for use of Eulerian CTH code.See Refs.［23，24］for use of the Arbitrary-Lagrangian-Eulerian ALE3D/CALE codes and Ref.［25］for semi-empiricalnumericalmethods.

The smooth particle hydrodynamics（SPH）method is a Lagrangian technique［26］.This grid-less technique does not suffer from the problem associated w ith the Lagrangian techniqueof grid tangling in large deformation problems.SPH is based on two main approximations of the continuum equations.First，an arbitrary scalar field variable is described by an integral over the space that is only approximate since a smoothed kernel is used in the integral instead of the exact Dirac delta function.Second，this integral is approximated by a discrete sum of a finite set of interpolation points（the particles）.In AUTODYN and LS-DYNA，SPH nodes interact w ith Lagrangian surfaces.This allows to model the regions which undergo small deformations using the Lagrangian processor，while those regions experiencing large deformations（i.e.the explosive）can bemodeled using SPH.The mostwell-known problem w ith SPH is loss of stability due to tensile instability and artificial fragmentation due to large particle spacing relative to the smoothing length.Regularized smooth particle hydrodynamics（RSPH）was developed to increase accuracy in shock wavemodeling［27］.In the current work，the original RSPH code has been extended to study the fragmentation of solids w ith a state of the art handling of tensile instability［28］and a sufficiently small ratio between the original particle spacing and the smoothing length.

We also apply the IMPETUSAfea node splitting algorithm and the corpuscularmodel.The corpuscularmethod does not start from the continuum equations，but postulate a number of particles that interact by collisions［29，30］.In the Lagrangian solver，instead of eroding cells that fails，the nodes can split，resulting in a sort of crack propagation.These cracks are constrained by themesh，or cell size.

2.Experimental setup and geom etrical data

Figs.1 and 2 show the setup.The brass tubesw ith constant outside diameter and variable inside diameter were loaded w ith the exp losive，which ism odeled as composition LX 10. Steel ringsweremanufactured from projectile bodiesof the inservice round.To find the velocity of the rings，the test item is placed such that the expansion of the ring is perpendicular tothe axisof a rotatingmirrorcamera.The fragmentation studies were a duplication of the streak camera studies.However，in this case the fragments were collected in a water tank.To be able to repeat the actual velocity-time conditions，the tubes and rings were allowed to expand first in a thin plastic bag filled w ith air that was submerged underwater.Thus the expansions and break-up occurred in air.The water barrel was then emptied and sieved，and the fragments were collected w ith amagnet.More than 95%of ringmasswas collected.

The explosivewas ignited atone end of the cylinderat time zero.The density of the explosive was 1.87 g/cm3，and the detonation velocity was 8820 m/s.The AUTODYN“burn on time”modelwasused for the explosive.The total length of the cy linderw ith exp losive is10.2 cm.The length of the steel ring is 1 cm and its thickness is 0.3 cm.Two different shots（loadings）were studied（Table 1）.

Fig.1.Thematerial locations and the geometrical setup for the expanding ring test.

3.Strength and fracture/failuremodel of steel ring

The uniaxial tensile testspecimensand two notched tensile specimenswere extracted from a heat-treated steelmaterial to establish a J-C strength and failure/fracturemodel.The steel alloy composition is provided in Table 2.The steel is first casted，then rolled and heat-treated by quenching.Finally it is tempered.The hardness is 530 Vickers（5.6 GPa）.

The tests were carried out at room temperature in a hydraulic test machine w ith a strain rate of approximately 5×10-4s-1（quasi static condition）.The numerical simulations of the mechanical tests were performed，assuming the properties of isotropic material.The results were compared w ith the experimental results［20］.

The J-C［16］strength model is

where Y（εp）is setasa piecew ise linear function of the plastic strain，εp，vs.stress，Y，in MPa to read：｛｛0，1352.5｝，｛0.005，1518.68｝，｛0.015，1680.13｝，｛0.025，1742.46｝，｛0.035，1775.2｝，｛0.045，1796.74｝，｛0.0460，1798.56｝，｛0.146001，1864.57｝，｛0.346001，1958.05｝，｛0.657603，2089.82｝，｛2，2580.7｝｝；c is the strain rate parameter that is set to zero for the quasistatic testsand as the baseline value；.εpis the plastic strain rate；.ε*pis the nominal plastic strain rate of 1/s.Troomis the reference temperature set to 300K；and Tmeltis themelting temperatureset to 1800 K.For the quasistatic tensile tests，we set that T=Troom.Other properties given for this steel is E=210 GPa as the elasticmodulus，ν=0.33 as the Poisson ratio andρ=7850 kg/m3as the density.

The J-C［16］dam age developmentmodel is

whereσ*is the triaxiality（negative value of pressure/M ises stress ratio）.The experimental results for this steel give D1=0.069，D2=10.8 and D3=4.8［20］.We set the strain rate parameter，D4=0，as baseline value.

The brass was simulated by using the Johnson-Cook strength function w ith yield strength of 112 MPa，the hardening constant of 505 MPa，and the hardening exponent n=0.42.

Fig.2.The expanding ring test.

4.Resu lts

Fig.3 show s a picture of the 3D simulation of the experim ental setup in AUTODYN.The simulations were perform ed in 3D using quarter symm etry.In AUTODYN and IMPETUS the unstructured grid w ith 4-noded tetrahedral elements are used.The SPH algorithm wasused for theexplosive，brassand ring.The very same simulation was performed in IMPETUSAfea；however，here the corpuscularmodel was used for the explosive and the Lagrangemodel for the steel ring and the brass.The node splitting algorithm wasused for the steel ring only.

Table 1Dimensions of brass cylinder.

Table 2Steel alloy composition in percent.

Fig.4 compares the streak camera recordings w ith the simulations for two different shots.

IMPETUS Afea and AUTODYN show more or less the same results.The experimental results are scattered but in good agreement w ith the simulations.The sharp increase in the m easured velocity for the high velocity shot，at around 25μs，is likely due to gas leakage that was recorded by the streak camera.

The initial velocity of the ring is due to the grazing detonation wave.As the explosive products continue to drive the ring，the fractures begin to form and the acceleration of ring takes place.As the cracks coalesce and the fragments are formed，the explosive products begin to leak between the fragmentsand the acceleration decreases.The fragments have thus reached their final velocity.

The fragmentation pattern may depend on the numerical solver.Only the rings were used in a numerical study.The expansion velocities of the rings in Fig.4 were input to new simulations where we apply different numerical techniques. 190m/sexpansion velocity wasused for the low velocity and 630 m/s for the high velocity shot.For the low velocity we apply an element size of 600 microns in 3D as baseline.For the high velocity we apply quarter of symmetry in 3D and an element size of 400 microns.The experimental results are compared w ith the numerical simulations using the AUTODYN，IMPETUS A fea and the regularized smooth particle（RSPH）method.In AUTODYN，we used element erosion at the strain of 1.5.In the IMPETUS A fea，erosion by material failurewas used for the brass tube.Themass of the steel ring is preserved due to the node splitting algorithm.However，for the high velocity case（only），erosion due to falling time step had to be utilized in order to avoid the num erical problem s，only about 1.4%was eroded in this case.No erosion was used for the steel ring in the low velocity case and hence the fracturing of the ring is solely due to the node splitting algorithm.

Fig.3.A picture of the simulation in AUTODYN.

Fig.4.The simulated and experimental velocities of the expanding rings.

Figs.5-8 show the simulation results for baseline parametric values w ith T=Troom.We apply symmetry and only simulate half of the ring along the axial axes.Fig.5 shows the results after 6μs for the high velocity shot.In AUTODYN，a layer connected to the inner surface of the ring is severely damaged and failed.Failure develops from this region and spreads outwards by tensile，or shear，failure.In IMPETUS A fea，the node sp litting algorithm controls the fracture.The node splitted region spreadsoutwardsand radially.The RSPH showsmuch the same behavior as in AUTODYN although the severe damaged region at the inner surface is not observed.

Fig.6 shows the resultsafter 20μs.The symmetry plane is clearly seen.The number of the larger fragments is around 30-300.

Figs.7 and 8 show the similar results for the low velocity shot in IMPETUSAfea and RSPH.AUTODYN did not show any fractures.In general，the number of fragments ism uch lower than that for the high velocity shot.The reduced number of fragments can be explained.The fragmentation process startsw ith the initiation of shear or tensile fractures at some random points.After fractures are initiated，loads decrease so stresses are not sufficient to trigger themultiple fracture surfaces.However，when the same ring is deformed athigh strain rate，the fragmentation number increases since a fracture that develops at one location can only influence the stress andstrain at a neighboring location after a finite delay time.This delayed interaction between initiation sites provides time for crack grow th at neighboring sites.

Fig.5.The fragmentation/failure patterns at 6μs w ith the ring expansion velocity of 630 m/s.

Fig.6.The fragmentation/failure patterns at 20μs w ith the ring expansion velocity of 630m/s.

In AUTODYN and RSPH，the temperature of the element cells was never increased to more than 100 K，and applying tem perature softening did not significantly influence thefragmentation pattern（data notshown）.In the AUTODYN and IMPETUS A fea simulations，the energy error was lower than one percent at the tim e of crack generation.The errorwas not increasing w ith time.

Fig.7.The fragmentation/failure patterns at 6μs w ith the ring expansion velocity of 190m/s.

Fig.8.The fragmentation patternsat30μsw ith the ring expansion velocity of 190m/s.

For strain ratesup to 103/s，it isbelieved that thedislocation motion is controlled by thermal activation，and a linear logarithm ic relationship has traditionally been used for strength as a function of strain rate［16］.Above the rate of 103/s，the strength of m aterials are often significantly enhanced［16，18，31］due to the changes of them icrostructure rate controllingmechanisms.Enhanced strain rate dependencymay be due to resistance to dislocationmotion in the lattice itself by phonon viscosity［32］.The stress is here found to be linearly proportional to the strain rate［33］.In Ref.［20］，the ringmaterial was studied，and it was found that the simulation resultsworsen when the strength increasesw ith strain rate.

Shear localization due to adiabatic shear band that ismuch smaller than the element size can soften thematerial and increase theductilitymaterial.Thisunstable thermoplastic shear occurs locally in the shear bands when the local flow stress decreases w ith the increase in strain.This happens when the rate of thermal softening due to the internally generated heat exceeds the rate of isothermalwork hardening.Thegreater the shear strain rate is，the larger the number of these shear bands becomes，and hence there isa lower stress fora given strain on the element level.

The thickness of adiabatic shear bandsmay be notmore than 10 microns.Thus our 3D grid was likely too coarse to resolve shear bands by the direct simulation.However，in an attempt to im prove the simulation，we applied RSPH-2D simulations（infinite long cy linder）w ith a grid size（particle size）of around 40m icrons.For the high velocity，we applied a quarter symmetry.

Comparing the 3D results in Figs.5 and 8 w ith the RSPH-2D results in Figs.9 and 10，it appears that the shear failure ismoresignificant in 2D.The reasonwhether this is due to the grid size per se or due to the 2D assumption per se is not explained.

Fig.9.The fragmentation/failure patterns at 20μs w ith the ring expansion velocity of 630 m/s.

Fig.10.The fragmentation pattern at30μsw ith the ring expansion velocity of 190 m/s.

Fig.9 show s，com pared w ith Fig.6，that the number of fragments increases when the ductility decreases w ith strain rate（D4=-0.08）.Figs.10 and 8 show for IMPETUS Afea that the simulation w ith the explosive is sim ilar to the simulation w ith the radial expansion velocity only.

Fig.11.The numberof fragments from experimentand software simulations.

Meyer and Brannon［1］used a Weibull distributing to generate statistical failure that predicts the size distribution of fragmentsbetter than a homogeneous failuremodel.Applying this approach to the ring may enhance the fragmentation of steel［2-6］.The AUTODYN low velocity results abovemay indicate that the statistical failure isnecessary for somesolvers if the elements are too small.

Fig.11 shows the simulated and experimental number of fragments.The experimentaland simulated results are sim ilar. However，in general the numberof fragments is too large.The exception is the low velocity AUTODYN simulation that showsonly one fragment.Fig.12 shows the variability of the fragment countwhen applying different smoothing lengths h and statistical failure in the RSPH.The D2parameter in the J-C model is stochastic according to a Weibull distribution（see Ref.［1］）.

Fig.12.Variability of fragment countw ith smoothing length（h）and statistical failure（M）.

5.Conclusions and discussion

The fracture behavior of the steel ringsmade of a casing of 25 mm warhead was studied experimentally and numerically. The parameters of a J-C strength and fracture model were established using the results from tensile tests of the smoothed barand two notched bars.The simulated expansion velocity of the ringsmatches the streak camerameasurements.

Two different expansion velocities of the steel ring were studied.The number of fragments increasesw ith the increase in expansion velocity of the rings.The number of fragments was similar to the experimental results.The RSPH shows the same results as those the AUTODYN gives for the high velocity shot.The IMPETUSAfea shows a somewhat different fragm entation characteristic due to the node sp litting algorithm that induces pronounced tensile splitting.

Applying 2D in RSPH simulationw ith a grid size of around 40 m icrons，i.e.one order of magnitude lower than in 3D，showed a highernumberof shear fracturesat the inner surface of the ring.The effectwasmore visible at the lower velocity. The low velocity shot in AUTODYN did not show any fractures for chosen grid.This may highlight the necessity of applying statistical failure instead of homogeneous failure.A numerical computed chaotic trajectory divergesexponentially from the true trajectory with the same initial condition. However，there existsan errorless trajectory（no computational error）w ith slightly different initial condition，which stays near（shadow s）the numerical computed one.Thus a computational solution w ith no variation in the initial conditionsmaym im ic the true solution with the variations in the initial conditions［34］.

The number of fragments in AUTODYN increased when the ductility decreased w ith strain rate.

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.Tel.：+47 63 807514，+47 63 807000；fax：+47 63 807509.

E-mail address：john-f.moxnes@ffi.no（J.F.MOXNES）.

Peer review under responsibility of China Ordnance Society.

1Tel.：+47 63 807000.

2Tel.：+47 61 153609.

http：//dx.doi.org/10.1016/j.dt.2014.08.006

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