mon

Hig

Numerical simulationConcreteFragment

rtanon oedale m

terial hot onlyy andice life

full-scale tests is often beyond affordability. The majority of blastmodels today are mainly based on empirical or semi-empiricalformulae. They tend to overlook the physical behavior of concretein the dynamic process in blasting. With the rapid development of

twin shear strength model. The dynamic properties of concretematerial are different from its static properties. From dynamicexperimental tests, it has been found that both the tensile strengthand the compressive strength of concrete are highly dependent onthe strain rate, i.e., the strain rate effect, which is usually modelledby a dynamic increase factor (DIF) to relate the dynamic strength tothe corresponding static strength. The strain rate effect for tensileand compressive strength is also different [711]. Usually the DIF isobtained from experimental tests. The dynamic strength criterion is

* Corresponding author. Tel.: 61 8 6488 1825; fax: 61 8 6488 1044.

Contents lists availab

International Journal o

lse

International Journal of Impact Engineering 36 (2009) 13151326E-mail address: hao@civil.uwa.edu.au (H. Hao).explosive loads. Industrial accidents can be a cause of such events.Some critical infrastructures, such as embassy buildings, bridgesand government buildings, might be targets of terrorist attack.Structural damage and personnel injury may be caused by directblast loading and the ying secondary fragments from the failedstructural and nonstructural components. It is of interest forresearchers to understand the damage process in order to predictfragments of concrete material under blast loads. Predicting theperformance of concrete structures to explosive loading through

1.1. Concrete material model under blast loading

One of the challenges for reliable numerical simulations is todevelop a proper material model for concrete. To model concretedamage and fragmentation, its strength criterion is the mostimportant. Based on static tests, many static strength criteria havebeen proposed in the past [36], such as MohrCoulomb Criterion,DruckerPrager Criterion, WilliamWarnke ve parameter Crite-rion, Ottosen Criterion, HsiehTingChen Criterion and the unied1. Introduction

Concrete as a main structural maover the world. It can be used in ndential buildings, but also in industrconcrete structures during their serv0734-743X/$ see front matter 2009 Elsevier Ltd.doi:10.1016/j.ijimpeng.2009.02.010damage material model is incorporated into the hydrocode AUTODYN. The dynamic damage process ofthe concrete slab under contact detonation is numerically simulated. Based on the numerical results, thefragment size distribution is estimated by an image analysis program. Two different random aggregatedistributions are assumed in the present simulations. Numerical results from the two different cases arecompared, and the results from the mesoscale model are compared with that from the homogeneousconcrete material model. The fragment size distributions obtained from numerical simulations are alsocompared with those from the empirical statistic formulae.

2009 Elsevier Ltd. All rights reserved.

as been widely used allcommercial and resi-

military facilities. Somemight be subjected to

computer technology and the advancement of numerical tech-niques, it makes the predictions through computer simulationviable [1,2]. Different numerical methods have already beenreported in the literature to model the damage and fragmentationof concrete materials under blast loading.Blast loadingMesoscale modelstructure components. Each coarse aggregate is assumed to be circular with a random radius in a givendistribution range following the Fullers curve. The mesoscale model together with a dynamic plasticKeywords:the high strength coarse aggregates and the low strength mortar matrix, randomly distributed in theMesoscale modelling and analysis of daof concrete slab under contact detonati

X.Q. Zhou, H. Hao*

School of Civil & Resource Engineering, The University of Western Australia, 35 Stirling

a r t i c l e i n f o

Article history:Received 12 February 2008Accepted 12 February 2009Available online 27 May 2009

a b s t r a c t

It is interesting and impopredict fragment distributiconcrete model is developdetonation. In the mesosc

journal homepage: www.eAll rights reserved.t for researchers to understand the damage process in order to reliablyf concrete material under blast loading. In the present study, a mesoscaleto simulate the dynamic failure process of a concrete slab under contactodel, the concrete material is assumed to consist of two phases, that is,age and fragmentation

hway, Crawley WA 6009, Australia

le at ScienceDirect

f Impact Engineering

vier .com/locate/ i j impeng

To analyze the static heterogeneous behavior of concrete, somemesoscale models for concrete have been developed [2226]. In

of Immost of these mesoscale models, the concrete is assumed con-sisting of three phases, that is, the coarse aggregates, the mortarmatrix with ne aggregate dissolved in it, and the interfacialtransition zones (ITZ) between the aggregate and the mortarmatrix. Based on the static experimental results, the behavior ofITZ does affect the mechanical properties of concrete. However, itis very difcult to obtain the mechanical parameters of ITZ.Therefore including ITZ in the model introduces some uncer-tainties. Moreover, considering ITZ in numerical model substan-tially increases the computational time and computer memoryrequirement. For these reasons, in some models the ITZs are notincluded in the numerical simulation [22,25], instead, the idealbond between the aggregates and the mortar matrix are assumed.To perform the mesoscopic study of concrete material, bothdiscrete element methods, such as lattice model [26] and trussmodel [27], and continuum nite element methods [2224] havebeen used. So far, this kind of mesoscale models has mainly beenapplied in static numerical simulations.

The present paper aims to construct a mesoscale heterogeneousmodel for concrete material under blast loading. In the mesoscalemodel, the concrete material is assumed consisting of two phases,that is, the high strength coarse aggregate and the low strengthcement paste, randomly distributed in the structure components.Perfect bond between aggregates and cement paste is assumed. Asa numerical example, the dynamic damage process of a concreteslab under blast loads studied by other researchers using a differentalso simply obtained by multiplying the static compressive, ortensile strength, by the respective DIF in practice.

Some material models have been constructed to simulate theconcrete behavior under dynamic loading conditions [1220]. In1993, Johnson and Holmquist [12] developed a brittle damagemodel for concrete [12]. Based on this model, RHT model [14],Gebbekens model [15], K&C model [16] and modied DruckerPrgager model [12,17] were developed. Recently, Leppanen [19]modied the RHT model by using a different DIF for tension anda bi-linear crack softening law. All the above-mentioned modelsbelong to the category of plastic damage model. In addition, somevisco-plastic models were also developed for concrete, for example,Gatuingt and Pijaudier-Cabot [21] developed a damage visco-plastic model for concrete. They considered the interactionbetween the spherical and deviatoric response. The constitutiverelation for concrete is based on visco-plasticity combined with therate-dependant continuum damage. The difference between theplastic damage model and the visco-plastic damage model is thatthe time history effect is considered in the later model. Theoreti-cally the visco-plastic damage model is more reasonable because itconsiders the time dependant plastic ow. However, the visco-plastic behavior of concrete is very complicated and it is not wellunderstood yet.

In those models, the concrete material is always assumed to beisotropic, continuous and homogeneous.

1.2. Mesoscale modelling of concrete

As is well known, concrete is a composite material, produced byadding the appropriate portions of coarse and ne aggregates,cement, water and some additives if necessary. Obviously theconcrete material is heterogeneous, and its heterogeneity makesthe behavior of concrete under blast loads rather complicated.Especially, the heterogeneity of concrete affects the crack patternand the fragment size distribution when it is under blast loads.

X.Q. Zhou, H. Hao / International Journal1316approach [28] is analysed.1.3. Concrete dynamic fragmentation

When an explosion occurs, secondary fragments and airbornedebris resulting from the damaged structural components maycause serious injury and damage. Therefore, it is of interest to knowthe fragmentation process of structural components and to predictthe size and velocity distribution of the fragments.

The processes of dynamic fragmentation within a concretemember are very complicated since discontinuities such as cleavagecracks and defects with different shapes and orientations arecommonly encountered in concrete material and they have signi-cant inuence on the failure of concrete. The actual process ofdynamic fragmentation is still not well understood, but some theo-retical and experimental efforts have provided useful insightregarding the distribution of fragments. Based on energy andmomentum balance principles, some models have been developedto predict average fragment size as a function of strain rate andmaterial toughness [2931]. To determine the distribution of frag-ment in mass or size, some statistical approaches have been devel-oped [32,33]. In those approaches, the intrinsic failure processleading to fragmentation is not modelled. To understand themechanisms of fragmentation, some theoretical models have beensuggested to correlate the dynamic fracture and fragmentation[34,35]. Recently, numerical modelling has been carried out tosimulate the dynamic deformation in the fragmentation process[3538]. There aremainly three differentmethods for fragmentationsimulation: 1) Interface elements were incorporated between stan-dard nite elements to serve as dynamic fracture paths [35]. Theprimary drawback of this method is that it cannot give reliablepredictions of the fragment size because the size and shape aredetermined by the pre-dened interface; 2) Damagematerial modelhas been put into smooth particle hydrodynamics (SPH method) tosimulate the fragmentation process [28,38]. The fragment distribu-tion was obtained by checking the radius of the fully damagedparticles. 3) Standard nite element method together with damagemechanics has been employed to model the dynamic deformationand to predict the fragment size [36,37]. The fragment size is pre-dicted by either the energy balance principal or relating the fulldamage to the fragmentation. All the previous models assumedhomogenous material properties. In the present study, the latermethod is adopted, however, the heterogeneous concrete materialproperties are considered. The mesoscale concrete model andAUTODYN [2] are rstly employed to model the dynamic deforma-tion of a concrete slab under a contact detonation. The fragment sizedistributions and ejection velocities are then predicted by relatingthe full damage to the fragmentation. Next, an image analysisprogram in MATLAB is used to estimate the fragment size distribu-tion by analyzing the numerical results based on the damagemechanics theory and nite element model.

2. Generation of coarse aggregate particles

The mesoscale concrete model requires the generation ofa random aggregate structure in which the shape, size and distribu-tion of the coarse aggregates closely resemble the real concrete in thestatistical sense [22]. The coarse aggregates generation method inthe present study is the popular take-and-placemethod. Firstly, thesize distribution of the coarse aggregate particles is determined byfollowing a certain given grading curve; and then the aggregateparticles are placed into the mortar matrix one by one at randomlydetermined locations in suchaway that nooverlappingwithparticlesalready placed. The similar method was also employed in [22,23,27].

Coarse aggregates are the particles whose diameters are greaterthan 4.75 mm. For most concrete, the coarse aggregates represent

pact Engineering 36 (2009) 131513264050% of the concrete volume [22]. In the present study, it is

assumed that the total percentage of the coarse aggregates is 40% ofthe concrete. The most common coarse aggregate types are gravelsand crushed stones. The shape of aggregate depends on theaggregate type. Generally, gravel aggregates have a rounded shapewhile crushed stone aggregates have an angular shape [23]. Severalmethods to characterize the shape of the coarse aggregates havebeen published [23,25]. For simplicity, in this study the shape of thecoarse aggregates is assumed to be circular in two-dimensional(2D) simulations. The slab is modelled as axisymmetric. It should bementioned that the dynamic failure processes in concrete materialare affected by three-dimensional (3D) distribution of the coarseaggregates. However, due to the computer memory limitation, inthis study only the axisymmetric model is used to do the mesoscalesimulation of the concrete slab.

One of the most popular aggregate distribution models is theFullers curve (Fig. 1) [22,25]. It is believed that the Fullers curverepresents a grading of aggregate particles resulting in an optimumdensity and strength of the concrete mixture. Fullers curve can be

Fig. 2. An example of a generated 2D aggregate distribution for an area of300 300 mm2.

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 13151326 1317described by a simple equation [22]

Pd d=dmaxn (1)where P(d) is the cumulative percentage passing a sieve withaperture diameter d, dmax is the maximum size of aggregateparticle and n is the exponent of the equation (n 0.450.70). Inpractical concrete construction, the typical maximum size ofaggregate is about 32 mm to obtain high quality concrete mix.Therefore, in the present study, dmax is assumed to be 32 mm, andn is taken as 0.5.

From Fig. 1, it can be found that the percentage for the coarseaggregates, i.e., aggregates with size larger than 4.75 mm, is about61.5% of the total aggregates. The total volume of the coarseaggregates is assumed to be 40% of the whole concrete volume.The grading curve is divided into several segments. In the presentstudy, three aggregate diameter ranges are selected, i.e., 4.7510 mm, 1020 mm and 2032 mm. According to Fullers curve, thepercentages for the three aggregate ranges are 17.37%, 23.17%, and20.96%, respectively, which correspond to 11.3%, 15.07% and 13.63%of the total concrete volume. The aggregate distribution processstarts with the grading segment containing the largest sizeparticle, that is, 2032 mm. Firstly, random position for the centreof the circular aggregate is determined; next the diameter withinthe grading segment (2032 mm) is randomly decided; thenoverlapping between any two aggregates and every aggregatewiththe slab boundary are checked. If overlapping between twoaggregates or part of an aggregate is outside of the slab boundary,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.125 0.25 0.5 1 2 4 8 16 32

Fuller's curve

Size (mm)

To

tal p

ercen

tag

e Fig. 1. Fullers grading curve.the next simulated aggregate size and position is used. Once thetotal area of the coarse aggregates in the size segment reaches thedesigned percentage, the iteration is stopped and the next gradingsegment is placed. An example of a generated 2D aggregateparticle distribution is shown in Fig. 2. Similar method can also beused to construct a 3Dmesoscale model. A 3D example is shown inFig. 3.Fig. 3. An example of a generated 3D aggregate distribution for a volume of200 200 200 mm3.

In this model, the material is assumed to be compacted from its

sive strength fc s1 s2 0; s3 fc; (4) conned compressivep p

Table 1Material parameters for mortar matrix.

Initial densityr0M (kg/m3) 2.33 103

Solid density rsM (kg/m3) 2.450 103

Initial sound speed C0M (m/s) 2.20 103Solid sound speed Cs (m/s) 2.80 103Initial compaction pressure peM (MPa) 24.0Solid compaction pressure psM(MPa) 250Shear modulus (GPa) 8.3Damage parameters at, ac 0.5Tensile damage threshold 3st0M 3.0 104Compressive damage threshold 3sc0M 3.0 103Tensile strength ftM (MPa) 4

Solid sound speed cs

1

23

4

5

Pressure P

Pmin0 s Density

Initial sound speed Cinit

Plastic compaction path

Elastic unloading/loading paths

Fig. 4. Pressure and density curve for piecewise linear porous equation of state.

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 131513261318state of initial density r0 along an elastic path dened by thedifferential equation

dpdr

c2init (2)

until the pressure reaches the plastic yield stress, which is denedby the value of the pressure in the rst (r; p) pairs. Cinit is the initialsound speed. Subsequent loading takes place along the plasticcompaction path until the material is fully compacted, at whichpoint further compression takes place according to the linear3. Material model for concrete

AUTODYN [2] is a hydrocode. In a hydrocode, thematerial modelcontains two parts: the equation of state (EOS) which is used todetermine the hydro pressure in terms of the local density and thelocal energy, and the strength criterion which controls the yieldstrength according to the rst stress invariant I1 and the secondinvariant of the stress deviator tensor J2. Different material modelsare adopted here for the aggregate and the mortar matrix becausethey have different mechanical behaviors.

3.1. Material model for mortar matrix

Typically the mortar matrix has a porosity of 10% and it hasa complex non-linear compression behavior, thus it can be consid-ered as a porous material. The equation of state used to model themortar matrix is a piecewise-linear porous model, in which theplastic compaction path is dened as a piece-wise linear path fromwhich unloading and reloading can occur along an elastic line asshown generically in Fig. 4. Similar material model was constructedin our previous work to model the concrete behavior [12].relationship

p

J2

0

Undamaged material D=0

Damaged material 0

High explosive

1200

103

320

75

Fig. 8. Test setup (mm).

Tensile DIF

0

1

2

3

4

5

0.000001 0.0001 0.01 1 100 10000

RouabhiChoWangCaiLokPresent model

Fig. 6. Tensile DIF of rock.

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 13151326 1319permanent plastic strain. The details on the plastic ow treatmentin AUTODYN can be found in [2]. The damage scalarD is determinedby Mazars damage model [40].

In the damage model, the two scalars, namely, Dt and Dc, whichcorrespond, respectively, to the damage in tension and compres-sion of concrete, are dened as

Dt~3t 1 eat~3t3t03t0 Dc~3c 1 eac~3c3c0=3c0 (6)

where at and ac are the damage parameters that depend on thematerial properties, the range for them are from 0 to 1. For moredetails of the determination of the parameters, please refer to ref[41]. In this study they are taken as 0.5, while 3to and 3co are thethreshold strains in the uniaxial tensile and compressive states. ~3tand ~3c are the equivalent tensile and compressive strains, dened as

~3t Xi1;3

3i2s

~3c Xi1;3

3i2s

(7)

where 3i is the positive principal strain. The means it vanishes ifit is negative. 3i is the negative principal strain, and the means itvanishes if it is positive.

The cumulative damage scalar can be determined by combiningthe tensile and compressive damageCompressive DIF

0

1

2

3

4

5

0.0001 0.01 1 100 10000

RouabhiLICaiLI XBLokPresent model

Fig. 7. Compressive DIF of rock.D AtDt AcDc; _Dt > 0; _Dc > 0 and At Ac 1 (8)the weights At and Ac are dened by the following expressions [40],

At Xi1;3

Hi3i3i 3i

~32

; Ac Xi1;3

Hi3i3i 3i

~32

(9)

where ~3 P

i1;33i 3i 2q

is the effective strain. Hi[x] 0 whenx< 0 and Hi[x] x when x 0. It can be veried that in uniaxialtension, At 1, Ac 0, DDt, and vice versa in compression.

Material parameters for mortar matrix are listed in Table 1.The dynamic yield strength surface is amplied from static

surface by considering the strain rate effect. Typically thecompressive (tensile) strength is multiplied by a compressive(tensile) dynamic increase factor (DIF). In the model, the DIFs arefrom the CEB recommendation for concrete [42], which areobtained from many experimental test data on concrete andcement mortar.

The compressive DIF recommended by CEB is,

CDIF fcdfcs

_3d_3cs

1:026afor _3d 30s1 (10a)

CDIF fcdfcs

g_3d13 for _3d > 30s1 (10b)where fcd is the dynamic compressive strength at the strain rate _3d(in the range of 301061000 s1), _3cs 30 106 s1,log g 6:156a 0:49; a 5 3fcu=41, fcs is the staticcompressive strength, and fcu is the static cube compressive

strength (in MPa).

Axis of symmetry

High explosive

Outflowboundary

Outflow boundary

Concrete Air

Euler-Lagrangeinterface

Fig. 9. Axisymmetric numerical model.

Fig. 10. Aggregate distribution (a) case I (b) case II.

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 131513261320The CEB recommendation for the tensile DIF is [42],

TDIF ftdfts

_3d_3ts

1:016dfor _3d 30s1 (11a)

TDIF ftdfts

b

_3d_3ts

1=3for _3d > 30s

1 (11b)

where ftd is the dynamic tensile strength at the strain rate _3d (in therange of 3106300 s1), fts is the static tensile strength at thestrain rate _3ts_3ts 3 106s1, and log b 7:11d 2:33, inwhich d 1=10 6f 0c=f 0co; f 0co 10MPa, f 0c is the static uniaxialcompressive strength (in MPa).Fig. 11. Density distributio3.2. Material model for coarse aggregate

The aggregate is assumed to suffer brittle failure witha minimum deformation. Therefore the simplest linear equation ofstate is adopted to calculate the pressure

p Km (12)

where p is the pressure, m (r/r0)1, and K is the material bulkmodulus.

Because rock behaves similarly as concrete under dynamicloading, the same strength criterion for the mortar matrix isadopted here to model the aggregates, only the material constantsare different. The parameters for the aggregates are given in Table 2.n (a) case I (b) case II.

(13b)

from the AUTODYN material library are utilized, that is, the airdensity r 1.225 kg/m3 and g 1.4. The air initial internal energy isassumed to be 2.068 105 kJ/kg.

4. Numerical simulation

In this study, a concrete slab tested under blast loading by otherresearchers [28] is modelled. In numerical simulation, the concreteslab is modelled by a Lagrange subgrid, in which the coordinatesmove with the material; while the air and high explosive are

of Impact Engineering 36 (2009) 13151326 1321CDIF 0:0225log _3 1:12 _3 10s1 (14a)

CDIF 0:2713log _320:3563log _31:2275 10s1 _3 2000s1 (14b)

Comparisons of the above empirical formulae with the availabletest results are shown in Figs. 6 and 7. Rouabhi et al. [49] performednumerical simulations to construct the material constitutive modelfor rock, the numerically derived tensile and compressive dynamicincrease factors are also shown in Figs. 6 and 7 for comparison. It isfound that the DIFs from the numerical model are lower than thoseobtained from test results.

3.3. Material model for high explosive and air

High explosives are typically modelled by using the JonesWilkinsLee (JWL) equation of state, which models the pressuregenerated by chemical energy in an explosion. It can be written inthe form,

p C1

1 u

r1v

er1v C2

1 u

r2v

er2v ue

v(15)

where p hydrostatic pressure; v specic volume; e specicinternal energy; and C1, r1, C2, r2 and u are material constants. Thevalues of the constants for many common explosives have beendetermined from dynamic experiments and are available inAUTODYN [2]. In the present simulation, C1, r1, C2, r2, and u areassumed as, 3.7377105 MPa, 4.15, 3.7471103 MPa, 0.9, 0.35,respectively.

Air is modelled by the ideal gas equation of state, which is one ofthe simplest forms of equation of state. The pressure is related tothe energy by

p g 1re (16)

where g is a constant, r is the air density and e is the specicThe DIFs for the coarse aggregates are based on some test resultson rock materials. Some researchworks have been done to study thestrain rate effect on both the tensile and the compressive strength ofthe rock material. Cho et al. [43] investigated the dynamic tensilestrength of Indian granite and Tage tuff, by using experimentalapproach based on Hopkinsons effect combined with the spallingphenomena. Wang et al. [44] used split Hopkinson pressure bar(SHPB) to impact a attened Brazilian disc of marble for testingdynamic tensile strength of rock material. Cai et al. [45] conductedboth the tensile and the compression SHPB tests to study thedynamicbehavior of Meuse/HauteMarne argillite. Lok et al. [46] used a sha-ped striker bar in a large diameter SHPB tests to obtain the dynamictensile and compression strengthof granite. Li et al. [47] also reportedsome compression test results by using the same equipment. Li et al.[48] obtained the dynamic compressive strength for different rockmaterials by using a rock dynamic testing systemwhich is driven byhydraulic and air. Based on the above-mentioned test results, thetensile and compressive DIFs are obtained as follows,

TDIF 0:0225 log _3 1:12 _3 0:1s1 (13a)

TDIF 0:7325log _321:235log _3 1:6 0:1s1 _3 50s1

X.Q. Zhou, H. Hao / International Journalinternal energy. In the simulation, the standard constants of air Fig. 12. Damage distribution (a) case I (b) case II.

modelled by Euler subgrid, in which the grid is xed and materialows through it. At the Euler-Lagrange interface, interaction isconsidered. The Lagrange subgrid imposes a geometric constraintto the Euler subgrid while the Euler subgrid provides a pressureboundary to the Lagrange subgrid.

4.1. Test setup

The dimension of the tested slab was 1.21.2 0.32 m3 and theconcrete had a static compressive strength of 48 MPa. This slab wasloaded by an explosive cone of TNT and Composition B, and the

equivalent charge weight was about 350 g. The inner cone con-sisted of TNT and the outer thin cone of composition B. Thedimensions are shown in Fig. 8. This slab was tested by otherresearchers [28]. It is employed here to validate the proposednumerical model.

4.2. Numerical model

The slab is approximately modelled as axisymmetric in thisstudy. It should be mentioned that the dynamic fragmentationprocess may be affected by the 3D aggregate distribution. The 3D

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 131513261322Fig. 13. Cracked mesh and aggregate distribution.

the mortar matrix around the aggregates. Therefore, the aggregatedistribution in this part highly affects the tensile damage.

Table 3Comparison of the numerical results and the experimental results.

Upper craterdiameter (mm)

Bottom craterdiameter (mm)

Experimental results 51 62Numerical results (case I) 49.5 61.5Numerical results (case II) 71.0 71.0Numerical results (homogeneous) 70.5 52.5

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 13151326 13234.3. Numerical results and fragment analysis

Typical numerical results on the damage process of the twodifferent aggregate distribution cases are shown in Fig. 12. From theFig., the damage of the slab caused by stress wave propagation canbe clearly seen. At 0.05 and 0.1 ms, the compressive stress wave hasnot reached the bottom of the concrete slab, the concrete damage isassociated with concrete crushing caused by the compressivestress; at 0.15 ms, the stress wave reaches the free boundary(bottom of the slab), a strong tensile stress wave is generated owingto the reection of the compressive stress wave, and the tensilestress is high enough to cause many cracks near the bottom surfaceeffect analysis will be carried out in our future research work withimproved computational power. The 2D axisymmetric numericalmodel is shown in Fig. 9. Two random aggregate distributions areshown in Fig. 10. The blue circles are the coarse aggregates and thewhite area is the mortar matrix. The density distributions for thetwo cases are shown in Fig. 11. In the numerical simulation, quad-rilateral elements are used. Therefore the circular geometry of theaggregate is only approximately modelled by quadrilateralelements. The element size in the 2D simulation is 2 2 mm. Themesh size was determined by convergence test in the homoge-neous model and proved yielding reliable predictions of the blasttesting results [12].

Fig. 14. Cracked mesh (homogeneous model).of the concrete slab; at 0.2 ms, the cracks extend further. From thegure, it can be found that both the aggregate and the mortarmatrix are fully damaged in the high compression zone, which isthe upper crater directly caused by the contact detonation.However, in the bottom spalling zone, the damage is caused owingto the tensile failure. Basically the damage (tensile crack) occurs in

Fig. 15. Test results [28] (a) toIn order to further study the fragment distribution, erosiontechnique is adopted. In this simulation the highly distortedelements and the elements with high tensile strains are eroded. Thecracks are estimated after the elements are eroded. It should bementioned that the fragment size less than the element size of2 mm cannot be obtained in the present numerical simulation dueto the limitation of the erosion technique. The cracked meshes andthe distributions of the aggregates for the two cases are shown inFig. 13. Comparison of the two cases shows that the aggregatedistribution does affect the crack position and the fragmentdistribution. The cracks basically occur at the boundary of thecoarse aggregate, therefore the size and the position of the coarseaggregates in the spalling area inuence the crack generation andpropagation. For comparison, the result from the homogeneousmodel is also obtained and shown in Fig. 14. In the homogeneousmodel, the same material properties as those of the mortar inthe mesoscale model are adopted. It should be mentioned that theactual strengths of the present homogeneous model and theheterogeneous model are not exactly the same, however, ourprevious research shows that the compressive and the tensilestrength of the mesoscale model are only slightly higher than thestrength of the mortar matrix. The experimental results are shownin Fig. 15. Comparison of Figs. 1315 shows that all the threemodels(two mesoscale model and one homogeneous model) successfullypredicted the perforation of the slab. However, the crater sizesobtained from different models differ a lot. Comparison of thecraters of test results and the numerical results are listed in Table 3.It can be found that the numerical results from the case I agree wellwith the test results. Themesoscalemodel II overestimates both thetop and the bottom craters, whereas the homogeneous modeloverestimates the top crater size but underestimates the bottomcrater size. The results indicate that the concrete slab damage canbe reliably predicted if a proper mesoscale model is used. However,the results very much depend on the aggregate distributions. Sincethe aggregate distribution is random in real concrete structures,probabilistic analysis is needed to determine the mean and varia-tion of the concrete structural damage levels. This will be our futureresearch topic.p view (b) bottom view.

From the numerical simulations, the ejecting velocity of thefragments can also be obtained. The highest ejecting velocities ofthe fragments from the bottom surface of the concrete are 20.6,24.0 and 19.8 m/s, for case I, case II and the homogeneous model,respectively. It should be noted that these ejecting velocities areassociated with the fragments with size larger than 2 mm. For the

estimated as the square root of the fragment area. The size distri-butions for the different cases are shown in Fig. 18. In the gure, thevertical axis corresponds to the mass percentage. In the masscalculation, it is assumed that the fragment size in the thirddirection is the same as that in the 2D plane, i.e., the square root ofthe fragment area. From Fig. 18, it can be found that the homoge-

Fig. 16. Fragment distribution (estimated from the numerical results, bottom spalling crater) (a) case I (b) case II (c) homogeneous model.

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 131513261324fragments smaller than 2 mm, the ejecting velocity might behigher, but as discussed above, the present numerical model is notcapable of generating fragments smaller than 2 mm because of theelement size used in the model. The results from the numericalsimulation also depend on the erosion criterion used. The erosioncriterion used here is a combination of the effective strain and thedamage value. If the tensile damage value is higher than 0.99 andthe effective strain is higher than 0.2, or if the effective strain ishigher than 2.0, the element is assumed to be fully damaged anderoded from the model. The erosion criterion adopted here ispartially based on ref [2], partially based on numerical trial. Nor-mally the erosion is mainly based on effective strain, only the highlydistorted elements are deleted to avoid numerical difculty, but inthe present study, tensile damage is also considered to model thepossible tensile cracks.

The results from the numerical simulation in Figs. 13 and 14 arethen treated by an image analysis program (a toolbox in MATLAB)to predict the fragment size distribution. Fig. 16 shows the frag-ments from the bottom spalling (estimated from Figs. 13 and 14). Itshould be mentioned that the fragments in the upper crater is notanalysed in the present study because a large number of theelements are eroded due to large element distortion in this region.In the image analysis program, the gures are loaded rst by usingthe function imread, then the imcomplement function in MAT-LAB is used to take the complement of the image; next, thebwmorph function is used to skeletonize the results, nally eachfragment boundary is found and the fragment area is obtained.After the treatment by the image analysis program, the fragmentdistributions can be seen more clearly in Fig. 17. This gure is thenused to extract the area distribution. For each fragment, the size isFig. 17. Fragment distribution (treatneous model predicts more small size fragments than the meso-scale model. It can also be found that the fragment sizedistributions are different for the two aggregate distribution cases,indicating again the inuence of the aggregate distribution on theconcrete slab damage to blast loads.

According to a statistical approach [33], the cumulative mass offragments with mass less than or equal to m is,

Mcm M1 em=ma (17)

accordingly, the cumulative size of fragments with sizes less than orequal to s is,

Mcs M1 es=sa3 (18)

where M is the total fragment mass, ma is the average fragmentmass, and sa is the average fragment size.

For comparison, the results based on statistical empiricalformulae (Eq. (18)) are also shown in Fig.18, inwhich Asize 30, 40, 50denote the average fragment size are 30, 40 and50 mm, respectively.From this gure, it can be found that the numerical simulationspredict that the average fragment size is in the range of 3050 mm,which is in the same order of the biggest coarse aggregate. It alsoindicates that the numerical results match the empirical statisticalpredictions when the fragment size is smaller than the averagefragment size, however, the numerical results and the statisticalpredictions differ when the fragment size is larger than the averagefragment size. Nonetheless the numerical results of fragmentdistribution are in the same range of the statistical prediction.Unfortunately, no experimental result on fragment size distributionis available from the experimental test analysed in this study.ed by image analysis program).

pe Asize 40

of Im5. Conclusions

In thepresent paper, a two-phasemesoscalemodel is developed tosimulate the dynamic damage and fragmentation of a concrete slabunder blast loading. The distribution of the coarse aggregates isassumed to follow the Fullers curve. Two different aggregate distri-bution cases are generated to simulate the example concrete slab. It isfound that the aggregate distribution signicantly affects the crackand fragment distribution. The cracks in general occur in the mortarmatrix around the coarse aggregates. The fragment size distributionsobtained from the two mesoscale models are compared with thatobtained from the homogeneous model and the results from thestatistical empirical formula. It is found that the estimated fragmentsize distributions from the two mesoscale models and the homoge-neous model are all comparable with the statistical predictions basedon the empirical formula. The fragment size is in the range between0 and about 6080mm. Themean size of the fragment is in the sameorder as the biggest coarse aggregate. Thepresent study demonstratesa practical method to predict the fragment size distribution using theimage analysis method and numerical simulations.

3D mesoscale numerical simulation will be carried out in ourfuture study. Experimental tests will also be carried out to obtainfragment size distribution and ejecting velocity to calibrate the

Fig. 18. Fragment size distribution.0

10

20

0 20 40 60 80 100 120fragment size (mm)

Asize 5030

40

50

60

70

80

90

100

rcen

tag

e p

assin

g (%

)

Case ICase IIHomo

Asize 30

X.Q. Zhou, H. Hao / International Journalpresent numerical model.

Acknowledgements

The authors would like to thank the Australian Research Councilfor nancial support under grant No. DP0774061 to carry out thisresearch work.

References

[1] LS-DYNA, Keyword users manual. Livermore, California, USA; LivermoreSoftware Technology Corporation; 2006.

[2] Autodyn, Century dynamics, Theory manual. Concord, California, USA:Century Dynamics; 2005

[3] Pinto PE. RC elements under cyclic loading, state of the art report. London:Thomas Telford; 1996.

[4] Chen ACT, Chen WF. Constitutive relations for concrete. Journal of EngineeringMechanics Division, ASCE 1975;101(EM4):46581.

[5] Kotsovos MD, Newman JB. Generalised stress-strain relations for concrete.Journal of Engineering Mechanics Divisions, ASCE 1978;104(EM4):84556.

[6] Ottesen NS. A failure criteria for concrete. Journal of the engineeringMechanics Division, ASCE 1977;103(EM4):52735.

[7] Bischoff PH, Perry SH. Compressive behaviour of concrete at high strain rate.Materials and Structures 1991;24:42550.[8] Fu HC, Erki MA, Seckin M. Review of effects on loading rate on concrete incompression. Journal of Structural Engineering, ASCE 1991;117(12):364559.

[9] Malvar LJ, Ross CA. Review of strain rate effects for concrete in tension. ACIMaterials Journal 1998;95(M73):7359.

[10] Eibl J, Schmidt-Hurtienne B. Strain-rate_sensitive constitutive law forconcrete. Journal of Engineering Mechanics 1999;125(12):141120.

[11] Schuler H, Mayrhofer C, Thoma K. Spall experiments for the measurement ofthe tensile strength and fracture energy of concrete at high strain rates.International Journal of Impact Engineering 2006;32:162550.

[12] Zhou,X.Q.,Hao,H.&Deeks,A.J.,Modellingdynamicdamageofconcrete slabunderblast loading. In: Proceedings of the 6th International Conference on Shock &impact Loads on Structures. Perth, Australia: 79, December, 2005, p. 703710.

[13] Holmquist, T.J. and Johnson, G.R. A Computational Constitutive Model forConcrete Subjected to Large Strains, High Strain Rates, and High Pressures. In:Proceedings of the 14th International Symposium on Ballistics. Quecbec,Canada: 1993. p. 591600.

[14] Riedel W., Thoma K., Hiermaier S. and Schmolinske E. Penetrating of rein-forced concrete by BETA-B-500 numerical analysis using a new macroscopicconcrete model for hydrocodes. In: Proceedings of 9th International Sympo-sium IEMS, Berlin: 1999. p. 315322.

[15] Gebbeken N, Ruppert M. A new material model for concrete in high-dynamichydrocode simulations. Archive of Applied Mechanics 2000;70:46378.

[16] Malvar LJ, Crawford JE, Wesevich JW, Simons D. A plasticity concretematerial model for DYNA3D. International Journal of Impact Engineering1997;19:84773.

[17] Katayama M, Itoh M, Tamura S, Beppu M, Ohno T. Numerical analysis methodfor the RC and geological structures subjected to extreme loading by energeticmaterials. International Journal of Impact Engineering 2007;34(9):154661.

[18] Rabczuk T, Eibl J. Simulation of high velocity concrete fragmentation usingSPH/MLSPH. International Journal of Numerical Methods in Engineering2003;56:141244.

[19] Leppanen J. Concrete subjected to projectile and fragment impacts: modelingof crack softening and strain rate dependency in tension. International Journalof Impact Engineering 2006;32:182841.

[20] Clayton JD. A model for deformation and fragmentation in crushable brittlesolids. International Journal of Impact Engineering 2008;35:26989.

[21] Gatuingt F, Pijaudier-Cabot G. Coupled damage and plasticity modelling intransient dynamic analysis of concrete. International Journal for Numericaland Analytical Methods in Geomechanics 2002;26:124.

[22] Wriggers P, Moftah SO. Mesoscale models for concrete: homogenisation anddamage behaviour. Finite Elements in Analysis and Design 2006;42:62336.

[23] Wang ZM, Kwan AKH, Chan HC. Mesoscopic study of concrete I: generation ofrandom aggregate structure and nite element mesh. Computers and Struc-tures 1999;70(533):544.

[24] Kwan AKH, Wang ZM, Chan HC. Mesoscopic study of concrete II: nonlinearnite element analysis. Computers and Structures 1999;70(545):556.

[25] Hafner S, Eckardt S, Luther T, Konke C. Mesoscale modelling of concrete:Geometry and numerics. Computers and Structures 2006;84:45061.

[26] VanMier JGM,VanVlietMRA. Inuence ofmicrostructure of concrete on size/scaleeffects in the tensile fracture. Engineering FractureMechanics 2003;70:2281306.

[27] Bazant ZP, Tabbara MR, Kazemi MT, Pijaudier-Cabot G. Random particle modelfor fracture of aggregate or bre composites. Journal of EngineeringMechanics, ASCE 1990;116:1686705.

[28] Rabczuk T, Eibl J. Modelling dynamic failure of concrete with meshfreemethods. International Journal of Impact Engineering 2006;32:187897.

[29] Grady DE. The spall strength of condensed fragmentation. Journal of theMechanics and Physics of Solids 1988;36:35384.

[30] Yew CH, Taylor PA. A thermodynamic theory of dynamic fragmentation.International Journal of Impact Engineering 1994;15:38594.

[31] Zhang YQ, Lu Y, Hao H. Analysis of fragment size and ejection velocity at highstrain rate. International Journal of Mechanical Sciences 2004;46:2734.

[32] Grady DE, Kipp ME. Geometric statistics in dynamic fragmentation. Journal ofApplied Physics 1985;58:121022.

[33] Grady DE. Particle size statistics in dynamic fragmentation. Journal of AppliedPhysics 1990;68:6099105.

[34] Grady DE, Kipp ME. Continuum modeling of explosive fracture in oil shale.International Journal of Rock Mechanics and Mining Science & GeomechanicsAbstracts 1980;17:14757.

[35] Espinosia HD, Zavattieri PD, Dwivedi SK. A nite deformation continuum/discrete model for the description of fragmentation and damage in brittlematerial. Journal of the Mechanics and Physics of solids 1998;46:190942.

[36] Liu LQ, Katsabanis PD. Development of a continuum damage model forblasting analysis. International Journal of Rock Mechanics and Mining Science1997;34(2):21731.

[37] Zhang YQ, Hao H, Lu Y. Anisotropic dynamic and fragmentation of rockmaterials under explosive loading. International Journal of EngineeringScience 2003;41:91729.

[38] Rabczuk T, Eibl J, Stempniewski L. Numerical analysis of high speed concretefragmentation using a meshfree Lagrangian method. Engineering FractureMechanics 2004;71:54756.

[39] ChenWF. Plasticity in reinforced concrete. NewYork:McGrawHill; 1982. p. 474.[40] Mazars J. A description of micro- and macroscale damage of concrete struc-

tures. Engineering Fracture Mechanics 1986;25(5/6):72937.

pact Engineering 36 (2009) 13151326 1325[41] Ma GW, Hao H, Zhou YX. Modeling of wave propagation induced by under-ground explosion. Computers and Geotechnics 1998;22:283303.

[42] Comite Euro-International du Beton. CEBFIP model code 1990. Trowbridge,Wiltshire, UK: Redwood Books; 1993.

[43] Cho SH, Ogata Y, Kaneko K. Strain-rate dependency of the dynamic tensilestrength of rock. International Journal of Rock Mechanics & Mining Science2003;40:76377.

[44] Wang QZ, Li W, Song XL. A method for testing strength and elasticmodulus of rock materials using SHPB. Pure and Applied Geophysics2006;163:1091100.

[45] Cai M, Kaiser PK, Sourineni F, Su K. A study on the dynamic behaviour of theMeuse/HauteMarne argillite. Physics and Chemistry of the Earth2007;32:90716.

[46] Lok TS, Zhao PJ, Li XB and Lim CH. Dynamic stress-strain response of granitefrom split Hopkinson Pressure Bar tests. In: Proceedings of the 5th Asia-Pacic

Conference on Shock & Impact Loads on Structures. November 1214, 2003,Changsha, Hunan, China: p. 27786.

[47] Li XB, Lok TS, Zhao J. Dynamic characteristics of granite subjected tointermediate loading rate. Rock Mechanics and Rock Engineering2005;38(1):2139.

[48] Li HB, Zhao J, Li JR, Liu YQ and Zhou QC. Experimental studies on the strengthof different rock types under dynamic compression. International Journal ofRock Mechanics and Mining Science. 2004, 41(3). Paper 1A 12 SINOR-OCK2004 Symposium.

[49] Rouabhi A, Tijani M, Moser P, Goetz D. Continuum modelling of dynamicbehaviour and fragmentation of quasi-brittle materials: application to rockfragmentation by blasting. International Journal for Numerical and AnalyticalMethods in Geomechanics 2005;29:72949.

X.Q. Zhou, H. Hao / International Journal of Impact Engineering 36 (2009) 131513261326

Mesoscale modelling and analysis of damage and fragmentation of concrete slab under contact detonationIntroductionConcrete material model under blast loadingMesoscale modelling of concreteConcrete dynamic fragmentation

Generation of coarse aggregate particlesMaterial model for concreteMaterial model for mortar matrixMaterial model for coarse aggregateMaterial model for high explosive and air

Numerical simulationTest setupNumerical modelNumerical results and fragment analysis

ConclusionsAcknowledgementsReferences