Dimensional analysis and extended hydrodynamic theory applied to long-rod penetration of ceramics

J.D.CLAYTON*

Impact Physics RDRL-WMP-C，US Army Research Laboratory，Aberdeen Proving Ground，Aberdeen，MD 21005-5066，USA A.James Clark School of Engineering，University of Maryland，College Park，MD 20742，USA

Dimensional analysis and extended hydrodynamic theory applied to long-rod penetration of ceramics

J.D.CLAYTON*

Impact Physics RDRL-WMP-C，US Army Research Laboratory，Aberdeen Proving Ground，Aberdeen，MD 21005-5066，USA A.James Clark School of Engineering，University of Maryland，College Park，MD 20742，USA

Principles of dimensional analysis are applied in a new interpretation of penetration of ceramic targets subjected to hypervelocity impact.The analysis results in a power series representation-in terms of inverse velocity-of normalized depth of penetration that reduces to the hydrodynamic solution at high impact velocities.Specif i cally considered are test data from four literature sources involving penetration of conf i ned thick ceramic targets by tungsten long rod projectiles.The ceramics are AD-995 alumina，aluminum nitride，silicon carbide，and boron carbide. Test data can be accurately represented by the linear form of the power series，whereby the same value of a single f i tting parameter applies remarkably well for all four ceramics.Comparison of the present model with others in the literature （e.g.，Tate’s theory）demonstrates a target resistance stress that depends on impact velocity，linearly in the limiting case.Comparison of the present analysis with recent research involving penetration of thin ceramic tiles at lower typical impact velocities conf i rms the importance of target properties related to fracture and shear strength at the Hugoniot Elastic Limit （HEL）only in the latter.In contrast，in the former （i.e.，hypervelocity and thick target）experiments，the current analysis demonstrates dominant dependence of penetration depth only by target mass density.Such comparisons suggest transitions from microstructure-controlled to density-controlled penetration resistance with increasing impact velocity and ceramic target thickness. Production and hosting by Elsevier B.V.on behalf of China Ordnance Society.

Ceramics；Terminal ballistics；Armor；Dimensional analysis；Hydrodynamics

1.Introduction

Ceramic materials are of keen interest for use in modern armor systems because of their high hardness，high stiffness，and relatively low mass density relative to traditional armor steels，for example.As reviewed in Reference 1，ceramic materials have been considered for personal and vehicular protection systems since the mid 20th century.Popular candidate ceramics for such systems include alumina，aluminum nitride，boron carbide，silicon carbide，and titanium diboride，among others（e.g.，transparent spinels ［2］and glass ［3］）.Despite signif i cant research progress，precise relationships among （micro）structures，properties，and dynamic performance of these often complex materials remain undef i ned or even contradictory among collective f i ndings of the engineering mechanics，materials science，and condensed matter physics communities.

A number of different experimental methods have been devised over the past six decades that probe properties or performance of ceramic materials at very high rates of loading pertinent to armor applications.These experiments involve impact and/or penetration of ceramic targets and commensurate shock wave propagation，progressive material degradation，and material failure.Two particular experimental target conf i gurations are of primary interest in this work.The f i rst，as explained in detail in References 4 and 5，consists of one or more ceramic tiles backed by a semi-inf i nite metal block.Performance of the ceramic armor tiles is measured by the depth of penetration of the projectile，often either a metallic bullet or rather short rod，into the backing metal.The second，also described in Reference 4 as well as References 6 and 7 involves thick（effectively semi-inf i nite）cylindrical ceramic targets conf i ned laterally，with or without a metallic cover plate.Performance of the thick ceramic is measured via depth of penetration of the projectile，often a metallic long rod.Dwell and interface defeat may be observed if the projectile velocity is too low to enable penetration into a hard，conf i ned ceramic［7］.

A vast ensemble of models have been developed，often in conjunction with experimental methods，for describing the mechanical response of armor ceramics.These can be organized in terms of scale of resolution.Quantum mechanical representations incorporating density functional theory ［8，9］have enabled quantif i cation of fundamental elastic and fracture properties of ceramic crystals and interfaces，as have empirical lattice statics/dynamics calculations ［10］.Mesoscale calculations ［11-13］incorporating cohesive f i nite elements ［14-16］or phase f i eld representations ［17，18］enable description of the effects of microstructure such as grain sizes and orientations on the response of ceramic polycrystals.Macroscopic constitutive models are used in hydrocodes for computing the response of protection systems subjected to impact，blast，and perforation. These models may be empirical ［19-23］or based on micromechanical principles ［24-26］.One-dimensional f i nite crystal mechanics models have also been developed for addressing planar shock experiments ［27-31］.Finally，analytical penetration models based on a one-dimensional momentum balance and various simplifying assumptions have been invoked to describe mechanics of shaped charge jets ［32-34］and long rods［35-38］piercing thick ductile metallic targets.These models，which tend to reduce to a Bernoulli-type equation at the stagnation point of impact，will be critically reviewed later in the present work.

The present paper provides a new description of penetration of semi-inf i nite ceramic targets by metallic long-rod projectiles using dimensional analysis with application to literature data［6，39-41］.This paper extends prior recent work ［5］that addressed，in an empirical manner，the f i rst experimental conf i guration discussed above，i.e.，metal-backed ceramic tiles.The present work includes new comparison of dimensional analysis with some one-dimensional penetration theories ［35，36，42］for example providing contextual insight into the target resistance term in Tate’s model ［36］，now applied for ceramic targets. Another goal is possible elucidation of the effects of differences，or lack thereof，among ceramic material properties（inherently related to microstructure）on penetration resistance.

The main body of this paper is structured as follows.In §2，one-dimensional penetration mechanics models are reviewed，including key assumptions and governing equations.In §3，dimensional analysis of the problem of penetration of thick ceramic targets is performed.In §4，application of this dimensional analysis to test data is reported.In §5，further analysis and comparison of the present results with one-dimensional penetration mechanics models and other experimental conf i gurations are undertaken.Conclusions follow in §6.

2.One-dimensional penetration mechanics

Presented next in §2.1 is a derivation of the governing equation for ideal hydrodynamic penetration of a semi-inf i nite target by a jet or rod.This derivation provides a basis for the extended descriptions and more elaborate models discussed in §2.2.

2.1.Ideal hydrodynamic theory

The ideal hydrodynamic theory of penetration of ductile targets by shaped charge jets was developed around the Second World War and years soon thereafter ［32］.The assumptions involved in the derivation are listed as follows:the penetration process is steady-state and one-dimensional；the target is semiin fi nite；the projectile is a continuous jet；and both target and jet materials are incompressible with null shear strength，thereby acting as perfect fl uids.

Let spatial coordinates at time t be denoted by xi=xi（ XI，t ），where reference coordinates of a material point are XIand in general，i， I =1， 2， 3.Particle velocity is

Letσijdenote the symmetric Cauchy stress tensor，and p =the Cauchy pressure.Let ρ denote the spatial mass density.The local balance of linear momentum in continuum mechanics is ［43］

In a Cartesian coordinate system，the particle acceleration is the following material time derivative of the velocity

For steady fl ow，υi= υi（xj）and the fi rst term on the right side of Equation （3）vanishes，leading to

For steady one-dimensional fl ow，i，I=1 and Equation （2）becomes，withσ（x）=-σ11（x ）the axial stress，positive in compression，

Now assume that a jet or rod with initial velocity V impinges on the target，an inf i nite half-space.The stagnation point between projectile and target recedes with velocity U. The axial stress P at the stagnation point in the projectile is then found by integrating Equation （5）with ρ0the projectile density，which is assumed constant for incompressible f l ow，leading to ［44］

Now considering the stagnation point in the target，which is assumed incompressible with mass density ρT，

Equating P from Equations （6）and （7），and letting p=P for inviscid f l ow （no shear stresses），Bernoulli’s equation for steady hydrodynamic penetration is

The time needed for a projectile of initial length L0to fully erode ist0= L0（U -V ），and the depth of penetration is P0=U·t0.Using Equation （8），the normalized depth of penetration can be expressed completely in terms of the ratio of densities of target and projectile

Though originally developed for penetration of metallic targets by metallic jets，this model has been used，often with success，for describing the steady penetration regime for longrod projectiles as well as brittle targets.Regardless，it serves as a useful basis of comparison with experimental data and predictions of more complex theories and/or numerical calculations.

2.2.Extended penetration models

Many analytical penetration mechanics models have used the hydrodyamic theory derived in §2.1 as a foundation or starting point.Birkhoffetal. ［32］ accounted forjet particulation via incorporation of a shape factor λ that has a value of one for continuous jets and two for dispersed particle jets

Pack and Evans ［33，45］extended Equation （9）to allow for secondary penetration （i.e.，afterf l ow in addition to primary penetration）r as well as an empirical correction for target strength YT

Here a is permitted to depend on target and jet densities，and aYT（ρ0V2）=kR ，with k an empirical factor and R the work per unit volume required for crater formation.Eichelberger［42］added to Equation （8）a statistical factor γ and the net strength difference YN=YT-Y0，with Y0the jet strength

In the late 1960s，Alekseevski ［35］and Tate ［36］independently derived theories for long-rod penetration of metallic targets that considered deceleration of the rod due to strengths of both projectile （Y0）and target （RT）.The governing equation for equal stresses in target and projectile at the stagnation point can be derived by modifying the limits of integration in Equations （6）and （7）such that steady f l ow does not commence until the stress reaches the strength/resistance of either material

Equating axial stresses P then gives Tate’s extended Bernoulli equation

The complete theory developed in References 35 and 36 includes differential equations for projectile deceleration and erosion that must be integrated numerically to obtain depth of penetration，except in very special/simple cases.However，if deceleration is ignored，then the analytical solution for penetration depth is

where

The same result can be recovered from Equation （12）when γ→ 1 and YN→ RT-Y0.Notice that Equation （16）reduces to the hydrodynamic result of Equation （9）for very high velocities or low material strengths，i.e.，for V2?A.

In Reference 46，the description of long-rod penetration was extended to account for a transition from plastic deformation to fl uid fl ow at the head of the rod，with a transition velocity derived depending on rod strength.In Reference 47，the governing Equation （15）for the steady penetration phase was re-derived in the context of an assumed f l ow f i eld for a perfectly plastic target.Extension of the analysis to phases of unsteady impact，plastic-wave dominated，and after-f l ow was presented［48］.Numerical simulations ［49］predicted that the entrance phase of penetration provides little net effect on penetration eff i ciency，while the end phase dominates overshoot of the hydrodynamic limit for ductile metals subjected to hypervelocity impact.

More elaborate analytical models considering momentum exchange that relax assumptions of Tate’s original theory were derived in References 34，37 and 50；the latter ［50］compares analytical predictions with numerical results for metal rods penetrating metal targets，demonstrating that target resistance RTdepends on the experimental conf i guration as well as target material properties.

Walker and Anderson ［38］derived a time-dependent model for unsteady long-rod penetration of semi-inf i nite targets.This model considers initial impact （requiring an initial interface velocity from the shock jump conditions）as well as rod deceleration；assumptions are made on the plastic f l ow f i eld in the target（from a dynamic cavity expansion analysis）and velocity prof i le in the projectile （from observations in numerical simulations）.For a limiting case，the analogy of Tate’s target resistance was found to vary with the dynamic ratio β of plastic zone size to cavity size，which decreases with increasing velocity V for metallic targets

A dimensional analysis of simulation results ［51］for metals showed that larger-scale targets tend to be weaker than their small-scale counterparts due to rate and time-to-failure effects，since longer times are available for damage mechanisms （shear bands，fractures，etc.）to incubate and propagate in larger targets.This phenomenon was also observed in experiments of penetration of layered ceramic-metal systems ［52］.Experiments and hydrocode simulations showed importance of strengths of both target and projectile over a range of impact velocities，with strength effects increasing with decreasing impact velocity ［53］.Simulations also demonstrated a relatively small effect of compressibility on penetration for very ductile metallic targets except at very high velocities ［54］.

In another analysis of long-rod impact data，a two-regime model was developed to study the transition from plastically deforming rods to eroding penetration ［55］.Recently，a threeregime model has been used to address the transition from rigid，to non-eroding but deforming，to eroding，long rods with increasing impact velocity ［56］.Both of the latter two mentioned approaches are most relevant for rod materials of higher strength than target materials （e.g.，tungsten impacting aluminum ［55］）and less so for hypervelocity impact of metal projectiles into harder ceramic targets of current interest ［6，39-41］wherein rod erosion dominates the nearly steady penetration process.

3.Dimensional analysis:general

Buckingham’s pi theorem is now applied toward dimensional analysis of mechanics of long-rod penetration.Buckingham’s pi theorem applied to any physical system can be described generically as follows ［57-59］.See also the discussion on similitude concepts in Reference 60.If an equation involving n variables is dimensionally homogeneous，it can be reduced to an equation among n-k independent dimensionless products，with k being the number of independent reference dimensions needed to characterize all of the variables.In dimensional，rather than dimensionless form，let

where y is the dependent variable and x2，...，xnare independent quantities，some of which may be independent variables and others constants in a given problem.Equation （19）can be converted to dimensionless form as

where Π1is the dimensionless analog of y and function φ depends on n-k-1 of other dimensionless products （i.e.，pi terms）constructed from the original set｛y， x2， x3，… ，xn｝. Normalization ofallvariablesentering φ should， for convenience，be completed via only the set｛x2，x3，… ，xn｝such that independent variable y appears only once，on the left side of the governing equation.

Fig.1.Reverse ballistic experiment.Long-rod penetration into encased thick ceramic target（s）；dimensions in inches （mm）.Reproduced from Reference 6 with permission from Elsevier Science.

In this paper，the ballistic penetration problem to which Buckingham’s theorem is applied is illustrated in Fig.1.The experimental set-up，as discussed for example in References 4， 6，7，39-41，consists of a rod of initial length L0impacting a thick （effectively semi-inf i nite）ceramic target at velocity V0，with the cylindrical ceramic target encased in a metallic sleeve and cover plate that provide conf i nement.The experiment is performed in reverse ballistic fashion，meaning that the ceramic target package is launched at the rod in a light gas gun.The present analysis is restricted to normal impact，i.e.，in principle null obliquity.Performance of the ceramic target is measured by residual penetration depth P0，with penetration resistance decreasing with increasing P0.

In dimensional form，penetration depth is expressed as the following function of impact velocity，geometric variables ｛g｝，and material property variables ｛m｝，letting respective subscripts 0 and T denote penetrator，ceramic target

Possible lateral and frontal con fi nement is included in the set of targetgeometric variables.In subsequentanalysis，application/ fi tting of Equation （21）is restricted to experimental data sets for which the projectile and target geometries remain fi xed or nearly so.Further letting｛g｝0→L0，Equation （21）reduces under these assumptions to

A reduced form of Equation （22）is required in the context of dimensional analysis.Presumably，this reduced equation should satisfy the following conditions: （i）it should be a dimensionless function of dimensionless quantities，（ii）it should yield the penetration depth into a perfect inviscid and incompressible f l uid as V0→ ∞，and （iii）it should satisfy observed physics of the problem，notably a decreasing primary penetration depth with decreasing V0relative to the ideal hydrodynamic solution given by Equation （9），as will be demonstrated explicitly in the context of experimental results analyzed later in §4.In order to satisfy these requirements with minimal complexity，the following material parameters are introduced:the strength Y0of the projectile，themassdensitiesoftheprojectileandtargetρ0and ρT，andaseriesof z dimensionless constantsαl，l = 0， 1， 2， … ，z that potentially depend only on the target material （i.e.，type of ceramic）. In other words，｛ m｝0→ ｛ρ0，Y0｝and ｛ m｝T→ ｛ρT，αl｝.In dimensional form，Equation （22）then becomes

Table 1Ballistic penetration experiments.

Each application of Equation （23）is further restricted to data sets for which the penetrator’s geometry is fi xed，such that L0=constant is used only to normalize P0as is conventional in analysis of ballistic data ［4，37，51］and，e.g.，in Equation （9）.The now-posited dimensionless form ［requirement （i）］of the penetration depth equation is

Recall μ is fi rst de fi ned in Equation （17）.Dimensionless variableχ=，a function only of penetrator characteristics，is de fi ned here.In the context of Buckingham’s theorem，three independent dimensions （mass，length，and velocity）enter the variables considered in Equation （23），so k=3.The total number of pi terms is n=7+z，and the number of independent dimensionless terms is thusn-k -1=3+z，consistent with the number of independent variables on the right side of dimensionless Equation （24）.Requirement （ii）implies

where φ∞is the dimensionless penetration depth in the limit of inf i nite impact velocity （χ → 0）given by Equation （9）. Requirement（iii）suggests a power series in χ of the following form

where α0=1 to satisfy Equation （25）.Note that if z=1，then α1＜ 0 to satisfy requirement（iii）.Application of Equation （26）to data in §4 demonstrates that z=1 is suff i cient to describe penetration depth over hypervelocity impact regimes for ceramic materials and target con fi gurations of present interest，as will be shown explicitly later.Notice also that making particular choicesα1= 0，α2=-aYTY0recovers the form in Equation （11）［33］in the absence of secondary penetration（r=0），for steady projectile velocity （V0=V）and a continuous projectile （λ =1）.

4.Dimensional analysis:application to experimental data

The dimensional analysis framework developed in §3 is now applied to the experimental penetration data of Subramanian and Bless ［6］and Orphal et al.［39-41］.As mentioned already in the context of Fig.1，all such reverse ballistic experiments involve long-rod hypervelocity impact and penetration into conf i ned cylindrical ceramic targets.Important characteristics of experiments reported in each reference are listed in Table 1. Projectile properties are listed in Table 2，corresponding to relatively pure polycrystalline tungsten.The strength value Y0=2.0 GPa is consistent with that used in References 39-41 and dynamic yield reported in Reference 14.With a few exceptions，most experiments consider long rods with length-todiameter ratios of L0/D0=20；impact velocities V0range from 1.5 to 5.0 km/s.Target resistance values RTentering Equation（15）as reported in the experimental references are listed in the rightmost column of Table 1.

Normalized primary penetration depths are fi t to cubic equations in References 39-41 with corresponding fi tting parameters βi，（i =0， 1， 2， 3）listed in Table 1

The same cubic equation is also used to f i t alumina data of Reference 6，with parameters βidetermined newly here by regression.Also listed in Table 1 is the value of dimensionless parameter α1of Equation （26）used to f i t the experimental data

As demonstrated in the context of Figs.2，3，4，and 5，truncation of Equation （26）at order one （i.e.，a linear f i t with respect to inverse impact velocity）is suff i cient to accurately f i tall experimental data.Specif i cally，α is f i t to each experimental data set by minimizing the error

Table 2Projectile material properties.

Table 3Representative ceramic material properties.

where （P0）modelis given by Equation （28）and （P0）cubicby Equation （27），and the domain of integration corresponds to velocity ranges listed in Table 1.Also considered later is a universal fi t of Equation （28）with a single best value of α =3.0 obtained by minimizing the sum of errors ［computed using Equation （29）］for all four data sets involving four different ceramic target materials.

Properties of ceramic materials comprising each target are given for reference in Table 3.Initial mass density ρT，elastic（Young’s）modulus E，Poisson’s ratio ν，fracture toughness KC，compressive strength σC，bending strength σB，and Vicker’s hardness HVare static material properties.The Hugoniot Elastic Limit （HEL）σHand dynamic shear strength τ of the shocked ceramic-the latter def i ned in Reference 61 from the intersection of the elastic line with the failed strength curve of the shocked material-are dynamic properties.Lattice parameters are a and c.Variations among ceramics’properties-which are intrinsically related to their microstructures-are common as evidenced by ranges reported in Reference 68，for example.All of these ceramics undergo fracture when subjected to dynamic loading of suff i cient magnitude；other known inelastic deformation mechanisms are listed in the second column from the right，where slip and twinning refer to dislocation glide and mechanical twinning ［43］，respectively.The phase change in AlN occurs at pressures around 20 GPa.Amorphization in B4C is a stress-induced change from trigonal crystal structure to a non-crystalline solid phase ［16］.

Experiments and model f i ts are compared in Fig.2 for alumina，F(xiàn)ig.3 for aluminum nitride，F(xiàn)ig.4 for silicon carbide，and Fig.5 for boron carbide.Raw data points are included in Fig.2 since these are tabulated in Reference 6；data points are not tabulated in other experimental sources ［39-41］and hence are not included in corresponding Figs.3，4，and 5.Importantly，for each material，the f i t using a universal value of α =3.0 is very nearly as accurate as that which minimizes the error（Equation 29）for each material individually.As reported in References 6，39-41，the following features also characterize most experiments analyzed here:the penetration process is steady；the relationship between target velocity U and projectile velocity V is usually linear；the relationship between penetration depth and time is usually linear；the rod is completely eroded by the end of each experiment；some secondary penetration occurs，such that the total penetration depth exceeds the primary penetration depth.Notice from Figs.2，3，4，and 5 that the hydrodynamic limit penetration depth of Equation （9）is approached with increasing V0but is never achieved for any target conf i guration over the range of impact velocities consid-ered.Setting P0/L0=0 in Equation （28）results in the following limit velocity below which penetration should not occur

Fig.2.Aluminum oxide.Comparison of experimental results-raw data of Reference 6 and cubic f i t（Equation 27）-with those of the presently proposed dimensionless model（Equation 28）.Hydrodynamic limit penetration depth is μP0/L0=1 （see Equation 9）.

Fig.3.Aluminum nitride.Comparison of experimental results-cubic f i t of Reference 39 and Equation （2）-with those of the presently proposed dimensionless model （Equation 28）.Hydrodynamic limit penetration depth is μP0/ L0=1 （see Equation 9）.

Fig.4.Silicon carbide.Comparison of experimental results-cubic f i t of Reference 40 and Equation （27）-with those of the presently proposed dimensionless model（Equation 28）.Hydrodynamic limit penetration depth is μP0/L0=1（see Equation 9）.

With α =3 and tungsten rod properties of Table 2，the predicted limit velocity is 0.97 km/s for the present targetprojectile combinations.It is cautioned，however，that the present analysis has only been verif i ed for impact velocities at or exceeding V0=1.5 km/s，so this limit is an obvious extrapolation of the model in Equation （28）.

5.Further analysis and discussion

Fig.5.Boron carbide.Comparison of experimental results-cubic f i t of Reference 41 and Equation （27）-with those of the presently proposed dimensionless model （see Equation 28）.Hydrodynamic limit penetration depth is μP0/ L0=1 （see Equation 9）.

The primary discovery of the present work is that normalized penetration data for all four ceramic target materialsalumina，aluminum nitride，silicon carbide，and boron carbidecan be described well using new dimensionless Equation （28）with a single f i tting parameter，α，taking a universal value of 3.0.With α thus independent of target material，penetration depth depends only on the （f i xed）properties of the projectile （Y0and ρ0）and the density of the ceramic target which enters the ratio μ =With this supposition in place，static and dynamic strength properties and mechanisms listed in Table 3 would seem to be of little inf l uence on penetration depth for the present test conf i guration，since otherwise α would tend to vary among materials with different physical properties，underlying microstructures，and dominant deformation mechanisms.Furthermore，α would seem to be unrelated to dynamic viscosity of the ceramic material，since penetration depth results reported for aluminum nitride ［39］and boron carbide ［41］suggest similar trends for larger-scale tests.If α were to depend on viscosity or failure kinetics，then test results would not demonstrate such scaling since larger scale tests involve longer time scales and lower average strain rates ［51］.Instead，α must depend on ceramic/target properties that do not vary strongly with material type or with average strain rate.Such properties are unresolved at present，and further speculation on the physical origin of resistance parameter α is not supported by the existing results.In future work，parametric numerical simulations wherein material properties can be varied systematically and at low cost should provide further insight.

Comparison of the present f i ndings with those of a recent dimensional analysis ［5］of a different target conf i guration and velocity regime is in order.The latter conf i guration，as explained in References 4 and 5，consists of one or more ceramic tiles backed by a semi-inf i nite metal block.Performance is measured by the depth of penetration of the projectile into the backing metal.In Reference 5，dimensional analysis determined that penetration of rather thin，metal-backed ceramic tiles could be described by two parameters that do depend on the type of ceramic material.The f i rst，needed for describing the effect of ceramic tile thickness，was found to be associated with the ratio of surface energy （i.e.，fracture toughness）to elastic modulus.The second，needed to describe the relationship between penetration depth and impact velocity，was related to the ratio of dynamic shear strength τ to target density.Impact velocities ranged from 0.6 to 3.0 km/s，and projectiles had typical L0/D0ratios on the order of 4 ［69］.In contrast，the present experiments involve thick，conf i ned ceramic targets （effectively semi-inf i nite） with somewhat longer projectiles and higher impact velocities，with penetration depth depending primarily on target material only through its mass density.Therefore，comparison of the dimensional analysis reported here and in Reference 5 suggests a transition from fracture-and dynamic strength-controlled target resistance to mass density-controlled target resistance with increasing impact velocity and ceramic target thickness or conf i nement.

It is instructive to consider the present penetration depth Equation （28）in the context of models of Eichelberger，Alekseevskii，and Tate ［35，36，42］.Def i ne RN=RT-Y0as a net target-projectileresistingstress.ThenEquation （15）canbewritten which is very similar to Equation （12）when γ=1.Using the series expansionnormalized penetration depth for steady penetration throughout the entiretime of target-projectile interaction （V=V0）is then calculated as

Then，equating like terms in Equations （28）and （32）gives，with α =- α1，

Thisresult， which relieson linearization and several intermediate mathematical reductions，can be positively compared with the experimental results for silicon carbide and boron carbide［40，41］，which show a nonlinearly increasing relation between RNand impact velocity，and contrasted with the Walker-Anderson theory in Equation （18）that predicts，for ductile metallic targets，a decreasing RTwith increasing impact velocity.The ceramic-dependent term in the last of Equation （34），ρT（1 +μ）2，varies only from 1.35 to 1.85 g/cm3for the different target materials considered herein.Therefore，consistent with experimental fi ndings，a nearly constant value of α among all four ceramics to be used in Equation （34）correlates well with the very similar ranges of target resistance RT≈ 5.0-9.0 GPa reported in experiments for all four ceramics as listed in Table 1.

For thick ceramic targets impacted at high velocities，the present analysis，as noted above，suggests that primary penetration depth P0depends on ceramic type almost exclusively on mass density ρT，speci fi cally viaConsider two target tiles of different ceramic materials of the same surface area.The ratio of masses of the two tiles is directly proportional to the ratio of areal densities ρT·h，with h the thickness.Now assume that h is suff i cient to enable semi-inf i nite conditions as addressed by the analysis herein，with each tile of the same（large）thickness.Then the ratio of penetration depths into the two different targets should scale simply and inversely with ratio of square roots of masses of the two targets.

It should be emphasized that the present analysis，including the main result given by Equation （28）with α =3 for all four ceramics considered，applies most strictly only for tests in which projectile and target geometries and projectile material are held f i xed.Furthermore，impact velocities to which the analysis holds are restricted to the range 1.5-5.0 km/s.Target materials are strong ceramics，such that penetration conditions are steady and eroding.Additional experiments and corresponding analysis are required to verify or refute the use of Equation （28）for other test conditions.

6.Conclusions

Dimensional analysis has been invoked in a new study of ballistic penetration of ceramic materials.In particular，data considered here involve primary penetration depths into thick ceramic targets of alumina （high purity），aluminum nitride，silicon carbide，and boron carbide struck by tungsten long rods at impact velocities spanning 1.5-5.0 km/s.Data sets from four independent experimental investigations from the literature have been analyzed.

Application of Buckingham’s pi theorem in conjunction with several physical assumptions has resulted in a dimensionless penetration depth depending on projectile mass density and strength，targetmassdensity，andapowerseriesofdimensionless parameters in terms of inverse impact velocity.The test data have been accurately f i t using only the f i rst（linear）term in this series，with a single value of the corresponding dimensionless f i tting parameter used successfully for all four ceramic target materials when other aspects of the test conf i guration （e.g.，penetrator and target dimensions）are held f i xed in an experimental series. Comparison with one-dimensional extensions to hydrodynamic theory has demonstrated a dependence ofTate’s target resistance on impact velocity，where a linear relation applies to f i rst order. Consideration of the current results with those of prior dimensional analysis of thin，metal-backed ceramic targets conf i rms existence of a transition from fracture-and strengthcontrolled resistance to mass density-controlled resistance with increasing average impact velocity and target thickness.

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Received 19 January 2016；revised 18 February 2016；accepted 19 February 2016 Available online 15 March 2016

Peer review under responsibility of China Ordnance Society.

*Tel.:410 278 6146.

E-mail address:john.d.clayton1.civ@mail.mil，jdclayt1@umd.edu （J.D. CLAYTON）.

http://dx.doi.org/10.1016/j.dt.2016.02.004

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