Inf l uence of conf i ning prestress on the transition from interface defeat to penetration in ceramic targets

Patrik LUNDBERG*，René RENSTR?M，Olof ANDERSSON

FOI，Swedish Defence Research Agency，SE-164 90 Stockholm，Sweden

Inf l uence of conf i ning prestress on the transition from interface defeat to penetration in ceramic targets

Patrik LUNDBERG*，René RENSTR?M，Olof ANDERSSON

FOI，Swedish Defence Research Agency，SE-164 90 Stockholm，Sweden

Replica scaled impact experiments with unconf i ned ceramic targets have shown that the transition velocity，i.e.，the impact velocity at which interface defeat ceases and ceramic penetration occurs，decreased as the length scale increased.A possible explanation of how this scale effect is related to the formation of a cone crack in the ceramic has been presented by the authors in an earlier paper.Here，the inf l uence of conf i nement and prestress on cone cracking and transition velocity is investigated.The hypothesis is that prestress will suppress the formation and growth of the cone crack by lowering the driving stress.A set of impact experiments has been performed in which the transition velocity for four different levels of prestress has been determined.The transition velocities as a function of the level of conf i ning prestress is compared to an analytical model for the inf l uence of prestress on the formation and extension of the cone crack in the ceramic material.Both experiments and model indicate that prestress has a strong inf l uence on the transition from interface defeat to penetration，although the model underestimates the inf l uence of prestress.

Impact；Ceramic；Armour；Interface defeat；Dwell；Conf i nement；Prestress

1.Introduction

The high strength of armour ceramics ［1-3］makes it possible to partially or totally defeat high velocity projectiles directly at the surface of the ceramic material.This phenomenon is called interface defeat or dwell［4-17］and is an important defeat mechanism in，e.g.，light armour applications.

One limitation when applying this in heavier armour designs is that it appears to be length scale dependent.Replica scaled impact experiments with unconf i ned ceramic targets show that the transition velocity，i.e.，the velocity at which interface defeat ceased and ceramic penetration occurred，decreased as the length scale increased ［11］.A probable explanation of the observed scale effect is that although maximum shear strength determines the upper bound for the transition from interface defeat to penetration，it is usually limited by the formation and growth of macroscopic cracks.Since the crack resistance of ceramic materials decreases with increasing length scale，in contrast to the otherwise scale-invariant stress f i eld，the extension of a crack to a critical size will occur at a lower impactvelocity in a larger target.An analytical model in ［11］for the inf l uence of length scale on the growth of a cone shaped modus I crack in thick unconf i ned ceramic targets gave reasonable results compared to the replica scaled impact experiments. The model showed that the projectile pressure at transition，i.e.，the impact velocity at which the contact pressure exceeds the strength of the ceramic material and penetration initiates，is proportional to one over the square root of the length scale.

A possible way to suppress the formation and growth of macroscopiccracksistoprestresstheceramicmaterial.Theinf l uence of prestress and the related failure modes of impacted ceramics have been studied by several authors.The papers ［18-20］report experimental data on small calibre projectiles impacting thin prestressed ceramics （i.e.，the thickness is of the same order as the diameter of the projectile）.These studies show that prestress reducesdamageintheformoffewermacroscopiccracksandthat the trajectory of possible cone cracks becomes shallower.An increase in protective performance was also observed.The papers ［8，9］report experimental data on model scale long rod projectiles impacting thick ceramic targets （i.e.，the thickness is much larger than the diameter of the projectile）.The experiments in ［8］with large and heavily conf i ned and prestressed targets showed similar interface defeat velocities as small，unconf i ned targetsin ［9］.Thisindicatestheneedofprestressinlargertargets.Holmquist and Johnson ［21］and later Runqiang et al.［22］conducted a computational study on the responses of a small scale thick prestressed ceramic target tested by Lundberg et al. ［7］. Various levels of prestress and stress states were simulated.Their studies showed that prestress enhanced the performance and that the velocity at which ceramic penetration occurred，i.e.，the transition velocity，could be increased by prestress.

This paper explores the inf l uence of a radial conf i ning prestress on the transition from interface defeat to penetration for a thick ceramic target.Although the physical background of the inf l uence of prestress on the transition velocity in ceramic targets is not fully explained，impact experiments as well as modelling indicate that it is intimately linked to ceramic fracture.A hypothesis proposed in ［11］is that the centre part of the ceramic suddenly loses radial support as a result of the cone cracking.A conf i ning prestress will suppress the growth of the cone crack by lowering the stress intensity over the crack tip.In order to overcome this virtual toughening of the ceramic，the projectile pressure on the surface of the target must be increased relative to that for an unconf i ned target in order to initiate critical fracture.A set of impact experiments have been performed in which the transition velocities for four different levels of prestress were determined.Two grades of silicon carbide ceramics with slightly different mechanical properties were used.The experimental technique used is presented in the paper together with the determined transition velocities versus radial conf i ning prestress.The experimental data are compared to an extended version of the model presented in ［11］.

2.Model of cone crack under conf i ning prestress

The inf l uence of a radial conf i ning prestress on the formation and extension of a cone shaped crack to a critical size is approximated in the model by the assumption that the crack extension occurs along a surface of principal stress.The normal stressonthissurfaceiscalculatedforthecaseofanaxi-symmetric contact pressure from a projectile in a state of interface defeat and for a radial conf i ning prestress，respectively.The critical normal stress for propagating the crack is determined through:（i）a stress intensity factor at the tip of the crack and （ii）a function of the inf l uence of external load and geometry on the path of the crack.

The detailed description of the present model is divided into four sections:2.1 Projectile contact pressure and stresses in the target，2.2 Principal surfaces and stresses，2.3 Crack initiation and propagation under conf i ning prestress and 2.4 Crack opening under conf i ning prestress.

2.1.Projectile contact pressure and stresses in the target

A long cylindrical projectile is assumed to fl ow axisymmetrically on the fl at and friction-free surface of an otherwise unbounded，but radially prestressed target，see Fig.1.The fl ow and the loading on the surface are steady，i.e.，the initial transient part of the impact process is not considered.The target material is linearly elastic with Young's modulus E and Poisson's ratio ν，respectively.The material of the projectile is linearly elastic and perfectly plastic with bulk modulus Kp，yield strengthσypand densityρp.

Fig.1.Projectile shape （solid line），axi-symmetric pressure of projectile （dotted line）and crack trajectory in target（gray）during a state of interface defeat.

With the assumption that the effects of yield strength and compressibility are small relative to that of inertia，the axisymmetric contact pressure of projectile load can be approximated ［7］by

whereq（ r）is the radial pressure distribution corresponding to inertia，

and

Herev0is the impact velocity of the projectile andqpis the stagnation pressure of an ideal fl uid with densityρp.The dimensionless parameters α ＞＞ 1 and β <<1 relate elastic and plastic effects to the effect of inertia.The in fl uence of β in Eq.（1）is evaluated from simulations in ［17］and the radial distribution ofq（ r）is taken from a low-velocity water jet ［7］.

The stress fi eld σij（r， z）of projectile pressure p in the semiin fi nite elastic target half-space，is expressed by a Boussinesq's potential as

whereIijis an in fl uence-function （negative in compression）for a point load of the amount of p? d ? d υat a surface point （υ， ?），with p according to Eq.（1）.Indicesi， jare the generic spatial variables r，φ，and z and A is the radius?of the circular limit of the distribution of projectile pressure.The radial stress componentσrris affected by the con fi ning prestresspcfso that the stress components from projectile and con fi nement together are expressed as

2.2.Principal surfaces and stresses

The assumed crack path along the principal surface z1（ r）is determined as

whereθ（r）is the angle betweenz1（ r）and the target surface z=0.The target surface is a principal surface since friction free，thus θ= π2forz=0.Rewriting Eq.（6），by the use of tan 2θ= 2 tan θ（1-tan2θ），Eq. （5） and the well-known expressions for stress-geometry on principal surfaces gives

where the sign is chosen so that the resulting surface crosses the target surface perpendicularly.The crack arc lengthc（ r）along the surfacez1（ r ），the principal stressesσ1，σ2andσ3（normal and tangential to this surface）and the strainε1，are expressed as

and

The principal surfacez1（ r ）is chosen in such a way that the stressσ1is tensile on z1（ r ）.

2.3.Crack initiation and propagation under con fi ning prestress

The maximum tensile stress at radiusr =rion the surface of the target initiates fracture for

and propagates the crack alongz1（ r ）as long asσ1exceeds a critical stressσc.The critical stress will be determined through a stress intensity factor for small crack extension c，as

where h is a function of external load and geometry.Although not explicitly mentioned in ［11］，h was found to correspond to unit value for pcf=0.Since the geometry and loading conditions here are similar to the one used in ［11］，except for the radial con fi ning prestress 0 ＜ pcf，h is sought as a function ofpcfand p0in such a way that the criterion of crack propagation and the critical stressσccan be written in comparison to that for pcf=0as

whereK1cis the fracture toughness of the ceramic material. Eqs.（7）-（12）result in an equation for critical crack extensions rcas roots of the equation

A more explicit expression of parameter in fl uence in Eq.（13）is achieved by the use of dimensionless variables.The principal stressσ1，con fi ning prestresspcf，crack arc lengthc and spatial coordinatesr，z are expressed in units of projectile pressure p0and radiusa through，pcf=p0c =，Eq.（8）andr =，z =，respectively.The overbar sign designates dimensionless.

Introducing dimensionless variables into Eq. （13）gives a relation between the projectile radiusaand a critical projectile pressure.This critical projectile pressure corresponds to the development of a cone crack with radial extensionrcand is named the transition pressure where the superscript*is used to denote critical/threshold quantities.The transition pressure corresponds to the transition velocityvia Eqs. （1）-（3）and is here assumed to be the lowest bound for transition from interface defeat to penetration for different levels of con fi ning prestresspcfaccording to

where the right hand member of Eq. （14）is valid for roots rc=of Eq. （13）.

The transition pressure for a con fi ned target relative to that of zero con fi nement is approximately expressed as

2.4.Crack opening under conf i ning prestress

Consider a sub-critical crack and its expansion to a stable，open crack and that the projectile force act to propagate thecrack and the force of con fi ning prestress is counteracting the crack propagation.The function of external load and geometry h in Eq. （11）is approximated through the opening process of the crack，that is，the changeΔU of the accessible partUof the internal elastic energy of the ceramic material.The energy term ΔUis the elastic energy released in crack-opening，so that the derivative ofΔUwith respect to crack propagation will be the relevant energy to drive the crack.Therefore，ΔUis proportional to the square of the stress intensity and thereby the square of h，Eq.（11）.

With the force of con fi ning prestress in place and with a projectile load close to maximum，but before cracking，the accessible internal energy may be expressed as

where F，P and ur（ F ），uz（F ），ur（ P）are the forces from projectile，con fi nement pressure and corresponding boundary displacements of the ceramic cylinder，respectively.The internal energy is expressed as if the ceramic body is fi rst squeezed at its cylindrical surface by a constant pressurepcfand then at its cross-end surfaces，each of area A0，by constant pressureFA0.The displacements in Eq. （16）may then be expressed as

whereRandL are radius and length of the ceramic cylinder and A0=πR2its cross-end surface.The differenceΔUcan be expressed as

whereU′and u′denote accessible internal energy and displacement as if the open stable crack is already present when F is applied.As a fi rst order approximation of Eq. （18），it is noted that（i）the volume of the open crack is small and（ii）at the moment of its instability，the critical geometry and volume at the crack are unaffected by P.It is here assumed thatΔuican be expressed as

where ηiare assumed to be small constants.Eq.（18）can now be expressed as

where，according to inferred assumptions and Eq. ［11］，

so that

Eq.（22）gives the effect of con fi ning prestress for a constant P and an approximate expression ofh.

Using Eq.（17），the projectile radiusa，the cross-end area of the ceramicA0=πR2and the expressions of force，viz.，F(xiàn)=p02πa2and P = pcfπ2RL ，the terms in the third of Eqs.（22）are expressed as

Finally，Eqs. （14）and （15）are substituted byEq. （23）to express the parameters of the transition pressurefor a confi ning prestress 0 ≤relative to that of zero con fi nement.

3.Experiments

The impact experiments were performed using a reverse impact technique.The stationary projectiles were suspended in blocks of Divinycell material （density 45 kg/m3）and mounted in front of the muzzle of the gun，see Fig.2.Two different qualities of silicon carbide materials have been used:SiC-B and SiC-X1，both materials are from CoorsTek （former BAE Systems Advanced Ceramics Division and Cercom Inc，Vista，CA）.The SIC-B material was initially delivered as large cylinders with diameter 50 mm and length 50 mm.From these，smaller cylinders with diameter 20 mm and length 20 mm were produced.The SiC-X1 material was delivered as cylinders with diameter 20 mm and length 20 mm.

Data on microstructure，Young's modulus and fracture toughness of the SiC-B used has been published byWereszczak et al. ［23］and properties for SiC-X1 has been provided by CoorsTek.The properties are given in Table 1.

Fig.2.Experimental set-up showing the stationary projectile suspended in a block of Divinycell material and mounted in front of the muzzle of the gun. Images plates and normal intensif i er screens and f i lm for the f l ash X-ray pictures are placed in the U-shaped cassette below the projectile.

Table 1Material data for SiC-B and SiC-X1.

Table 2Hardness and fracture toughness for SiC-B and SiC-X1.

The Vickers hardnessH and the fracture toughnessKIcwere estimated by means of a Wholpert macro hardness indenter equipped with video-system and imaging software. SiC-B and SIC-X1 samples were carefully polished using a semi-automatic polishing machine，and each material was indented 9 times.The fracture toughness was determined according to Anstis et al.［24］.The estimated Vickers hardness and fracture toughness of SiC-B and SiC-X1，normalised to these quantities for SiC-B，are given in Table 2.

The projectiles were fl at-ended cylinders made of a sintered tungsten alloy，Y925 from Kennametal HertelAG ［25］.Material data for the projectile are given in Table 3.

The target con fi nement consisted of a steel tube （B?hler W725， Poisson'sratio νs=0. 3andYoung'smodulus Es=186GPa ）.The external diameter of the steel tube was originally 30 mm and its internal diameter was 0.07 mm smaller than the diameter of the ceramic cylinder.Shrink fi t was achieved by heating the steel tube to about 500 °C before inserting the ceramic cylinder.After cooling，the con fi nement was machined to fi nal shape with tube wall thicknessest=1，2 and 4 mm，respectively.The shrink fi t resulted in a con fi ning prestress pcfon the ceramic cylinder estimated by

whereδis the difference between the radius of the ceramic cylinder and the interior radius of the con fi ning steel tube.Both the ceramic cylinders and the con fi ning steel tubes were ground to fi nal dimensions.The variation in ceramic and steel tube diameters gave a maximal variation inδof the order of 0.005 mm.

The front surface of the ceramic target was protected with a circular copper cover.The copper cover is expanded in its central part to a cylinder.It was glued onto the front surfacealong the rim.The geometries of the four target types used are shown in Fig.3.The different target types are labelled （a）-（d），corresponding to uncon fi ned，1，2 and 4 mm thick con fi nement，respectively.The different test series are summarised in Table 4.

Table 3Material data for the projectile material Y925.

The impact velocity and target response were evaluated from fl ash X-ray pictures.A 150 kV X-ray system was used together with image plates for the velocity pictures taken before the interaction.For the penetration pictures，a 450 kV X-ray system was used together with both the image plates and normal intensi fi er screens and X-ray fi lm.The 450 kV fl ashes were positioned at the same distance from the barrel and radially separated by 30°.The image plate picture was digitised using a laser scanner.The X-ray fi lms were digitised using a fl at-bed scanner.Enhancement was achieved by image-processing （contrasts，edges etc.）before evaluation.The uncertainty in the impact velocity evaluation，due to limited X-ray image resolution，measurement errors etc.，was within ±5 m/s.

The transition velocity was estimated by systematically varying the impact velocityv0.Ideally，the transition occurs at a well-de fi ned impact velocity，but in practice it is only possible to determine a velocity interval in which the transition occurs. The lower and upper limits of this interval correspond to the highest impact velocity observed without penetration and the lowest impact velocity with penetration.The transition velocity was estimated as the centre-point of this interval.The velocity interval was indicated by adding ± half the length of the interval to the estimated transition velocity.Typically 4-8 impact tests were needed in order to determine the transition velocitywithin ± 50 m/s，viz.，a total number of 44 impact experiments has been performed.

Fig.3.Target types: （a）unconf i ned，（b）conf i ned t=1 mm，（c）conf i ned t=2 mm，（d）conf i ned t=4 mm.Dimensions are in mm.

Table 4Targets and projectiles used in the impact experiments.

4.Results

Examples of X-ray pictures from the impact tests are shown in Figs.4 and 5.Fig.4 shows an uncon fi ned SiC-X1 target under a state of interface defeat while Fig.5 illustrate the phenomena of interface defeat for four different levels of prestress:pcf=0，56，101 and 16?8 MPa.

Fig.4.X-ray pictures of Y925 projectile and uncon fi ned SiC-X1 target at four different times after impact.The time interval between the pictures is 10 μs.The? i mpact velocity v0 = 967m sis slightly below the transition velocityv0 = 982 ±15ms.

Fig.5.X-ray pictures of Y925 projectiles and SiC-X1 targets with four levels of prestress under a state of interface defeat: （a）t=0 mm，v0 = 967m s，（b）t=1mm，v0 = 1347m s，（c）t=2 mm，v0 = 1508m s and （d）t=4 mm，v0 = 1485m s.

The transition velocitiesv0determined from the impact experiments and corresponding projectile pressures at transitioncalculated from Eqs. （1）-（3）are shown in Table 5 together with estimated levels of con fi ning prestresspcfaccording to Eq. （24）.

The principal surfacez1a used as crack-path in the model and the principal stressσ1p0along the crack are shown in Fig.6 versus radial crack extensionra for different levels of con fi ning prestresspcfp0，Eqs.（4），（5）and （7）-（9）.The critical crack extensionrca ，determined as a root of Eq.（14），is illustrated in Fig.7（a）for an uncon fi ned target and Fig.7（b）show the function1h versus the con fi ning prestress pcfp0.

The in fl uence of the fracture toughness KIcof the ceramic material on the transition pressure is illustrated in Fig.8. Experimentally determined as well as estimated transition velocities and pressures versus prestress are shown in Fig.9. Unless otherwise stated，the material data for SiC-B in Table 1 have been used in Figs.5-8.

5.Discussion

Fig.4 shows the development of interface defeat at an impact velocity just below the transition velocity in an unconfi ned SiC-X1 target.The copper cover in front of the ceramic reduces the initial effect of impact by establishing erosion of the projectile before the latter reaches the ceramic surface.Other studies ［10，26］have shown that after radial fl ow had been established，it continued steadily for a long time，i.e.，the projectile load can be seen as quasi-static.Although the transition velocity increases with prestress，the velocity interval from interface defeat to penetration remains narrow and is not affected by the prestress and the fl ow onto of the ceramic surface look similar for the different targets，see Fig.5.

The effect of prestress is clearly seen in the experiments；the transition from interface defeat to penetration is moved to signi fi cantly higher levels as the con fi ning pressure increases.The velocity at transition in uncon fi ned SiC-X1 target was found to be 982 m/s whereas a con fi ning pressure of pcf≈56MPa increased this transition velocity to 1367 m/s.This relatively low level of prestress almost doubles the projectile pressure at transition.The two leftmost open circles in Fig.9 correspond to these values.Further increase of the con fi ning pressure did not show the same strong in fl uence and a transition velocity of≈ 1500 ms was found unaffected in spite of increased confi ning pressure from 100 to 168 MPa.The SiC-B seams to behave in a similar way as the SiC-X1，though no data is available in-between uncon fi ned and 168 MPa of prestress.Although the number of experiments is not suff i cient for a statistical analysis，the SiC-B seems to perform slightly better than SiC-X1.This could be a result of the slightly higher fracture toughness for SiC-B or due to natural variations in the properties of the projectile which determines the transition pressure.Use of Eqs. （1）-（3）and typical uncertainties for the material data in Table 3 give an indication of an maximum error in transition pressure within ±1.7 GPa，i.e.，the observed differences between SiC-B and SiC-X1 may simply be within statistical f l uctuations.

Table 5Transition velocities and corresponding estimates of the projectile pressure.The superscript*is used to denote critical/threshold quantities.

The SiC-X1 experiments indicate a shift in behaviour for a conf i ning prestress around 100 MPa.This is illustrated with two grey sectors in Fig.9（a）and （b）.The shift may indicate that another fracture mode，e.g.，modus-II cracking，has been activated.A higher transition velocity for a similar combination of ceramic and projectile materials has been reported ［8］but thenwith strong axial and radial conf i nement.Such a high overall conf i ning pressure could affect both modus-I and II fracture.

Fig.6. （a）Crack path z a and （b）principal tensile stress along crack path σ1 p0versus radial crack extension r a for different levels of con fi ning prestress.

Fig.7. （a）Principal tensile stressσ1 p0（solid curve）and critical stress σc/p0（dashed curve）along the crack （r/a）for an estimated stress state at the point of transition in an uncon fi ned targetpcf p0 =0.（b）The function1 h （solid curve）versus con fi ning prestresspcf p0.

Fig.8.Estimate of maximum projectile pressure at transition p?0versus confi ning prestress pcffor three different levels of ceramic fracture toughness: KIc=3（dashed curve），4 （solid curve）and 5 MPam?（dotted curve），respectively.

The model for the projectile pressure in Eqs. （1）-（3）gives similar results as numerical simulations in ［17］where more detailed material models for both the tungsten and the SiC material were used.The projectile pressure distribution used here is not identical to the one in ［17］but gives only minor changes in the overall stress distribution and does not appreciably change the results.

The inf l uence of the conf i ning prestress in the model consists of two parts:（i）the change in tensile stress f i eld due to the prestress and （ii）the inf l uence of prestress on the opening process of the crack itself.The model for the tensile stress over the crack assumes that the stress f i eld will be unaffected by the crack.This is a simplif i cation but does not signif i cantly change the general behaviour of the model.Fig.6 shows that the principal tensile stress over the crack decreases and changes direction towards the impact surface as a result of the con fi ning prestress.This will gradually reduce the effect of a radial prestress on the crack propagation.

A valid solution rca of Eq. （13）is assumed to exist for aprojectile pressure suf fi cient to satisfy Eq. （12）and so that is the largest of the three-root solution.The critical crack length is therefore a constant for each grade of prestress since

in Eq. （13）is a fi x function.In agreement with this assumption，F(xiàn)ig.7（a）shows the estimated critical radial crack extensionrca =6. 4for an uncon fi ned target.

The in fl uence of prestress on the opening of the crack is approximated by an energy model，that is，the changeΔU of the accessible partU of the internal elastic energy of the ceramic material.The effect is described by the function of external load and crack geometry h，see Fig.7（b）.

Eq.（14）and Fig.8 show that the model in fl uence of fracture toughness on the transition from interface defeat to penetration is strong；the transition pressure at transition is directly proportional to the fracture toughness of the ceramic material.This is in line with the experimental fi ndings in ［10］where the transition velocity was determined for four different silicon carbide materials with slightly different fracture toughness.

The model shows an approximately linear relation between con fi ning prestress and the transition pressure and that this corresponds to the experimental data for prestress 0 ≤pcf≤100MPa .But the model underestimates the in fl uence of prestress with a factor of two relative to the experiments in this interval.The main hypothesis for the deviation between model and experiments is that the real con fi ning prestress is higher than estimated by Eq. （24）.The reason for this is that dynamic effects，e.g.，the inertia of the con fi nement，have not been taken into account，neither in the model nor in Eq.（24）.A dynamic effect would probably shift the experimental data towards higher prestress values and at the same time increase the gradient of the function h.A more detailed analysis of a possible dynamic effect is not within the scope of this paper and will probably require some type of continuum mechanic code.

Fig.9. （a）Estimated maximum projectile pressure at transitionand （b）transition velocityversus con fi ning prestresspcf.Experiments with SiC-B are marked with fi lled circles and SiC-X1 with open circles，respectively.Error bars indicate maximum and minimum values for projectile velocity，projectile pressure and con fi ning prestress，respectively.Solid curve:model estimate，grey sectors:graphical illustration of a possible two mode behaviour，respectively.

6.Conclusions

The main conclusions of this study can be summarised as follows: （i）An analytical model for the relation between projectile pressure and propagation of a cone crack under a state of interface defeat in a ceramic target has been formulated.The model connects the effects of fracture toughness and con fi ning prestress of the ceramic to the transition from interface defeat to penetration. （ii）The model shows a strong in fl uence of radial prestress；the projectile pressure at transition increases linearly with the level of prestress. （iii）Impact experiments with four different levels of prestress show that prestress has a stronger in fl uence than predicted by the model；the maximum possible contact pressure at transition is more than doubled if prestress is increased from zero to around 100 MPa.（iv）Further increase of the con fi ning pressure did not show the same in fl uence and a transition velocity of ≈ 1500 ms was found unaffected in spite of increased conf i ning pressure. （v）The model shows a similar behaviour compared to the lower interval of prestress，but underestimates the inf l uence of prestress with a factor of two.（vi）The main hypothesis for the deviation between model and experiments is due to dynamic effects which will increase the conf i ning prestress relative to the one estimated by the quasistatic model.

Acknowledgements

This research was funded by the SwedishArmed Forces and by the Army Research Laboratory through US Naval Regional Contracting Centre，Contract No.W911NF0810271.

［1］Vogler TJ，Chhabildas LC.Strength behaviour of materials at high pressure.Int J Impact Eng 2006；33:812-25.

［2］Kanel GI，Zaretsky EB，Rajendran AM，Razorenov SV，Savinykh AS，Paris V.Search for conditions of compressive fracture of hard brittle ceramics at impact loading.Int J Plasticity 2009；25:649-70.

［3］Paris V，F(xiàn)rage N，Dariel MP，Zaretsky E.Divergent impact study of the compressive failure threshold in SiC and B4C.Int J Impact Eng 2011；38:228-37.

［4］HauverGE，NetherwoodPH，BenckRF，KecskesLJ.Ballistic performance of ceramic targets.Army Symposium on Solid Mechanics，USA；1993.

［5］Hauver GE，Netherwood PH，Benck RF，Kecskes LJ.Enhanced ballistic performance of ceramic targets.19thArmy Science Conference.USA；1994.

［6］Rapacki EJ，Hauver GE，Netherwood PH，Benck RF.Ceramics for armours-a material system perspective.7thAnnual TARDEC Ground Vehicle Survivability Symposium.USA；1996.

［7］Lundberg P，Renstr?m R，Lundberg B.Impact of metallic projectiles on ceramic targets:transition between interface defeat and penetration.Int J Impact Eng 2000；24:259-75.

［8］Hauver GE，Rapacki EJ，Netherwood PH，Benck RF.Interface defeat of long rod projectiles by ceramic armor.ARL-TR-3590；2005.

［9］Holmquist TJ，Anderson CE Jr，Behner T.Design，analysis and testing of an unconf i ned ceramic target to induce dwell.22ndInternational Symposium on Ballistics，Vancouver，BC，Canada，14-18 November；2005.

［10］Lundberg P，Lundberg B.Transition between interface defeat and penetration for tungsten projectiles and four silicon carbide materials.Int J Impact Eng 2005；31:781-92.

［11］Lundberg P，Renstr?m R，Anderson O.Inf l uence of length scale on the transition from interface defeat to penetration in unconf i ned ceramic targets.J Appl Mech 2013；80（3）:031804-1-9.

［12］Behner T，Anderson CE Jr，Holmquist TJ，Wickert M，Templeton DW. Interface defeat for unconf i ned SiC ceramics.Proc 24thInt Symp Ballistics 2008；1:35-42.

［13］Anderson CE Jr，Behner T，Orphal DL，Nicholls AE，Holmquist TJ，Wickert M.Long rod penetration into intact and pre-damaged SiC ceramic.Proc 24thInt Symp Ballistics 2008；2:822-9.

［14］Anderson CE Jr，Behner T，Holmquist TJ，Orphal DL，Wickert M Dwell，Interface Defeat，and Penetration of Long Rods Impacting Silicon Carbide，Southwest Research Institute Technical Report 18.12544/008.

［15］LaSalviaJC，HorwathEJ，RapackiEJ，ShihCJ，MeyersMA. Microstructuraland micromechanicalaspectsofceramic/long-rod projectiles interactions:dwell/penetration transitions.In:Staudhammer KP，Murr KP，Meyers MA，editors.Fundamental issues and applications of shock-wave and high-strain-rate phenomena.New York:Elsevier Science.2001 p.437-46.

［16］Renstr?m R，Lundberg P，Lundberg B.Stationary contact between a cylindrical metallic projectile and a f l at target under conditions of dwell. Int J Impact Eng 2004；30:1265-82.

［17］Lundberg P，Renstr?m R，Lundberg B.Impact of conical tungsten projectiles on f l at silicon carbide targets:transition from interface defeat to penetration.Int J Impact Eng 2006；32:1842-56.

［18］Sherman D，Ben-Shushan T.Quasi-static impact damage in conf i ned ceramic tiles.Int J Impact Eng 1998；21:245-65.

［19］Sherman D.Impact failure mechanisms in alumina tiles on f i nite thickness support and the effect of conf i nement.Int J Impact Eng 2000；24:313-28.

［20］Savio SG，Ramanjaneyulu K，Madhu V，Bhat TB.An experimental study on ballistic performance of boron carbide tiles.Int J Impact Eng 2011；38:535-41.

［21］Holmquist TJ，Johnson GR.Modelling prestressed ceramic and its effect on ballistic performance.Int J Impact Eng 2005；31:113-27.

［22］Runqiang C，Ahmad S，Idapalapati S，Tan G.Pre-stress effect on conf i ned ceramic armor ballistic performance.Int J Impact Eng 2015；84:159-70.

［23］Wereszczak AA，Johanns KE，Jadaan OM.Hertzian ring crack initiation in hot-pressed silicon carbides.J Am Ceram Soc 2009；92（8）:1788-95.

［24］Anstis GR，Chantikul P，Lawn BR，Marshall DB.A critical evaluation of indentation techniques for measuring fracture toughness:I direct crack measurements.J Am Ceram Soc 1981；64（9）:533-8.

［25］Skoglund P.Constitutive modelling of a tungsten heavy metal alloy. J Phys IV 2003；110:207-12.

［26］Lundberg P，Renstr?m R，Holmberg L.An experimental investigation of interface defeat at extended interaction times.19thInt Symp Ballistics，Switzerland 2001；3:1463-9.

Received 15 September 2015；revised 23 January 2016；accepted 15 February 2016 Available online 2 March 2016

Peer review under responsibility of China Ordnance Society.

*Corresponding author.Tel.:+46 709277154.

E-mail address:patrik.lundberg@foi.se （P.LUNDBERG）.

http://dx.doi.org/10.1016/j.dt.2016.02.002

2214-9147/? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.

? 2016 China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.