10th International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-X

G. Pijaudier-Cabot, P. Grassl and C. La Borderie (Eds)

MODELLING THE DYNAMIC RESPONSE OF CONCRETE WITH THEDAMAGE PLASTICITY MODEL CDPM2

PETER GRASSL∗University of Glasgow

Glasgow, UKe-mail: peter.grassl@glasgow.ac.uk

Key words: Fracture, Concrete, Dynamic loading, Reinforcement, Damage-Plasticity

Abstract. Combinations of damage mechanics and the theory of plasticity have shown to be suitablefor modelling concrete’s quasi-brittle response in tension and low-confined compression, as well asthe ductile response in confined compression. In this work, the recently proposed damage-plasticityconstitutive model CDPM2 has been extended to the modelling of plain and reinforced concretesubjected to dynamic loading by making the damage part dependent on the strain rate. The approachis then applied to a dynamic compact tension test and a slab subjected to air blast loading for both ofwhich experimental results have been reported in the literature.

1 INTRODUCTIONConcrete structures subjected to dynamic

loading exhibit a complex nonlinear failure re-sponse which differs significantly from the onedue to quasi-static loading. For instance, struc-tures subjected to impact and blast can ex-hibit localised shear failure for loads, which,if they were applied slowly, would result inbending failure. Furthermore, dynamic load-ing produces compressive shock waves whichcan cause tensile spalling of the concrete cover,once they are reflected at free boundaries.

For improving design approaches for con-crete structures subjected to dynamic loading,numerical techniques are useful since they al-low for elucidating processes, which would notbe easily measured by experiments. However,experiments are required to check that numer-ical techniques can reproduce important physi-cal processes correctly.

Here, numerical techniques consisting of theexplicit finite element method with an extendedversion of the damage-plasticity model CDPM2reported in Grassl et al. (2011) and Grassl

et al. (2013) are investigated. CDPM2 hasbeen shown to provide good agreement withexperimental results of concrete subjected toquasi-static loading in Grassl et al. (2013) andXenos and Grassl (2016). However, the partof CDPM2 for modelling the dynamic responseof concrete, which was originally proposed inGrassl et al. (2011), requires enhancementswhich are presented here.

For modelling the dynamic response ofstructures, an extension of CDPM2 is proposed.The performance of new model is then assessedby modelling a plain concrete compact tensiontest in Ozbolt et al. (2013) and a reinforced con-crete slab subjected to air blast reported in Thi-agarajan and Johnson (2014). The compact ten-sion experiment, in which the influence of ap-plied displacement rates on crack patterns andload capacity was studied, has been analysedbefore using 3D microplane models in Ozboltet al. (2013) and plain strain damage modelsin Pereira et al. (2017). The reinforced con-crete slab was part of a blind simulation compe-tition, which was discussed in Schwer (2014b).

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It was also analysed in Ekstrom (2016) with theold version of CDPM2 proposed in Grassl et al.(2011). This reinforced concrete slab exhibits acomplex response, which is strongly influencedby the properties of the reinforcement.

Each test on its own might not be sufficient toassess the performance of a concrete constitu-tive model. For instance, computationally verydemanding models might be capable of repro-ducing well experimental results of small scaleplain concrete specimens, but not be applicableto three-dimensional analyses of structural con-crete components. On the other hand, the over-all response of structural components might notprovide enough information to discriminate theperformance of the concrete constitutive model.In this paper, it is aimed to analyse both exper-iments with the same calibration strategy to as-sess if the new version of CDPM2 provides sat-isfactory results for both of these experiments.

2 MODELFor the nonlinear finite element approach, an

explicit solution method is applied. Concrete ismodelled using constant strain tetrahedra withthe proposed extension of the damage-plasticitymodel CDPM2 (Grassl et al. 2011, Grassl et al.2013). For the reinforced concrete slab, steel re-inforcement is modelled using frame elementswith the modified Johnson-Cook plasticity ma-terial model (Johnson and Cook 1985). For theinteraction between steel and concrete, a bond-slip law based on Model Code 2010 (CEB-FIP2012) is used. The models are embedded inthe finite element program LS-DYNA. For steeland bond, the models are readily available inthe software. For concrete, the extension ofCDPM2 was incorporated by means of a user-subroutine. In the following sections, the mainfeatures of the constitutive model CDPM2, andthe steel and bond model are discussed.

2.1 CONCRETE MODELIn CDPM2, the stress evaluation is based on

the damage mechanics concept of nominal andeffective stresses. The nominal stress is eval-uated by a combination of damage mechanics

and elasto-plasticity. The effective stress is thestress in the undamaged material, which is de-termined from the elasto-plasticity part alone.For the nominal stress evaluation, tensile andcompressive damage variables are applied topositive and negative components of the prin-cipal effective stress, respectively.

The plasticity part of the model is formulatedin the effective stress space by means of Haigh-Westergaard stress coordinates, which are thevolumetric effective stress σv, the length of thedeviatoric effective stress ρ and the Lode an-gle θ (Jirasek and Bazant 2002). The yieldsurface is based on an extension of the staticstrength envelope in Menetrey and Willam(1995), which is known to reproduce well ex-perimental results of concrete subjected to mul-tiaxial stress states. This static strength enve-lope is characterised by curved meridians anddeviatoric sections varying from almost trian-gular in tension to circular in highly confinedcompression. A hardening function qh is usedto model the evolution of the yield surface inthe pre- and post-peak regimes as shown inFigure 1. Here, these regimes are defined bythe static strength envelope, which forms thepeak. In the pre-peak regime, the yield sur-face is capped both in hydrostatic tension andcompression. At peak, the static strength enve-lope proposed in Menetrey and Willam (1995)is reached, which is open in hydrostatic com-pression. In the post-peak regime, the yield sur-face further extends with the shape being simi-lar to the strength envelope.

The hardening function qh in the post-peakregime is controlled by the hardening modulusHp. The greater the value of Hp, the smalleris the contribution of plasticity in the post-peakregime. In quasi-static simulations, damage isinitiated once the strength envelope is reached.The damage initiation at a point is a functionof maximum equivalent strains in tension andcompression reached in the history. For the dy-namic simulations in the present study, in whichthe strain rate dependence of strength is mod-elled, damage initiation is made dependent onthe strain rate. The greater the strain rate is, the

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4

2

0

2

4

-4 -3 -2 -1 0

(a) (b)

Figure 1: Evolution of the yield surface for varying values of the hardening variable qh (from 0.3 to 2) which is less thanone in the pre-peak and greater than one in the post-peak regime: (a) Deviatoric section for a constant volumetric stressof σv = −fc/3 (b) meridians at θ = π/3 (compression) and θ = 0 (tension). Pre- and post-peak regimes are defined bythe static strength surface (qh = 1), which is shown by a thicker line.

more plastic strains are generated before dam-age is initiated. This is achieved by dividingthe rates of the equivalent strains by a strainrate dependent factor, which is equal to one forquasi-static loading and increases with increas-ing strain rate. Since the plastic part exhibitshardening (controlled by Hp), the strength atwhich damage (and softening) is initiated ex-ceeds the static strength of the envelope. Thismeans that for dynamic loading, the pre- andpost-peak regimes differ from those defined bythe static strength envelope. For this techniqueto produce reasonable results in compression,it is required that the hardening modulus Hp

is set to a large enough value so that the plas-tic strain before the onset of damage remainssmall. For dynamic simulations in which strainrate dependence of strength is considered, therecommended value of the hardening modulusis Hp = 0.5.

The strain rate dependent factors for ten-sion and compression versus the strain rate areshown in Figure 2a. They are based on the ex-pressions in Model Code 10 (CEB-FIP 2012).However, only the initial branch of the expres-sions are used, since the steep branch for highstrain rates is contributed to damage induced in-ertia effects as discussed in Cusatis (2011) andOzbolt et al. (2014). It is assumed that these

effects can be reproduced in finite element sim-ulations automatically. This part of the modeldiffers from the original proposal in Grassl et al.(2011), in which also the steep part was in-cluded.

Once damage is initiated, the response is acombination of the theory of plasticity and dam-age mechanics. Evolution laws for tensile andcompressive damage are formulated as func-tions of positive and negative parts of the prin-cipal effective stress so that tensile and com-pressive softening responses can be describedindependently of each other. The function forthe tensile damage variable is derived froma bilinear stress-crack opening (σ-wc) curve,so that the results of analyses of tensile fail-ure in which strains localise in mesh-dependentregions are independent of the finite elementmesh (Pietruszczak and Mroz 1981, Bazant andOh 1983, Willam et al. 1986). This bilinearcurve is defined by the tensile strength ft atzero crack opening, the stress threshold ft1 atthe crack opening threshold wf1 and the crackopening threshold wf at zero stress. In the crackband approach, the finer the mesh is, the largerthe strain at constant damage is. This mesh de-pendent strain would result in a strong depen-dence of the mesh size on the strain rate ef-fect on strength and fracture energy. Therefore,

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1

1.25

1.5

2

1 × 10-6

1 × 10-4 0.01 1 100

str

ain

ra

te fa

cto

rsα

rt,α

rc

strain rates ε.t, ε

.c [1/s]

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

stress/ft

wc/wf

(a) (b)

Figure 2: Strain rate dependence in CDPM2: (a) strain rate factor in uniaxial tension and compression modified fromCEB-FIP (2012), (b) Bilinear stress-crack opening curve for quasi-static loading and moderate strain rate.

once damage is initiated, the strain rate depen-dent factor which is used to determine strainrate effect is set constant.

The tensile fracture energy of concrete GF isdefined as the area under the stress-crack open-ing curve. Not many experimental studies onthe effect of strain rate on fracture energy areavailable (de Pedraza et al. 2018). Therefore, itis decided to keep the fracture energy indepen-dent of the strain rate. Within CDPM2, this isachieved by dividing the crack opening thresh-olds wf and wf1 by the square of the rate factor,so that the area below the stress-crack openingcurve remains constant (Figure 2b). This ap-proach differs from the original formulation inGrassl et al. (2011), where the crack thresholdwas not modified so that the model predicted astrong dependence of fracture energy on strainrate, which is not supported by experimental re-sults.

For the present bilinear curve, the fractureenergy is the area under stress-crack openingcurve which is GF = (ftwf1 + ft1wf)/2. Forthe default in CDPM2, ft1 = 0.3ft and wf1 =0.15wf , so that GF = 0.225ftwf . Thus, thecrack opening threshold is related to fractureenergy as wf = 4.444GF/ft. In the present fi-nite element analyses, a crack band approach isapplied for which the inelastic strain is deter-mined as εc = wc/he, where he is a measure ofthe element length determined as a function of

the volume of the element as he = β 3√Ve. For

direct tension tests using tetrahedral elements,factor β must be chosen greater than one, oth-erwise the fracture energy obtained in simula-tions will be overestimated (Jirasek and Bauer2012). In Grassl (2016), it was shown that fortetrahedral meshes used in a three-point bend-ing analysis, β = 1.79 provides a good repre-sentation of fracture energy, which is used inthis study. The compressive damage variable islinked to a stress-inelastic strain curve, insteadof a stress-crack opening curve, since the de-formation patterns in the compressive zones ofbending dominated applications are often mesh-size independent (Grassl et al. 2013).

CDPM2 requires many input parameters,which can be divided into groups related to theelastic, plastic and damage parts of the model.In the present work, most of these parametersare set to their default values provided in Grasslet al. (2013), where it was shown that theyprovide a good match with experimental re-sults. Some of the parameters which are di-rectly linked to experimental results, such asdensity ρ, Young’s modulus E, Poisson’s ra-tio ν, tensile strength ft, compressive strengthfc, fracture energy GF were adjusted to matchmaterial data available for the different groupsof analyses. Furthermore, the default value forthe inelastic strain threshold εcf results in a verybrittle response in compression. Therefore, it

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is sometimes required to choose a more ductilecompressive response to avoid premature fail-ure in regions close to supports or applied loadsby choosing a greater value for εcf than the de-fault. This was done in the present study for thetwo tests.

2.2 STEEL CONSTITUTIVE MODELFor the reinforcement steel, the modified

Johnson-Cook constitutive model is used. Thelinear elastic part of the model requires as inputparameters the Young’s modulus E and Pois-son’s ratio ν. In the plastic part, the equation ofthe rate dependent yield strength is

σy =(A+B (εpeff)N

)(1 + C ln ε) (1)

where parameters A, B and N determine thehardening response and C is used to model thestrain rate dependence.

2.3 BOND CONSTITUTIVE MODELThe bond-slip law describes the evolution of

the bond stress with slip so that the force Ftransferred by bond is

F = τb(s)πφhm (2)

Here, the displacement s is the slip, i.e. rel-ative displacement, between the reinforcementand the concrete, φ is the diameter of the rein-forcement and hm is the average length of thereinforcement bar over which bond is acting.The function τb(s) describes the bond-slip law,which is chosen as

τb =

τmax

(s

smax

)0.4

if s < smax

τmax if s ≥ smax

(3)

where τmax is the bond strength and smax is theslip at which the bond strength is reached. Thefirst part of (3) for s < smax is identical to theone in Model Code 2010 (CEB-FIP 2012) forribbed bars with good bond conditions. Thesecond part was chosen here to be constant, asdone previously in Grassl et al. (2018). Accord-ing to the Model Code 2010, the bond strengthis τmax = 2

√fck. Here, the characteristic com-

pressive strength fck is obtained by subtracting8 MPa from the mean compressive strength.

3 ANALYSES AND RESULTS

Two concrete tests subjected to dynamicloading are analysed. In the first set of analy-ses, a compact tension specimen subjected to arange of prescribed displacement rates reportedin Ozbolt et al. (2013) is studied. The exper-iments in Ozbolt et al. (2013) showed that thecrack patterns in this compact tension test de-pended strongly on the displacement rate. Thesecond test is a reinforced concrete slab sub-jected to air blast. The experimental results forthis test were reported in Thiagarajan and John-son (2014). In the following sections, the anal-yses of these tests are described.

3.1 COMPACT TENSION TEST

The first set of analyses is for a plain con-crete compact tension test for which the exper-imental results were reported in Ozbolt et al.(2013). The geometry and loading setup of thetest is shown in Figure 3.

The setup consists of a notched concretespecimen loaded by two steel brackets con-nected to steel bars. The end of one of thebars is supported. To the other end a con-stant displacement rate is applied. In total,four displacement rates of v = 0.01, 0.5, 1.4and 4.3 m/s are simulated in separate analyses.Both concrete specimen and loading bracketsare meshed using tetrahedral elements of threedifferent edge length of he = 20, 10 and 5 mmto check that the model is able to reproduce theexperimental results independent of the meshsize.

The input parameters for this test are deter-mined using Model Code 2010 (CEB-FIP 2012)based on the uniaxial compressive strength fc =53 MPa reported in Ozbolt et al. (2013). Theresulting input parameters are E = 37 GPa,ft = 3.8 MPa, Gf = 149 J/m2. Poisson’s ra-tio is set to ν = 0.2. Furthermore, the strainthreshold for compressive softening is set toεfc = 1 × 10−3 to avoid premature crushing atthe load application point. The other parame-ters are set to their default values for the strainrate dependent model withHp = 0.5. The value

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concrete

steel

Reaction

Load

Figure 3: Geometry and loading setup of the compact tension test according to Ozbolt et al. (2013).

of the fracture energy GF determined from theModel Code 2010 is greater than the value usedin the analyses reported in Ozbolt et al. (2013).

In Figure 4, the reaction force versus timeof the analyses for the four displacement ratesand three meshes are shown and compared tothe available experimental results for v = 1.4and v = 4.3 m/s. The experimental curves areshifted along the time axis, so that the times ofthe first peak for analyses and experiments coin-cide with the one of the experiments. The modelprovides mesh insensitive results for the firstpart of the analyses. There is also a reasonableagreement with the experimental results. In par-ticular, for the highest displacement rate, the ex-perimental results agree very well both with re-gard to the reaction force and the time evolu-tion of this force. The model predicts also wellthe rate dependence of the maximum reactionforce. For the quasi-static case (v = 0.01 m/s),the maximum reaction force is 2.52 kN. For thehighest displacement rate (v = 4.3 m/s), thisvalue increases to 5.32 kN.

In the analyses, cracks are modelled as re-gions of localised strains. In Figure 5, the crackpatterns for the fine meshes for the four dis-

placement rates are shown in the form of max-imum strain contour plots. For v = 0.5, 1.4and 4.3 m/s, the experimental crack patterns re-ported in Ozbolt et al. (2013) are shown aswell.

The crack patterns obtained in the modelare strongly influenced by the displacementrate. For the smallest displacement rate (v =0.01 m/s), the crack is aligned perpendicular tothe loading direction as expected from a quasi-static compact tension test. As the displace-ment rate increases, the crack direction devi-ates slightly from this perpendicular direction.For v = 1.4 m/s crack branching occurs andfor v = 4.3 m/s, two cracks start at the notchtip. These numerical crack patterns are simi-lar to those observed in experiments. However,in the experiments, crack branching only occursfor v = 4.3 m/s. It is expected that fracture en-ergy has a strong effect on the displacement rateat which crack branching is observed. There-fore, it is planned to investigate further the in-fluence of fracture energy on the crack patternsin these tests.

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]

time [ms]

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FineMediumCoarse

Exp

(c) (d)

Figure 4: Reaction force versus time for the compact tension test with velocities (a) v = 0.01, (b) v = 0.5, (c) v = 1.4and (d) 4.3 m/s. The experimental results are from Ozbolt et al. (2013).

Reaction

Loading

Reaction

Loading

Reaction

Loading

Reaction

Loading

(a) (b) (c) (d)

Figure 5: Crack patterns for (a) v = 0.01, (b) v = 0.5, (c) v = 1.4 and (d) 4.3 m/s for the fine mesh in the form ofmaximum strain contour plots. Here, black indicates a maximum principal strain which corresponds to crack openingsgreater than 0.1 mm. Experimental crack patterns are shown in blue.

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3.2 REINFORCED CONCRETE SLABThe second test consists of a reinforced con-

crete slab subjected to air blast for which theexperimental results were reported in Thiagara-jan and Johnson (2014). This setup was part ofa blind simulation competition which was dis-cussed in detail in Schwer (2014a) and was alsoanalysed with the old version of CDPM2 in Ek-strom (2016). The slab has a two-directional re-inforcement arrangement as shown in Figure 6.In the experiment, the slab was positioned verti-cally in a test frame, with the side which is awayfrom the pressure resting on a support frame.This support frame is modelled in the analysesby the supporting beams shown in Figure 6. Toavoid that the slab falls out of the frame dur-ing the experiment, it was kept in position bytwo additional support beams on the pressureside. In Thiagarajan and Johnson (2014), it wasstated that gap between the additional supportbeams on the pressure side and the slab was solarge that it could be assumed that the slab wassimply supported in the experiments. There-fore, in the analysis setup in Figure 6, the sup-port beams on the pressure side are not mod-elled. However, the pressure which is used torepresent the air blast is not applied over thearea in which these additional support beamsare positioned in the experiments.

Concrete is modelled again by tetrahedralconstant strain elements using the new versionof CDPM2 described in section 2.1. Only onemesh size is used for these analyses with an el-ement edge length of 10 mm. This correspondsto the medium mesh size used for the compacttension test in section 3.1. The input parame-ters for CDPM2 are determined based on a uni-axial compressive strength of fc = 34.5 MPareported in Schwer (2014a). Model Code 2010(CEB-FIP 2012) is then used to determine theother material properties asE = 32.5 GPa, ft =2.7 MPa, Gf = 138 J/m2. Poisson’s ratio is setto ν = 0.2. The strain threshold for compres-sive softening is set to εfc = 1 × 10−2 to avoidpremature compressive failure on the pressureside. The other parameters are set to their rec-ommended default values for the strain rate de-

pendent version of CDPM with Hp = 0.5.For the reinforcing steel, circular cross-

section frame elements are used with a diameterof φ = 9.5 mm. The constitutive model for thesteel is the modified Johnson-Cook model de-scribed in section 2.2. The elastic input param-eters are chosen as E = 200 GPa and ν = 0.3.The parameters for the hardening response areset to A = 415 MPa, B = 550 MPa, N = 0.21to match the stress-strain curves of the steel re-ported in Schwer (2014a). The parameter forthe rate dependence of steel is set to C = 0.017.The supports are assumed to be linear-elasticwith E = 200 GPa and ν = 0.3.

The effect of the air blast is modelled by ap-plying a time dependent pressure over the areaof the slab shown in Figure 6. The evolution ofthe pressure with time shown in Figure 7a wastaken from Schwer (2014a).

The results are shown in form of the verticalmid-point deflection at the pressure side of theslab versus time in Figure 7b and crack patternsin Figure 8.

From Figure 7b, it can be seen that theinitial response of the slab obtained in themodel agrees well with the experimental resultsreported in Thiagarajan and Johnson (2014).However, the maximum displacement obtainedwith the model is less than the one recorded inthe experiment.

The crack pattern in Figure 8 at a displace-ment marked in Figure 7b shows tensile cracksat the bottom of the slab, which are generatedat an early state of the analysis. Then, at a laterstage, shear cracks are created which are vis-ible in the side view of the cracks pattern. Ashear failure occurs on the left hand side of theslab. Although the bending cracks agree wellwith experimental results, this shear failure wasnot reported in Thiagarajan and Johnson (2014).In future studies it is intended to investigate thisaspect further.

4 CONCLUSIONSIn this work, the damage-plasticity model

CDPM2 is extended to the modelling of con-crete subjected to dynamic loading and applied

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concrete support

uniform pressure

s

Figure 6: Geometry and loading setup of the single reinforced concrete slab subjected to blast loading according toThiagarajan and Johnson (2014).

0

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re [

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a]

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100

120

0 10 20 30 40 50 60 70 80

dis

pla

ce

me

nt

[mm

]

time [ms]

MediumExp 1Exp 2

(a) (b)

Figure 7: Reinforced concrete slab subjected to air blast: a) Pressure versus time used to model the effect of the air-blast, b) model midpoint displacement versus time compared to experimental results reported in Thiagarajan and Johnson(2014).

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Figure 8: Reinforced concrete slab subjected to air blast: Contour plots of the maximum principal strain representingcrack patterns at the side and bottom of the slab. Here, black shows a maximum principal strain which corresponds tocrack openings greater than 0.5 mm.

to the analyses of a dynamic compact tensiontest and a reinforced concrete slab subjected toair blast. The new version of CPDM2 repre-sents overall well the rate dependent crack pat-terns observed in the experiments of the com-pact tension test. The reaction versus time re-sponse is independent of the mesh size for thefirst part of the analysis. For the reinforced con-crete slab, the bending cracks and initial dis-placement versus time response observed in theexperiments are well represented. However, themaximum deflection of the slab is underesti-mated. Also, the model predicts a shear failurewhich has not been reported in the experiments.

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ory for fracture of concrete. Materials and Struc-tures 16, 155–177.

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de Pedraza, V. R., F. Galvez, & D. C. Franco (2018).Measurement of fracture energy of concrete athigh strain rates. In EPJ Web of Conferences,

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blast loading. Licentiate thesis, Chalmers Uni-versity of Technology.

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Johnson, G. R. & W. H. Cook (1985). Fracture char-acteristics of three metals subjected to variousstrains, strain rates, temperatures and pressures.Engineering fracture mechanics 21(1), 31–48.

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ods in Engineering 17(3), 327–334.Schwer, L. (2014a). Blind blast simulation: Predic-

tion of air blast loaded concrete slab response us-ing six ls-dyna concrete models. In 13th Interna-tional LS-DYNA Conference, pp. 1–32.

Schwer, L. (2014b). Modeling rebar: The forgot-ten sister in reinforced concrete modelling. In13th International LS-DYNA Conference, pp. 1–28. Dynamore.

Thiagarajan, G. & C. F. Johnson (2014). Experi-mental behavior of reinforced concrete slabs sub-jected to shock loading. ACI Structural Jour-nal 111(6), 1407–1417.

Willam, K., N. Bicanic, & S. Sture (1986). Com-posite fracture model for strain-softening andlocalised failure of concrete. In E. Hinton andD. R. J. Owen (Eds.), Computational Modellingof Reinforced Concrete Structures, Swansea, pp.122–153. Pineridge Press.

Xenos, D. & P. Grassl (2016). Modelling the failureof reinforced concrete with nonlocal and crackband approaches using the damage-plasticitymodel CDPM2. Finite Elements in Analysis andDesign 117, 11–20.

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