Numerical study of size effect in concrete penetration with LDPM

Jun Feng,Wei-wei Sun,Bo-ming Li

aNational Key Laboratory of Transient Physics,Nanjing University of Science and Technology,Nanjing,210094,China

bDepartment of Civil Engineering,Nanjing University of Science and Technology,Nanjing,210094,China

Keywords:Penetration of concrete Size effect Lattice discrete particle model Target resistance Abnormal nose projectile

ABSTRACT Projectile size effect is of great importance since the scaling researches are extensively applied to concrete penetration investigations.This paper numerically deals with the projectile size effect on penetration resistance via the recently developed Lattice Discrete Particles Model(LDPM)which is featured with mesoscale constitutive laws governing the interaction between adjacent particles to account for cohesive fracture,strain hardening in compression and compaction due to pore collapse.Simulations of two different penetration tests are carried to shed some light on the size effect issue.The penetration numerical model is validated by matching the projectile deceleration curve of and predicting the depth of penetration(DOP).By constant velocity penetration simulations,the target resistance is found to be dependent on the projectile size.By best fitting numerical results of constant velocity penetration,a size effect law for target resistance is proposed and validated against literature data.Moreover,the size effect is numerically obtained in the projectile with longer extended nose part meanwhile the shorter extended nose is found to improve the DOP since the projectile nose is sharpened.

1.Introduction

The response of high speed penetration of concrete target by projectiles has been a research hot spot for several decades,due to its relevance for both industry and military establishments.Usually,the concrete penetration investigations are carried out in reduced geometrical scale since the ballistic tests are dangerous and money consuming.Issues concerning penetration of massive concrete targets by rigid projectiles have been dealt with empirical formulae,analytical model as well as the numerical modelling[1-7].However,almost all of these methods fail to account for the size effect of concrete structure during penetration analyses.The studies on size effect of concrete structures mainly were conducted in the area of fracture[8-10],and the understanding of the size effect in concrete penetration is still largely lacking.

With the concept of non-dimensional formula based on the dynamic cavity expansion model,Chen and Li[4,5]developed DOP prediction model for penetration with extensive boundary conditions which has been widely used by researchers in concrete penetration area.Although the non-dimensional formula for DOP significantly contributes to the promotion of concrete penetration investigations,there might be some problem when researchers try to extend the small scale penetration mechanism to the larger(real)scale experiment scenarios.The reason of the projectile diameter scale effect on the penetration response was explained by Rosenberg and Kositski[11]that the harder aggregate may pose more resistance to the smaller diameter projectile during its movement inside the target.On the other hand,size effect theory proposed and explained by Bažant[9]indicates that the nominal structure strength decreases as the structure size increases.Forrestal et al.[12]noticed there might be size effect between the projectile diameter and the target resistance Rtsince the experimental data of 76.2 mm diameter projectile penetration doesn't match the empirical formula for the target resistance.Then Bludau et al.[13]showed that the resistance to penetration of concrete targets is dependent on both the aggregate size,through the ratio of projectile diameter and aggregate size,and the strength of the aggregate material.Similar conclusions were drawn via experiments by Dancygier et al.[14].

Although the projectile size effect was reported,few efforts have been made to address this issue.Rosenberg and Kositski[11]attempted to develop a semi-empirical model for both concrete penetration and perforation by including the projectile diameter size effect.More recently,Peng et al.[15]developed a mesoscale concrete model for penetration considering both aggregate,cement and ITZ.Their numerical results suggest the penetration resistance is strongly dependent on the projectile relative size against the aggregate.With reference to the cavity expansion analysis,a modified penetration resistive force formulae was proposed with aggregate strength and projectile size taken into account.

In the present paper,the numerical simulation of rigid projectile penetration is performed with Lattice Discrete Particle Model(LDPM)which has been proven a robust model for concrete penetration and perforation in our previous study[16-18].The numerical study of LDPM[19]shows a good capacity of concrete size effect modelling of 3-point bending,splitting and even compression tests.Two penetration tests concerning projectile size effect are selected for LDPM numerical study.The penetration model is validated by predicting the projectile deceleration history as well as DOP where data scattering is discussed.The extensive simulations of constant velocity penetration in concrete give the target resistance of different projectile geometries.Then,the size effect law for target resistance is proposed and validated.Finally,the size effect of abnormal nose projectile penetration is numerically captured and discussed.

2.Lattice Discrete Particle Model（LDPM)introduction

As a synthesis of the Confinement Shear Lattice model and discrete method,the theoretical framework of Lattice Discrete Particle Model was developed to simulate the mechanical interaction of coarse aggregate pieces embedded in a binding matrix[20-22].The geometrical representation of concrete mesostructure is constructed by randomly introducing and distributing spherical shaped coarse aggregate particles inside the volume of interest and zero-radius aggregate particles on its surface as shown in Fig.1(a).With Delaunay tetrahedralization of the generated particle centers,a three-dimensional domain tessellation creates a system of polyhedral cells(see Fig.1(b))interacting through triangular facets and corresponding lattice system.To construct the mesoscale framework,concrete mix design parameters are needed including:maximum/minimum aggregate particle size da/d0;cement content c water-to-cement ratio wt/c;and the coefficient for the classical Fuller curve nF.

2.1.LDPM kinematics

The triangular facets forming the rigid polyhedral particles are assumed to be the potential material failure locations.Each facet is shared between two polyhedral particles and is characterized by a local system of reference featured by a unit normal vector n and two tangential vectors m and l as noted in Fig.1(b).The deformation of the lattice particle system is described by the rigid body kinematics whereas the displacement jump 〚uC〛at each triangular facet centroid is used to define the mesoscale strain component as:

where l is the length of the tetrahedron edge.The strain definitions in Eq.(1)have been proven by Cusatis et al.[23]to be consistent with the definition of strain in classical continuum mechanics.The corresponding normal and shear stress are calculated through LDPM mesoscale constitutive laws and the equilibrium are imposed via the principle of virtual work[20].

2.2.LDPM constitutive equations

In the elastic regime,the normal and shear stresses are proportional to the corresponding strains:σN=EN·εN;σM=ET·εM;σL=ET·εL,where EN=E0,ET= αE0,E0is the effective normal modulus,andαis shear-normal coupling parameter.On the facets,the reversible elastic behavior is limited by a number of nonlinear boundaries which are featured by softening for pure tension and shear-tension,as well as plastic hardening for pure and shear compression.

2.2.1.Fracturing behavior

For fracture behavior characterized by normal strains （εN＞0）,the fracturing evolution is formulated through the effective strainand the effective stress σ=The tensile boundaryσbtevolves exponentially as a function of the maximum effective strain of the loading history εmax=maxτ≤t[ε（τ）]

where Macaulay bracket 〈x〉=max｛x，0｝,,andReaching elastic limit ε0(ω),the fracturing damage decays the boundaryσbcwith the post peak softening modulus defined as H0（ω）=Ht（2ω/π）nt,where Htis the softening modulus in pure tension(ω=π/2)expressed as Ht=2E0/（lt/l-1）;character length lt=2E0Gt/σ2;and Gtis the mesoscale fracture energy.LDPM provides a smooth transition between pure tension and pure shear(ω=0)with parabolic variation for strength given by

where rst= σs/σtis the ratio of shear strength to tensile strength.

2.2.2.Pore collapse and subsequent compaction

Normal stresses for compressive loading(εN＜0)are computed meeting the inequality-σbc（εD，εV）≤ σN≤ 0 where σbcis the boundary function of the volumetric strain εVand the deviatoric strain εD.Beyond the mesoscale compressive yield stress σc0,-σbcmodels pore collapse as a linear evolution of stress for increasing volumetric strain with stiffness Hcfor- εV≤ εc1= κc0εc0:

where Hc（rDV）=Hc0/（1+ κc2〈rDV- κc1〉）and κc1,κc2are deviatoric parameters.Beyond pore collapse- εV≥ εc1,compaction and rehardening occur.In this case one has

where σc1rDV= σc0+ （εc1- εc0）HcrDV.

2.2.3.Friction due to compression-shear

The incremental shear stresses rates are computed asandwhere=,andλis the plastic multiplier.The plastic potential is defined aswhere the nonlinear frictional law for the shear strength is assumed to be

whereσN0is the transitional normal stress,μ0and μ∞are the initial and final internal friction coefficients.

2.2.4.Strain rate effect in LDPM

The LDPM formulation is extended to incorporate rate dependent fracture mechanisms associated with the interpretation of thermally activated phenomena which is governed by the classical Maxwell-Boltzmann equation[24].The crack opening rate dependent cohesive behavior can be expressed as:σch（w，）=[1+c1a sin h（/c0）]f（w）whereσchis the cohesive stress,w is the crack opening,˙w is the crack opening rate,f（w）is the cohesive law under static condition,c0is the reference crack opening rate and c1is the strain rate coefficient.

Compared to the crack opening rate,the elastic strain rate can be negligible[25],hence the LDPM effective strain can be written as.Substituting˙w by l˙ε,LDPM boundary condition can be modified as:

3.Numerical analysis of Sandia Lab penetration tests[27]

A serious of experimental investigations on penetration in concrete targets with different projectile nose geometry,striking velocities were carried out by M.J.Forrestal et al.[12,27,28]to study the response of projectile penetration in concrete target.The 76.2-mm-diameter ogival nose projectiles were designed with a singlechannel acceleration data recorder.Both penetration depth and projectile deceleration data during penetration were measured and provided in details.With 3.0 and 6.0 caliber-radius-head(CRH),ogival nose projectiles were launched with striking velocities between 140 and 460m/s to impact the concrete targets with 23 MPa compressive strength.These experiments,including the triaxial compression and penetration tests,have been numerically studied in our previous work[26]whereby the calibration and validation of the penetration model are conducted.In this section,we further analyze the size effect phenomenon with respect to the penetration test in Ref.[27].

3.1.LDPM simulation of concrete penetration

3.1.1.LDPM parameter calibration

The concrete mixture proportion by weight is characterized by cement content c=310 kg/m3,water to cement ratio wt/c=0.84,aggregate to cement ratio a/c=5.2,fuller coefficient nF=0.5 which are input information for LDPM particle generation.The nominal compressive strength was obtained from uniaxial compression test with 50.8mm(diameter),114mm(height)cylinder specimens which were also used for triaxial compression(TXC)tests.Calibration of LDPM parameters are conducted with respect to hydrostatic test,triaxial compression tests with different confinement pressure and uniaxial strain compression test.The hydrostatic simulation is conducted by applying an increasing hydrostatic pressure phyon the cylinder surface.For triaxial compression simulations,the top surface is driven by a rigid top block with velocity control corresponding to the hydrostatic pressure,and the bottom surface is assigned frictional contact with a fixed block.After the lateral surface reaches the confinement pressure,the to p block keeps moving with a constant velocity while the confinement is kept constant.The uniaxial strain test is modelled with the top surface moving at a low constant velocity and the lateral surface radial boundary fixed.The calibrated results are shown in Fig.2,which compares the numerical and experimental curves for hydrostatic compression and TXC tests.In Fig.2(b),the damaged concrete cylinders under 50 and 200MPa confinement are comparatively plotted where comminution occurs for the triaxial response.The detailed response of TXC simulation is depicted in Fig.2(c),where the red circles represent the loading condition transforming from hydrostatic compression to triaxial compression.With reference to Ref.[26],the LDPM parameters are calibrated as follows:normal elastic modulus E0=16500 MPa,densified normal modulus Ed= 4.2E0,shear-normal coupling parameterα=0.25,tensile strength σt=2.7MPa,yielding compressive stressσc0=42 MPa,shear strengthσs=2.0σt,tensile characteristic length lt=100mm,softening exponent nt=0.3,initial hardening modulus Hc0=0.46E0,transitional strain ratio κc0= 6,initial internal friction coefficient μ0=0.4,internal asymptotic friction coefficient μ∞=0,transitional stressσN0=400MPa,deviatoric-to-volumetric strain ratio κc1=2,and κc2=2.

Subsequent to calibration,the obtained LDPM parameters are used to simulate the response of concrete target subjected to ogival nose projectile impact.According to the penetration experiments[12],the projectiles were launched to normal impact the concrete targets and the ballistic tunnels measured after tests were almost straight within 0.6opitch and yaw.Furthermore,abrasion on the projectile surface was also reported negligible,thus rigid projectile assumption can be made in this numerical simulation.The calibrated LDPM parameters are utilized for concrete target modelling and a penalty interaction[29]between rigid projectile and LDPM particles is assigned.

3.1.2.Penetration simulation validation

The penetration prediction results are listed in Table 1,which indicates that the depth of penetration(DOP)is within 10%error with respect to the tests data reported[27].Moreover,the projectile deceleration during penetration is another important ballistic property needed to be validated.Fig.3 illustrates the numerical predicted deceleration curves during the impact process in blue lines and the measured data in black lines where G is the gravitational acceleration.Firstly,the deceleration is characterized with a quick increase corresponding to the projectile nose length.After the nose entirely enters the target,the deceleration value keeps almost constant with a scatter.The large scattering of the numerical deceleration data may due to the updating interaction between LDPM aggregate particles with projectile.As explained in 2D diagram(see Fig.4),the blue circles represent concrete aggregates in contact with the projectile surface in Fig.4(a).As the projectile moves forward,the interacted aggregates get updated according to the relative distance between aggregates and projectile.As shown in Fig.4(b),the aggregate in purple starts to contact with projectile while the aggregate in green becomes too far away from the projectile and no more interaction exists in between.The constant contact pair updating leads to the rough resistive force which is different from FEM results in Ref.[30].As the projectile velocity reaches zero,the plateau is followed by a sudden drop.This phenomenon indicates that the resistance is not only determined by the impact velocity,rather there should be a material dependent constant term which was called target resistance by Forrestal et al.[12,31].This ballistic property is fully captured by LDPM numerical prediction and both DOP and deceleration data are in agreement with the test data.

3.2.Size effect of concrete penetration

To investigate the projectile size effect on penetration resistance,projectile constant velocity penetration of LDPM simulation is further conducted in this work to obtain the resistive force acting on the projectile nose.Right now,researchers deem that the target resistance Rtdepends on the ratio dp/da[11,15]where dais the maximum aggregate size identical to the notation in LDPM introduction.Combining the cavity expansion with projectile penetration in concrete,the ballistic tunnel expansion is driven by the cavity wall with radius R=dpΨ,namely ogival nose radius.There might be a possibility that the size effect in concrete penetration is related to the ratio R/da.Moreover,penetration tests revealed the 6.0 CRH projectile penetration is characterized with lower Rtin Ref.[12],as shown in Table 2 where Rtis the averaged value of Rt.Thus,this section tries toexplorethe size factors which might affect the target resistance.

Projectiles with 30 mm,60 mm and 80 mm diameters are chosen herein for numerical study where common CRH values like 3.0,4.0 and 6.0 are determined.Constant impact velocity vcranging from 200 to 600m/s are assigned to the hard projectiles to penetrate the thick concrete target,and their resistive forces are then obtained through LDPM simulations where Fig.5 gives the projectile resistive forces under 400m/s projectile impact.For each penetration simulation,the resistive force corresponding to vc=400m/s(dash lines in Fig.5)is calculated as the averaged values when the projectile nose part fully enters the target.

According to the widely recognized cavity expansion analysis developed by Forrestal et al.[31,32],the resistive force on the projectile nose should be:

where dpis the projectile diameter,V is the projectile impactvelocity and N is the “nose factor”reflecting the geometrical characteristics of the projectile nose,forogival nose projectileΨis caliber-radius-head value;the concrete target is described by initial densityρ0and target resistance Rt.This postulation assumes that the resistance is only attributed to the compressive pressure acting on the normal direction of projectile nose surface.The target resistance Rtnumerically obtained can be estimated via Eq.(8)which are plotted in Fig.6(a).It is observed that the target resistance somehow increases with the impact velocity which might be caused by the rate effect term of the penetration resistance,as indicated by some cavity expansion models with three resistance terms,i.e.,inertial term,rate effect term and static term(namely Rt)[6,33].Since this work concentrates on the size effect rather than the resistance mechanisms,we choose to select the Rtof vc=400 m/s for further study.Also,it is interesting to find that the larger diameter projectiles tend to suffer less target resistance.But the Rtof 60 mm dpwith 6.0 CRH seems to be close to the Rtof 80 mm dpwith 3.0 CRH.These two cases have same ogival nose radius R,thus we prefer to believe the size effect of Rtdepends on R=dpΨ.The Rtbest fit is then performed with respect to R/daas shown in Fig.6(b).With reference to Ref.[34],the size effect law for target resistance is expressed as Rt=Sfcwith S proposed as:

Table 1 Comparison of Sandia Lab penetration tests and LDPM predictions.

Table 2 Target resistance of Forrestal et al.[12].

where there is no aggregate material strength parameter involved,because the aggregate material properties is related to the concrete mechanical property which is already accounted for in the parameter fc.This controversial issue needs to be further studied.

The validation is then conducted with reference to the penetration tests with fc=23 MPa and 39 MPa in Refs.[12,27].As mentioned by Forrestal et al.[12],the empirical equation S=82.6f-0.544ccould not be applied to penetration tests with 76.2 mm projectile diameter.By extensive mesoscale numerical simulation,Peng et al.[15]gave the size dependent target resistance function withRosenberg and Kositski[11]summarized penetration data and empirically come up with the expressionThese two models both gained success in Rtsize effect prediction.For comparison,these two models are utilized herein to predict the target resistance of 76.2mm diameter projectile penetration in 23 MPa and 39 MPa concrete.In Fig.7(a),the proposed model is able to predict the target resistance for penetration in 23 MPa strength concrete,but the size effect is somehow exaggerated.In Fig.7(b),the normalized target resistanceis defined asand the size effect due to CRH change is only captured by the proposed model due to the richer factors accounted for.And this phenomenon needs to be further validated through experimental tests.

4.LDPM simulation of abnormal nose projectile penetration[35]

A series of penetration test of 14.8MPa compressive strength concrete targets impacted by abnormal nose projectiles was carried out by Chai et al.[35].The projectiles used for penetration were made with a special nose shape,as shown in Fig.8 the first one is ogival nose with CRH=2 while the others are featured with a cylinder at in the middle of the ogival nose part.The ogival nose projectile(ONP)is of 40 mm diameter and the abnormal nose projectile#1(ANP1)has a cylinder of 10mm length in the nose part while the nose shape of ANP2 is characterized with a 30 mm length cylinder.The projectiles were designed to penetrate the concrete target with 440 and 610 m/s striking velocities.With the size effect modelling ability,the LDPM is used to capture the size effect which might occur in this penetration condition.

No apparent projectile deformation or erosion was reported by the authors and rigid projectile assumption is also made herein.With reference to the calibrated LDPM parameters for different strength concrete[19],parameters for fc=14.8 MPa are estimated as:normal elastic modulus E0=21000MPa,densified normal modulus Ed=1.0E0,shear-normal coupling parameterα=0.25,tensile strength σt=2.0MPa,yielding compressive stress σc0=30 MPa,shear strength σs=2.0σt,tensile characteristic length lt=160 mm,softening exponent nt=0.2,initial hardening modulus Hc0=0.54E0,transitional strain ratioκc0=4,initial internal friction coefficient μ0=0.4,internal asymptotic friction coefficient μ∞=0,transitional stressσN0=600MPa,deviatoric-to volumetric strain ratio κc1=1 and κc2=5.

After LDMP simulation,the predicted DOP are comparatively listed with the test results in Table 3 where the shot numbers are labelled with projectile name plus “1”and “2”representing the striking velocity of about 440 and 610m/s.It is interesting to find that under similar conditions the ANP1 tends to improve the DOP meanwhile the ANP2 doesn't contribute to the DOP even though they have similar nose shape.In general,the LDPM simulation derived DOP agrees with the experimental data in terms of DOP.

The numerical results of damage contour of the impact surface of shot ONP-2 is plotted in Fig.9 whereby the left half surface is covered by the experimental picture.The front crater shape as well as size are almost the same as the test but more radial cracks are obtained by LDPM simulation.Also the cross section views of the projectile penetration process of shot ONP-2 are plotted in Figs.10 and 11 at every 0.5 ms.The crack opening distribution of the cross section plane of the trajectory is shown in Fig.10 where the red cracks can be considered as the pulverized concrete material.In Fig.11,the velocity distribution of the cracks in the cross section plane shows the history of the cracks velocity during penetration whereby the cracks near the projectile nose tip always have the highest velocity until the projectile stops in the target in Fig.11(d).

Table 3 Comparison of ANP penetration tests and LDPM predictions.

Similarly,the projectile constant velocity penetration in concrete is simulated to achieve the penetration resistance.For ONP,the cavity expansion analysis can be applied to get the target resistance Rt.The best fitted curve against the LDPM simulation results are plotted in Fig.12 where Rt=253.5 MPa is estimated.Since the aggregate information is missing in Ref.[35],we take da=9.5mm and Eq.(9)suggests 267MPa for target resistance which is quiet close to the best fit value.Therefore,the Forrestal model prediction for DOP of shot ONP-1 and ONP-2 are 0.388 and 0.682m,matching the test data well.

More importantly,the constant velocity(vc=200 and 600 m/s)penetration simulation results are comparatively given in Fig.13 where the mean resistive force after the projectile nose part fully enters the target is denoted in bold dash line.It is worth noting that under same impact velocity and target condition,the mean resistive force of ANP1 seems to be smaller comparing to ONP.This can be explained that the cylinder part of ANP1 in the nose part actually extends the length of the projectile nose height and in general leading to a sharper nose shape.However,the ANP2 suffer more resistance than ONP which may attribute to the long cylinder in the nose part resulting in greater resistance stress acting on the front nose part before the cylinder.Since the front nose+cylinder part is significantly important in dimension,the size effect of the penetration resistance will pose greater resistance stress on the front nose.This numerical result agrees with the conclusion drawn in Ref.[35].And this interesting projectile nose shape is worthy to be further studied.

5.Conclusions

The numerical simulation of rigid projectile penetration is performed with Lattice Discrete Particle Model.Extensive numerical simulations of penetration are conducted to explore the projectile size effect on target resistance.The main conclusions are drawn as follows:

(1)The LDPM model with calibrated parameters can successfully simulate the projectile penetration in 23 MPa compressive strength concrete.The size effect phenomenon,i.e.,larger projectile suffers less resistance,can be captured through the LDPM simulation.

(2)The size effect law for target resistance Rtdepending on the ratio of ogival nose radius and maximum aggregate size is proposed which is then validated against test data.This view point might be different from existing models which believe both projectile diameter and aggregate strength should count.Thus,large amount of scaling penetration tests are needed to better understand the size effect in penetration.

(3)For the abnormal nose projectile penetration in concrete,LDPM simulation prediction agrees well with the experimental data.Also,the size effect can be numerically obtained for the projectile with longer extended nose part meanwhile the shorter extended nose can improve the DOP since the projectile nose is sharpened.

Acknowledgement

This effort was supported by the Natural Science Foundation of Jiangsu Province(No.BK20170824)and the Fundamental Research Funds for the Central Universities(No.30917011343).Prof.Gianluca Cusatis from Northwestern University is gratefully acknowledged for the timely helps in LDPM study.