Research Article Impact Resistance of Concrete...

Research ArticleImpact Resistance of Concrete Structures

R Drathi

IIT Kanpur Kanpur Uttar Pradesh 208016 India

Correspondence should be addressed to R Drathi 478923adminiitdacin

Received 13 February 2015 Revised 14 May 2015 Accepted 28 July 2015

Academic Editor Xiaoying Zhuang

Copyright copy 2015 R Drathi This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present simulations on the resistance of concrete structures due to impact loading Therefore a mesh-free method is exploitedin combination with a viscous damage model that accounts for strain rate effects and high pressures that commonly occur in thoseevents In contrast to many other studies in the literature we account for uncertainties in the material parameters by subjectingthem to a probability distribution function Furthermore we also consider the geometric correlation inside the concrete and showthat this geometric correlation has a minor influence on the predicted results All numerical results are compared to the impactexperiments performed by Hanchak and coworkers as well as our own impact experiments

1 Introduction

Predicting the impact resistance of concrete and reinforcedconcrete structures has been of interest for many yearsConstitutive models that account for strain rate effectshigh pressure and the associated dynamic strength increaseare the key to successful predictions Constitutive modelsaccounting for strain rate effects can be classified into damagemodels plasticity models and damage-plasticity models [1ndash6] Damage models for concrete accounting for high strainrates were developed for instance by [7ndash10] A popular plas-ticity model for concrete was proposed by [11] as an extensionof the Johnson-Cook [12] model originally developed formetals In this model the dynamic strength increase wasrelated to the current strain rate However memory effectswere not accounted for Furthermore discontinuities in thestress-strain response are allowed which are physically notmeaningful In [13] the dynamic buckling strengthwas there-fore dependent on the strain rate history and an average strainrate A viscous damage-plasticity model accounting for highstrain rates and high pressures was proposed by [14] There-fore a dynamic damage variable was introduced that decayedthe static damage evolution The decaying function dependson previous damage-rates or damage increments The scalardamage was later on extended to a damage tensor accountingfor the anisotropy for concrete under tensile loading [15]while a scalar damage was maintained in compression

Most of the studies on the impact resistance of concretewere done with the finite element method [16ndash24] Com-monly element-deletion techniques [25 26] were applied toallow for large deformations and perforations A strong com-petitor to finite elements is mesh-free methods [27ndash37] thatallow for a more robust and accurate prediction of dynamicfracture and fragmentation see [38] for a recent overviewof mesh-free methods and their applications Reference [39]for instance was able to predict the impact resistance of theexperiments carried out by [40] with mesh-free methodsquite accurately while the FE simulations done in [40] ledalways to an underestimation of the impact resistance Alsopenetration was not predicted correctly in those simulationsDynamic fracture in early mesh-free methods was treatedby simple separation of particles [41ndash47] However [30 48]showed that care needs to be taken as the fracture often iscaused by numerical artefacts More advanced methods suchas the extended finite element method [49 50] extendedmesh-free methods [34 51ndash59] the phantom node method[60ndash64] or efficient remeshing techniques [65ndash70] have beenapplied to dynamic fracture of concrete as well [71 72]However they do not seem suitable tomodel the impact resis-tance of concrete with large deformations and an enormousnumber of cracks A simpler and more efficient pathway isthe Cracking Particles Method (CPM) [73 74] which modelsthe crack as set of crack segments Similarmethods were usedby [75ndash80] Other researchers try to combine the advantages

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 494617 8 pageshttpdxdoiorg1011552015494617

2 Mathematical Problems in Engineering

of mesh-free methods and finite elements Therefore onlythe area around the perforation was modelled by a mesh-free method whereas a finite element discretization wasemployed elsewhere reducing computational cost [31 80]Most simulations were based on pure deterministic modelsthat can lead to unrealistic crack patterns as the ones shownin [39] In these simulations the radial main cracks appeartoo close to each other In order to achieve unsymmetricalcrack patterns [81] introduced randomness in the strengthto avoid the unrealistic symmetric patterns they modelledbefore in [82ndash84] However all other parameters remaineddeterministic

Several benchmark experiments have also been con-ducted to test the influence of the thickness of the specimenor the speed of the impactor on the impact resistanceof concrete or reinforced concrete structures [85ndash91] Oneclassical experiment was done by [92] who subjected concrete

structures to impact with different speeds of the impactorThey measured the residual velocity of the impactor Therecent experiments by [40] also report penetration depth

This paper presents one of the first studies on stochasticmodelling of dynamic fracture of concrete Our constitutivemodel is based on the model described in [14] and combinedwith a Lagrangian mesh-free method for the computationalstudies Fracture will be modelled by a simple particle split-ting as described in [77]This methodology will be describedin the first sections before we present computational resultsthat will be compared to the experiments of Hanchak et al[92] We also carried out our own experiments that will serveas validation

2 Governing Equations and Discretization

The governing equation to be solved is the balance of linearmomentum that is given in weak form of the updateddescription of motion by

120575119882int minus 120575119882ext = 120575119882kin 120575119882int = intΩ

120575120598119894119895 120590119894119895119889Ω 120575119882ext = int

Γ119905

120575119906119894119905119894119889Γ + int

Ω

120575119906119894119887119894119889Ω 120575119882kin = int

Ω

984858119906119894119894119889Ω (1)

where 120575119882int 120575119882ext and 120575119882kin denote the internal externaland kinetic energy respectively The domain is denoted byΩ whereas Γ

119905cup Γ119906

= Γ with Γ119905cap Γ119906

= 0 denotes theLipschitz continuous boundary 120598

119894119895and 120590119894119895indicate the strain

tensor and Cauchy stress tensor 119887119894and 119905

119894are the body

forces and tractions respectively 119906119894is the displacement field

and 984858 is the density while 120575 denotes the first variation andthe superimposed dote material time derivatives which areidentical to the partial derivatives in an updated Lagrangianframework

We have employed a stabilised [48] nodally regularised[30] element-free Galerkin method [93] Therefore the dis-cretization of the displacement field is given by

u (x 119905) = sum

119868isinS

119873119868 (x) u119868 (119905) (2)

where u119868denote the nodal parameters which are not equal to

the physical displacement value at that point that is u(x119868) =

u119868 and119873

119868(x) indicate the shape functions They are derived

from minimising a discrete weighted norm as explained indetail for example in [93]

N (x) = p119879 (x)Aminus1 (x)B (x) (3)

where A(x) = sum119899

119868=1119908(X minus X

119868)p(X119868)p119879(X

119868) is called the

moment-matrix and the matrix B(x) is obtained by

B (x)

= [119908 (x minus x1) p (x1) 119908 (x minus x

2) p (x2) sdot sdot sdot 119908 (x minus x

119899) p (x119899)]

(4)

119908(X minus X119868) denoting the kernel function and p119879(X) the

polynomial basis To ensure Galilean invariance we opted

to choose a linear basis p119879(x) = [1 119909 119910] The exponentialweighting function has been employed in all our simulations

Substituting the discretization of the displacement fieldand virtual displacement field into the weak form yields thewell-known equation of motion that is solved for the nodalparameter u

119868(119905) that is stored in the global vectorD

MD = Fext minus Fint (5)

with the mass matrix M = intΩ984858N119879N 119889Ω and the vector of

external and internal forces

Fext = intΩ

984858N119879b 119889Ω + intΓ119905

N119879t 119889Γ

Fint = intΩ

B119879120590119889Ω(6)

where N and B are matrices containing the mesh-free shapefunctions and their spatial derivatives respectively Theintegrals are evaluated by stabilised nodal integration whichensures bending exactness in a Galerkin method The use ofan updated Lagrangian kernel formulation as suggested in[39] ensures the stability of the method while maintainingthe applicability to extremely large deformations needed fordynamic fracture and fragmentation

3 Constitutive Model

We have used a modified version of the constitutive modelpresented in [14] Instead of using a viscous damage-plasticitymodel we removed the plasticity part of this model asit adds material parameters that are difficult to calibrateFurthermore preliminary studies indicated that the plastic

Mathematical Problems in Engineering 3

178mm

Thickness 610mm

610mm

Figure 1 Hanchak impact experiment

response is of minor importance as the results in the nextsectionwill showThe basic idea of the damagemodel is givensubsequently For more detailed information the reader isreferred to [14]

4 Results

Two types of experiments are employed in order to validatethe proposed computational framework

(i) the impact experiments carried out by Hanchak [92](ii) contact explosion experiments done at the laboratory

of our university

Hanchak tested the impact resistance of concrete struc-tures of different concrete strength The main goal of theexperiments of Hanchak studies was to test their high per-formance concrete compared to standard concrete The highperformance concrete had a compressive strength of 140MPAwhile the standard concretersquos compressive strengthwas nearly50MPA Therefore concrete specimens were subjected toimpacts of different projectile velocities The experimentalset-up is illustrated in Figure 1 As our focus is not on highperformance concrete but rather on validation of our frame-work we present only the results of the impact simulationson standard concrete In the results all plates were perforatedand the residual impact velocity was measured We will focuson experiments with penetration later

A full three-dimensional discretization has been uti-lized Preliminary studies showed substantial differences inresults obtained from 2D simulations based on plane strainassumptions Around 2 million nodes were needed in orderto get convergence in the residual impact velocity andsimultaneously in the damage pattern The (deterministic)parameters of the constitutive model are listed in Table 1They are classified in different categories as indicated by thedifferent lines (1) bulk parameters that is Youngrsquos modulusPoissonrsquos ratio and density (2) damage parameters 119890

0 119890119889 and

119892119889determining the ldquostaticrdquo damage evolution the key param-

eters are 119890119889and 119892

119889controlling the ductility and maximum

strength of the concrete and the parameters determiningthe dynamic part of the damage responses 119888

1199051and 1198881199052 they

determine how the dynamic damage is decaying over time

Table 1 Parameters of the constitutive model

Youngrsquosmodulus 119864[GPa]52

Poisson ratio022

Density [gmm3]2700

1198900

119890119889

119892119889

1198881199051

1198881199052

00001 00003 23 0032 0551198881

1198882

1198883

1198884

001234 00252 07821 03464119903119905

119903119888

12 188119886V 119887V 119890V 119890V119905ℎ

07 34 002 0007

(3) the parameters of the damage surface 1198881to 1198884 (4) the

reduction factors 119903119888and 119903119905controlling howmuch compressive

damage can be done by tensile loading and vice versaand (5) the parameters of the Hugoniot-curve accountingfor the pressure dependence of concrete under high strainrates and high volumetric strains All parameters except theones associated with the damage surface are assumed tobe stochastic in our computations and are subjected to auniform probability distribution function with a scatter of5 and 10 respectively In order to identify the influenceof each physical mechanism we modified only parametersassociated with this physical mechanism according to theprobability distribution function while keeping all otherparameters constant We are aware that a global sensitivityanalysis as proposed for example by [94ndash97] in the context ofstudies on polymeric composites or by [98] in the context ofoptimizationwill providemore quantitative results Howeversuch studies are out of the scope of this paper and left for thefuture

Before discussing the results presented in Table 2 let usdiscuss the different physical phenomena and the relatedinput parameters

(1) The parameters related to the first line of 1 govern theinitial elastic response of the material in the bulk

(2) The parameters 1198900 119890119889 and 119892

119889determine the stress-

strain response under static conditions The firstparameter determines the transition to nonlinearmaterial behaviour while the other two parametersdetermine the tensilecompressive strength and fail-ure strain

(3) The parameters 1198881199051 1198881199052govern the dynamic damage

evolution and are responsible for the strength increaseunder high strain rates

(4) The parameters 119903119905 119903119888are responsible for the amount

of compressive damage caused under tensile loadingand vice versa

(5) The last set of parameters in Table 1 account for thepressure dependence of the concrete

The results in Table 2 show the lower and upper bounds ofthe residual impact velocity for the different input parameters


Table 2 Residual velocity ([ms]) of the impactor in dependence of the impact velocity ([ms]) and the scatter in the input parameters thefirst line of the same impact velocity refers to a 5 scatter while the second line refers to the 10

Impact speed Determ Set 1 Set 2 Set 3 Set 4 Set 5 Experiment360 75 74-75 73ndash76 74ndash76 75-76 74ndash76 67360 74-75 73ndash77 72ndash77 75-76 73ndash76430 223 222-223 220ndash229 219ndash231 221ndash224 221ndash230 214430 222ndash224 215ndash230 213ndash234 221ndash225 215ndash232750 628 625ndash631 618ndash634 615ndash635 623ndash632 619ndash631 615750 624ndash632 610ndash639 601ndash645 621ndash633 611ndash6411060 957 952ndash961 942ndash971 938ndash974 950ndash962 943ndash973 9471060 951ndash961 934ndash980 928ndash989 949ndash963 932ndash984

Table 3 Penetration depth and damage diameter for the explosion experiment the first line refers to a scatter of 5 while the second linerefers to the 10 scatter in the input

Deterministic Set 1 Set 2 Set 3 Set 4 Set 5 ExperimentDiameter [cm] 57 567ndash57 56ndash572 556ndash575 565ndash57 564ndash573 56Diameter [cm] 567ndash57 555ndash573 55ndash578 561ndash573 563ndash573Penetr depth [cm] 28 278ndash28 273ndash282 27ndash284 275ndash281 277ndash28 26Penetr depth [cm] 278ndash28 270ndash285 263ndash288 270ndash283 276ndash28

for different impact velocitiesThe values of the deterministicsimulation are depicted in Table 2 as well As can be seenuncertainties in the input substantially influence the residualvelocity of the impactor While parameter sets 1 (parametersrelated to the initial elastic response) and 4 barely havean effect on the residual velocity (the scatter in the outputis much smaller than the scatter in the input) the staticdamage evolution (parameter set 2) and pressure dependence(parameter set 5) have a substantial influence in the orderof the scatter in the input This is observed consistently forboth 5 and 10 scatter The biggest influence is achievedby the parameters related to the dynamic strength increasewhich is more pronounced for increasing scatter Finally weremark that we also varied two sets of input parameters whilekeeping the other parameters constant Since parameter sets3 and 5 had the biggest influence on the output we variedthose parameters However the upper and lower boundsof the residual velocity barely changed The scatter in theoutput is slightly higher than the scatter in the input Notethat the effect gets more pronounced with increasing impactvelocity Overall one can see an excellent agreement betweencomputational and experimental results

The second test example is a concrete slab under con-tact explosion as depicted in Figure 2 The explosion wasmodelled by a (JWL) John-Wilkonson-Lee equation of stateThe explosive is also modelled with a mesh-free Lagrangianmethod which is one of the advantages of the proposedmethod the fluid-structure interaction remains fairly simplecompared to a coupled Lagrangian-Eulerian coupling such as(ALE) which is required for finite elements for instance

In the experiment a penetration was observed and thepenetration depth as well as the diameter of the crater atthe top surface was measured We used the same concreteas in the experiments performed by Hanchak and thereforethe same material parameters of the constitutive model were

500mm

Thickness 1200mm

1200mm

Figure 2 Explosion experiment

used Figure 3 shows the damage shape towards the end ofthe simulation the damage is indicated by the red colorThe full deformation has not been reached yet Howeverthe damage evolution is fully completed already Table 3summarizes the key results of the simulationWe tested againthe influence of uncertain input parameters for the sameprobability distribution function and scatter as discussedpreviouslyThe observations are somewhat similar comparedto the observations for the Hanchak experiments Howeverthe uncertainties in the input parameters seem to be lesspronounced hereThe largest effect is attributed to parameterset 3 related to the dynamic damage evolution followed byparameter sets 2 and 5 Parameter sets 1 and 4 have nearlya negligible influence on the penetration depth and damagediameter

5 Discussions

We presented a computational framework for predictingthe impact resistance of reinforced concrete structure Thisframework is based on a mesh-free Lagrangian method


Y

Z

X

Figure 3 Concrete under explosive loading

which guarantees the applicability to large deformations asthey occur under high velocity impact explosion dynamicfracture and fragmentation A viscous dynamic damagemodel is exploited which accounts for the strength increaseof concrete at high strain rates and the pressure dependentresponse of concrete Furthermore we consider the influenceof stochastic input parameters

The framework is validated by comparison of numericalpredictions to experimental data (1) the famous Hanchakimpact experiments which measure the residual velocity and(2) our own explosion experiment comparing the penetra-tion depth and damage diameter at the top surface Ourresults show excellent results with the experimental resultsFurthermore we account for stochastic influence Thereforewe classify the parameters of the constitutive model into 5categories related to (1) the initial mechanical properties inthe bulk (2) static damage evolution (3) dynamic damageevolution (4) reduction factor of compressive damage attensile load and vice versa and (5) pressure dependenceBy varying each set of parameters while keeping the othersconstant we show that the key factor influencing our outputis related to parameter set 3 We also reveal that the scatter inthe output is not small

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

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[2] Z P Bazant ldquoWhy continuum damage is nonlocal microme-chanics argumentsrdquo Journal of Engineering Mechanics vol 117no 5 pp 1070ndash1087 1991

[3] Z P Bazant andM Jirasek ldquoNon-local integral formulations ofplasticity and damage survey of progressrdquo Journal of Engineer-ing Mechanics vol 128 no 11 pp 1119ndash1149 2002

[4] W F Chen Constitutive Equations for Engineering Materialsvol 2 of Plasticity and Modeling Elsevier New York NY USA1994

[5] I Carol and Z P Bazant ldquoDamage and plasticity in microplanetheoryrdquo International Journal of Solids and Structures vol 34no 29 pp 3807ndash3835 1997

[6] W Han and B D Reddy Plasticity Mathematical Theory andNumerical Analysis vol 9 of Interdisciplinary Applied Mathe-matics Springer-Verlag New York NY USA 1999

[7] J W Tedesco J C Powell C A Ross and M L HughesldquoA strain-rate-dependent concrete material model for ADINArdquoComputers amp Structures vol 64 no 5-6 pp 1053ndash1067 1997

[8] J F Georgin and J M Reynouard ldquoModeling of structuressubjected to impact concrete behaviour under high strain raterdquoCement and Concrete Composites vol 25 no 1 pp 131ndash1432003

[9] L J Malvar and C A Ross ldquoReview of strain rate effects forconcrete in tensionrdquo Materials Journal vol 95 pp 735ndash7391995

[10] J-F Dube G Pijaudier-Cabot and C La Borderie ldquoRatedependent damage model for concrete in dynamicsrdquo Journal ofEngineering Mechanics vol 122 no 10 pp 939ndash947 1996

[11] G R Johnson and T J Holmquist ldquoA computational consti-tutive model for brittle materials subjected to large strainsrdquo inShock Wave and High-Strain-Rate Phenomena in Materials MA Meyers L E Murr and K P Staudhammer Eds pp 1075ndash1081 Marcel Dekker New York NY USA 1992

[12] G R Johnson andWHCook ldquoFracture characteristics of threemetals subjected to various strains strain rates temperaturesand pressuresrdquo Engineering FractureMechanics vol 21 no 1 pp31ndash48 1985

[13] T Rabczuk J Y Kim E Samaniego and T BelytschkoldquoHomogenization of sandwich structuresrdquo International Journalfor Numerical Methods in Engineering vol 61 no 7 pp 1009ndash1027 2004

[14] T Rabczuk and J Eibl ldquoSimulation of high velocity concretefragmentation using SPHMLSPHrdquo International Journal forNumericalMethods in Engineering vol 56 no 10 pp 1421ndash14442003

[15] T Rabczuk and J Eibl ldquoModelling dynamic failure of concretewith meshfree methodsrdquo International Journal of Impact Engi-neering vol 32 no 11 pp 1878ndash1897 2006

[16] G T Camacho and M Ortiz ldquoComputational modelling ofimpact damage in brittle materialsrdquo International Journal ofSolids and Structures vol 33 no 20ndash22 pp 2899ndash2938 1996

[17] X Teng T Wierzbicki S Hiermaier and I Rohr ldquoNumericalprediction of fracture in the Taylor testrdquo International Journalof Solids and Structures vol 42 no 9-10 pp 2929ndash2948 2005

[18] A Plotzitza T Rabczuk and J Eibl ldquoTechniques for numericalsimulations of concrete slabs for demolishing by blastingrdquoJournal of Engineering Mechanics vol 133 no 5 pp 523ndash5332007

[19] T Boslashrvik A H Clausen M Eriksson T Berstad O SHopperstad and M Langseth ldquoExperimental and numericalstudy on the perforation of AA6005-T6 panelsrdquo InternationalJournal of Impact Engineering vol 32 no 1ndash4 pp 35ndash64 2005

[20] G Ruiz A Pandolfi and M Ortiz ldquoThree-dimensional cohe-sive modeling of dynamic mixed-mode fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 52 no 1-2pp 97ndash120 2001


[21] G Ruiz M Ortiz and A Pandolfi ldquoThree-dimensional finiteelement simulation of the dynamic Brazilian tests on concretecylindersrdquo International Journal for Numerical Methods inEngineering vol 48 no 7 pp 963ndash994 2000

[22] M Polanco-Loria O S Hopperstad T Boslashrvik and T BerstadldquoNumerical predictions of ballistic limits for concrete slabsusing a modified version of the HJC concrete modelrdquo Interna-tional Journal of Impact Engineering vol 35 no 5 pp 290ndash3032008

[23] T Boslashrvik A G Hanssen M Langseth and L OlovssonldquoResponse of structures to planar blast loadsmdasha finite elementengineering approachrdquo Computers and Structures vol 87 no 9-10 pp 507ndash520 2009

[24] W Riedel M Noldgen E Straszligburger K Thoma and EFehling ldquoLocal damage to ultra high performance concretestructures caused by an impact of aircraft engine missilesrdquoNuclear Engineering and Design vol 240 no 10 pp 2633ndash26422010

[25] S R Beissel G R Johnson and C H Popelar ldquoAn element-failure algorithm for dynamic crack propagation in generaldirectionsrdquo Engineering Fracture Mechanics vol 61 no 3-4 pp407ndash425 1998


[27] S Beissel and T Belytschko ldquoNodal integration of the element-free galerkin methodrdquo Computer Methods in Applied Mechanicsand Engineering vol 139 no 1ndash4 pp 49ndash74 1996

[28] L D Libersky P W Randles T C Carney and D L Dickin-son ldquoRecent improvements in SPH modeling of hypervelocityimpactrdquo International Journal of Impact Engineering vol 20 no6ndash10 pp 525ndash532 1997

[29] G A Dilts ldquoMoving-least-squares-particle hydrodynamics IConsistency and stabilityrdquo International Journal for NumericalMethods in Engineering vol 44 no 8 pp 1115ndash1155 1999

[30] R Vignjevic J R Reveles and J Campbell ldquoSph in a totalLagrangian formalismrdquo Computer Modeling in Engineering andSciences vol 14 no 3 pp 181ndash198 2006

[31] T Rabczuk and T Belytschko ldquoAdaptivity for structured mesh-free particle methods in 2D and 3Drdquo International Journal forNumerical Methods in Engineering vol 63 no 11 pp 1559ndash15822005

[32] M A Puso J S Chen E Zywicz andW Elmer ldquoMeshfree andfinite element nodal integrationmethodsrdquo International Journalfor Numerical Methods in Engineering vol 74 no 3 pp 416ndash446 2008

[33] P C Guan S W Chi J S Chen T R Slawson and M JRoth ldquoSemi-Lagrangian reproducing kernel particle methodfor fragment-impact problemsrdquo International Journal of ImpactEngineering vol 38 no 12 pp 1033ndash1047 2011

[34] F Amiri DMillan Y Shen T Rabczuk andMArroyo ldquoPhase-field modeling of fracture in linear thin shellsrdquo Theoretical andApplied Fracture Mechanics vol 69 pp 102ndash109 2014

[35] Y Wu D D Wang and C-T Wu ldquoThree dimensional frag-mentation simulation of concrete structures with a nodallyregularized meshfree methodrdquoTheoretical and Applied FractureMechanics vol 72 no 1 pp 89ndash99 2014

[36] M Hillman J S Chen and S W Chi ldquoStabilized and varia-tionally consistent nodal integration for meshfree modeling ofimpact problemsrdquo Computational Particle Mechanics vol 1 no3 pp 245ndash256 2014

[37] S-W Chi C-H Lee J-S Chen and P-C Guan ldquoA levelset enhanced natural kernel contact algorithm for impact andpenetration modelingrdquo International Journal for NumericalMethods in Engineering vol 102 no 3-4 pp 839ndash866 2015

[38] V P Nguyen T Rabczuk S Bordas and M Duflot ldquoMeshlessmethods a review and computer implementation aspectsrdquoMathematics and Computers in Simulation vol 79 no 3 pp763ndash813 2008

[39] T Rabczuk and T Belytschko ldquoA three-dimensional largedeformation meshfree method for arbitrary evolving cracksrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 29-30 pp 2777ndash2799 2007

[40] M Unosson and L Nilsson ldquoProjectile penetration and per-foration of high performance concrete experimental resultsand macroscopic modellingrdquo International Journal of ImpactEngineering vol 32 no 7 pp 1068ndash1085 2006

[41] A G Petschek and L D Libersky ldquoCylindrical smoothedparticle hydrodynamicsrdquo Journal of Computational Physics vol109 no 1 pp 76ndash83 1993

[42] W Benz and E Asphaug ldquoSimulations of brittle solids usingsmooth particle hydrodynamicsrdquo Computer Physics Communi-cations vol 87 no 1-2 pp 253ndash265 1995

[43] P W Randles and L D Libersky ldquoSmoothed particle hydro-dynamics some recent improvements and applicationsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 139no 1ndash4 pp 375ndash408 1996

[44] L D Libersky A G Petschek T C Carney J R Hipp and FA Allahdadi ldquoHigh strain lagrangian hydrodynamics a three-dimensional SPH code for dynamic material responserdquo Journalof Computational Physics vol 109 no 1 pp 67ndash75 1993

[45] G R Johnson and S R Beissel ldquoNormalized smoothingfunctions for sph impact computationsrdquo International Journalfor Numerical Methods in Engineering vol 39 no 16 pp 2725ndash2741 1996

[46] G R Johnson S R Beissel and R A Stryk ldquoA generalizedparticle algorithm for high velocity impact computationsrdquoComputational Mechanics vol 25 no 2 pp 245ndash256 2000

[47] T Rabczuk J Eibl and L Stempniewski ldquoNumerical analysis ofhigh speed concrete fragmentation using ameshfree lagrangianmethodrdquo Engineering Fracture Mechanics vol 71 no 4ndash6 pp547ndash556 2004

[48] T Rabczuk T Belytschko and S P Xiao ldquoStable particlemethods based on Lagrangian kernelsrdquo Computer Methods inAppliedMechanics and Engineering vol 193 no 12ndash14 pp 1035ndash1063 2004

[49] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[50] T Belytschko and T Black ldquoElastic crack growth in finiteelements with minimal remeshingrdquo International Journal forNumerical Methods in Engineering vol 45 no 5 pp 601ndash6201999

[51] T Rabczuk and P Areias ldquoA meshfree thin shell for arbitraryevolving cracks based on an extrinsic basisrdquoComputerModelingin Engineering amp Sciences vol 16 no 2 pp 115ndash130 2006

[52] T Rabczuk and G Zi ldquoA meshfree method based on thelocal partition of unity for cohesive cracksrdquo ComputationalMechanics vol 39 no 6 pp 743ndash760 2007

[53] T Rabczuk P M A Areias and T Belytschko ldquoA simplifiedmesh-free method for shear bands with cohesive surfacesrdquo


International Journal for Numerical Methods in Engineering vol69 no 5 pp 993ndash1021 2007

[54] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[55] S Bordas T Rabczuk and G Zi ldquoThree-dimensional crackinitiation propagation branching and junction in non-linearmaterials by an extended meshfree method without asymptoticenrichmentrdquo Engineering Fracture Mechanics vol 75 no 5 pp943ndash960 2008

[56] G Zi T Rabczuk and W Wall ldquoExtended meshfree methodswithout branch enrichment for cohesive cracksrdquoComputationalMechanics vol 40 no 2 pp 367ndash382 2007

[57] T Rabczuk S Bordas and G Zi ldquoOn three-dimensionalmodelling of crack growth using partition of unity methodsrdquoComputers and Structures vol 88 no 23-24 pp 1391ndash1411 2010

[58] T Rabczuk and P M A Areias ldquoA new approach for mod-elling slip lines in geological materials with cohesive modelsrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 30 no 11 pp 1159ndash1172 2006

[59] H Talebi C Samaniego E Samaniego and T Rabczuk ldquoOnthe numerical stability and mass-lumping schemes for explicitenriched meshfree methodsrdquo International Journal for Numeri-cal Methods in Engineering vol 89 no 8 pp 1009ndash1027 2012

[60] A Hansbo and P Hansbo ldquoA finite elementmethod for the sim-ulation of strong and weak discontinuities in solid mechanicsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 33ndash35 pp 3523ndash3540 2004

[61] J-H Song P M A Areias and T Belytschko ldquoA methodfor dynamic crack and shear band propagation with phantomnodesrdquo International Journal for Numerical Methods in Engi-neering vol 67 no 6 pp 868ndash893 2006

[62] T Rabczuk G Zi A Gerstenberger and W A Wall ldquoA newcrack tip element for the phantom-node method with arbitrarycohesive cracksrdquo International Journal for NumericalMethods inEngineering vol 75 no 5 pp 577ndash599 2008

[63] T Chau-Dinh G Zi P-S Lee J-H Song and T RabczukldquoPhantom-nodemethod for shell models with arbitrary cracksrdquoComputers amp Structures vol 92-93 pp 242ndash256 2012

[64] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA phantom-nodemethod with edge-based strain smoothing for linear elasticfracture mechanicsrdquo Journal of Applied Mathematics vol 2013Article ID 978026 12 pages 2013

[65] P Areias and T Rabczuk ldquoFinite strain fracture of plates andshells with configurational forces and edge rotationrdquo Interna-tional Journal for Numerical Methods in Engineering vol 94 no12 pp 1099ndash1122 2013

[66] P Areias T Rabczuk and D Dias da Costa ldquoAssumed-metricspherically-interpolated quadrilateral shell elementrdquo Finite Ele-ments in Analysis and Design vol 66 pp 53ndash67 2013

[67] P Areias D Dias-Da-Costa J M Sargado and T RabczukldquoElement-wise algorithm for modeling ductile fracture with therousselier yield functionrdquoComputationalMechanics vol 52 no6 pp 1429ndash1443 2013

[68] P Areias T Rabczuk and D Dias-da-Costa ldquoElement-wisefracture algorithm based on rotation of edgesrdquo EngineeringFracture Mechanics vol 110 pp 113ndash137 2013

[69] P Areias T Rabczuk and P P Camanho ldquoInitially rigid cohe-sive laws and fracture based on edge rotationsrdquo ComputationalMechanics vol 52 no 4 pp 931ndash947 2013

[70] P Areias T Rabczuk and P P Camanho ldquoFinite strain fractureof 2D problems with injected anisotropic softening elementsrdquoTheoretical and Applied Fracture Mechanics vol 72 pp 50ndash632014

[71] G Liu Y Hu Q Li and Z Zuo ldquoXFEM for thermal crack ofmassive concreterdquo Mathematical Problems in Engineering vol2013 Article ID 343842 9 pages 2013

[72] T Rabczuk S Bordas and G Zi ldquoA three-dimensional mesh-free method for continuous multiple-crack initiation propa-gation and junction in statics and dynamicsrdquo ComputationalMechanics vol 40 no 3 pp 473ndash495 2007

[73] T Rabczuk and T Belytschko ldquoCracking particles a simplifiedmeshfree method for arbitrary evolving cracksrdquo InternationalJournal for Numerical Methods in Engineering vol 61 no 13 pp2316ndash2343 2004

[74] T Rabczuk J-H Song andT Belytschko ldquoSimulations of insta-bility in dynamic fracture by the cracking particles methodrdquoEngineering FractureMechanics vol 76 no 6 pp 730ndash741 2009

[75] H X Wang and S X Wang ldquoAnalysis of dynamic fracturewith cohesive crack segment methodrdquo Computer Modeling inEngineering amp Sciences vol 35 no 3 pp 253ndash274 2008

[76] N Sageresan and R Drathi ldquoCrack propagation in concreteusing meshless methodrdquo Computer Modeling in Engineering ampSciences vol 32 no 2 pp 103ndash112 2008

[77] T Rabczuk G Zi S Bordas and H Nguyen-Xuan ldquoA simpleand robust three-dimensional cracking-particle method with-out enrichmentrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 37ndash40 pp 2437ndash2455 2010

[78] T Rabczuk RGracie J-H Song andT Belytschko ldquoImmersedparticle method for fluid-structure interactionrdquo InternationalJournal for Numerical Methods in Engineering vol 81 no 1 pp48ndash71 2010

[79] F Caleyron A Combescure V Faucher and S PotapovldquoDynamic simulation of damage-fracture transition insmoothed particles hydrodynamics shellsrdquo InternationalJournal for Numerical Methods in Engineering vol 90 no 6pp 707ndash738 2012

[80] Y Chuzel-Marmot R Ortiz and A Combescure ldquoThreedimensional SPH-FEM gluing for simulation of fast impacts onconcrete slabsrdquo Computers and Structures vol 89 no 23-24 pp2484ndash2494 2011

[81] T Rabczuk G Zi S Bordas and H Nguyen-Xuan ldquoA geomet-rically non-linear three-dimensional cohesive crackmethod forreinforced concrete structuresrdquo Engineering Fracture Mechan-ics vol 75 no 16 pp 4740ndash4758 2008

[82] T Rabczuk and J Eibl ldquoNumerical analysis of prestressedconcrete beamsusing a coupled element freeGalerkinfinite ele-ment approachrdquo International Journal of Solids and Structuresvol 41 no 3-4 pp 1061ndash1080 2004

[83] T Rabczuk J Akkermann and J Eibl ldquoA numerical model forreinforced concrete structuresrdquo International Journal of Solidsand Structures vol 42 no 5-6 pp 1327ndash1354 2005

[84] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1ndash4 pp 19ndash49 2006

[85] R John and S P Shah ldquoMixed mode fracture of concretesubjected to impact loadingrdquo ASCE Journal of Structural Engi-neering vol 116 no 3 pp 585ndash602 1990

[86] P Rossi ldquoA physical phenomenon which can explain themechanical behaviour of concrete under high strain ratesrdquoMaterials and Structures vol 24 no 6 pp 422ndash424 1991


[87] M J Forrestal D J Frew S J Hanchak and N S BrarldquoPenetration of grout and concrete targets with ogive-nose steelprojectilesrdquo International Journal of Impact Engineering vol 18no 5 pp 465ndash476 1996

[88] D J Frew S J Hanchak M L Green and M J ForrestalldquoPenetration of concrete targets with ogive-nose steel rodsrdquoInternational Journal of Impact Engineering vol 21 no 6 pp489ndash497 1998

[89] T Boslashrvik A H Clausen O S Hopperstad and M LangsethldquoPerforation of AA5083-H116 aluminium plates with conical-nose steel projectilesmdashexperimental studyrdquo International Jour-nal of Impact Engineering vol 30 no 4 pp 367ndash384 2004

[90] W Riedel M Wicklein and K Thoma ldquoShock properties ofconventional and high strength concrete experimental andmesomechanical analysisrdquo International Journal of Impact Engi-neering vol 35 no 3 pp 155ndash171 2008

[91] P Forquin W Riedel and J Weerheijmh ldquoDynamic testdevices for analyzing the tensile properties of concreterdquo inUnderstanding the Tensile Properties of Concrete pp 137ndash180181e Woodhead Publishing 2013

[92] S J Hanchak M J Forrestal E R Young and J Q EhrgottldquoPerforation of concrete slabs with 48MPa (7 ksi) and 140MPa(20 ksi) unconfined compressive strengthsrdquo International Jour-nal of Impact Engineering vol 12 no 1 pp 1ndash7 1992

[93] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods in Engi-neering vol 37 no 2 pp 229ndash256 1994

[94] N Vu-Bac T Lahmer Y Zhang X Zhuang and T RabczukldquoStochastic predictions of interfacial characteristic of carbonnanotube polyethylene compositesrdquo Composite Part B Engi-neering vol 59 pp 80ndash95 2014

[95] N Vu-Bac T Lahmer H Keitel J Zhao X Zhuang and TRabczuk ldquoStochastic predictions of bulk properties of amor-phous polyethylene based onmolecular dynamics simulationsrdquoMechanics of Materials vol 68 pp 70ndash84 2014

[96] N Vu-Bac M Silani T Lahmer X Zhuang and T RabczukldquoA unified framework for stochastic predictions of mechanicalproperties of polymeric nanocompositesrdquoComputationalMate-rials Science vol 96 pp 520ndash535 2015

[97] M Silani H Talebi S Ziaei-Rad P Kerfriden S P Bordas andT Rabczuk ldquoStochastic modelling of clayepoxy nanocompos-itesrdquo Composite Structures vol 118 pp 241ndash249 2014

[98] H Ghasemi R Rafiee X Zhuang J Muthu and T RabczukldquoUncertainties propagation in metamodel-based probabilis-tic optimization of CNTpolymer composite structure usingstochastic multi-scale modelingrdquo Computational Materials Sci-ence vol 85 pp 295ndash305 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of mesh-free methods and finite elements Therefore onlythe area around the perforation was modelled by a mesh-free method whereas a finite element discretization wasemployed elsewhere reducing computational cost [31 80]Most simulations were based on pure deterministic modelsthat can lead to unrealistic crack patterns as the ones shownin [39] In these simulations the radial main cracks appeartoo close to each other In order to achieve unsymmetricalcrack patterns [81] introduced randomness in the strengthto avoid the unrealistic symmetric patterns they modelledbefore in [82ndash84] However all other parameters remaineddeterministic

Several benchmark experiments have also been con-ducted to test the influence of the thickness of the specimenor the speed of the impactor on the impact resistanceof concrete or reinforced concrete structures [85ndash91] Oneclassical experiment was done by [92] who subjected concrete

structures to impact with different speeds of the impactorThey measured the residual velocity of the impactor Therecent experiments by [40] also report penetration depth

This paper presents one of the first studies on stochasticmodelling of dynamic fracture of concrete Our constitutivemodel is based on the model described in [14] and combinedwith a Lagrangian mesh-free method for the computationalstudies Fracture will be modelled by a simple particle split-ting as described in [77]This methodology will be describedin the first sections before we present computational resultsthat will be compared to the experiments of Hanchak et al[92] We also carried out our own experiments that will serveas validation

2 Governing Equations and Discretization

The governing equation to be solved is the balance of linearmomentum that is given in weak form of the updateddescription of motion by

120575119882int minus 120575119882ext = 120575119882kin 120575119882int = intΩ

120575120598119894119895 120590119894119895119889Ω 120575119882ext = int

Γ119905

120575119906119894119905119894119889Γ + int

Ω

120575119906119894119887119894119889Ω 120575119882kin = int

Ω

984858119906119894119894119889Ω (1)

where 120575119882int 120575119882ext and 120575119882kin denote the internal externaland kinetic energy respectively The domain is denoted byΩ whereas Γ

119905cup Γ119906

= Γ with Γ119905cap Γ119906

= 0 denotes theLipschitz continuous boundary 120598

119894119895and 120590119894119895indicate the strain

tensor and Cauchy stress tensor 119887119894and 119905

119894are the body

forces and tractions respectively 119906119894is the displacement field

and 984858 is the density while 120575 denotes the first variation andthe superimposed dote material time derivatives which areidentical to the partial derivatives in an updated Lagrangianframework

We have employed a stabilised [48] nodally regularised[30] element-free Galerkin method [93] Therefore the dis-cretization of the displacement field is given by

u (x 119905) = sum

119868isinS

119873119868 (x) u119868 (119905) (2)

where u119868denote the nodal parameters which are not equal to

the physical displacement value at that point that is u(x119868) =

u119868 and119873

119868(x) indicate the shape functions They are derived

from minimising a discrete weighted norm as explained indetail for example in [93]

N (x) = p119879 (x)Aminus1 (x)B (x) (3)

where A(x) = sum119899

119868=1119908(X minus X

119868)p(X119868)p119879(X

119868) is called the

moment-matrix and the matrix B(x) is obtained by

B (x)

= [119908 (x minus x1) p (x1) 119908 (x minus x

2) p (x2) sdot sdot sdot 119908 (x minus x

119899) p (x119899)]

(4)

119908(X minus X119868) denoting the kernel function and p119879(X) the

polynomial basis To ensure Galilean invariance we opted

to choose a linear basis p119879(x) = [1 119909 119910] The exponentialweighting function has been employed in all our simulations

Substituting the discretization of the displacement fieldand virtual displacement field into the weak form yields thewell-known equation of motion that is solved for the nodalparameter u

119868(119905) that is stored in the global vectorD

MD = Fext minus Fint (5)

with the mass matrix M = intΩ984858N119879N 119889Ω and the vector of

external and internal forces

Fext = intΩ

984858N119879b 119889Ω + intΓ119905

N119879t 119889Γ

Fint = intΩ

B119879120590119889Ω(6)

where N and B are matrices containing the mesh-free shapefunctions and their spatial derivatives respectively Theintegrals are evaluated by stabilised nodal integration whichensures bending exactness in a Galerkin method The use ofan updated Lagrangian kernel formulation as suggested in[39] ensures the stability of the method while maintainingthe applicability to extremely large deformations needed fordynamic fracture and fragmentation

3 Constitutive Model

We have used a modified version of the constitutive modelpresented in [14] Instead of using a viscous damage-plasticitymodel we removed the plasticity part of this model asit adds material parameters that are difficult to calibrateFurthermore preliminary studies indicated that the plastic


178mm

Thickness 610mm

610mm



4 Results



of our university



0 119890119889 and


eters are 119890119889and 119892



1199051and 1198881199052 they




Poisson ratio022

Density [gmm3]2700

1198900

119890119889

119892119889

1198881199051

1198881199052

00001 00003 23 0032 0551198881

1198882

1198883

1198884

001234 00252 07821 03464119903119905

119903119888

12 188119886V 119887V 119890V 119890V119905ℎ

07 34 002 0007























500mm

Thickness 1200mm

1200mm



5 Discussions



Y

Z

X






References














































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










178mm

Thickness 610mm

610mm



4 Results



of our university



0 119890119889 and


eters are 119890119889and 119892



1199051and 1198881199052 they




Poisson ratio022

Density [gmm3]2700

1198900

119890119889

119892119889

1198881199051

1198881199052

00001 00003 23 0032 0551198881

1198882

1198883

1198884

001234 00252 07821 03464119903119905

119903119888

12 188119886V 119887V 119890V 119890V119905ℎ

07 34 002 0007























500mm

Thickness 1200mm

1200mm



5 Discussions



Y

Z

X






References














































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















500mm

Thickness 1200mm

1200mm



5 Discussions



Y

Z

X






References














































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Y

Z

X






References














































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Impact Resistance of Concrete...

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