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

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

SIMULATIONS OF SPLIT HOPKINSON PRESSURE BAR BY DISCRETEMESOSCALE MODEL

JOSEF KVETON∗ AND JAN ELIAS∗

∗Brno University of TechnologyCzech Republic

e-mail: kveton.j@fce.vutbr.cz, jan.elias@vut.cz

Key words: Particle model, Concrete fracture, Strain-rate, Dynamics

Abstract. The contribution presents simulations of concrete fracture under high strain rates. For rela-tively low rates (below 0.1 m/s) rate dependency is attributed mostly to creep phenomenon, whereasfor higher rates the leading phenomenon is inertia. Discrete meso-scale model is used to representmaterial behavior. Thanks to the explicit representation of mesoscale structure, the main inertia effectsshould be captured automatically. However, the inertia due to smaller omitted particles must bephenomenologically represented as a rate dependent component of the constitutive relation. In thepresented study, several model parameters settings are investigated and the results of the numericalsimulations are compared with the experimental evidence.

1 INTRODUCTIONThe concrete fracture is phenomenon stud-

ied in detail for several decades. Its character-istic quasi-brittle and size dependent behaviorbrings complications that make investigationschallenging. Another difficulty is rate depen-dency of the behavior. It is well understoodthat the resistance of material increases withincreasing loading rate [1]. This behavior isattributed to several phenomena, major one athigh loading rates being the inertia. Usual wayto describe the rate effects is via dynamic in-crease factor (DIF).

Experimental data regarding increase incompressive strength are quite abundantly re-ported in literature already since the half of 20th

century, e.g. [12]. On the other hand, data qual-ifying tensile strength of concrete under dif-ferent strain-rates, especially when concerningmore information than simply dynamic tensilestrength, are quite limited.

Using conventional techniques for testing ofconcrete tensile properties (e.g. uni-axial ten-

sile test), one is limited by the presence ofsupports, more precisely by their unsatisfyingtoughness. For example experiments on con-crete L-shape corners or compact tension testscan be found in [15]. These test are performedin dynamic regime under relatively low loadingrates, up to 2.4 m/s.

More convenient technique was reported e.g.in [10] where long concrete bar was first loadedin compression along the bar length, and simul-taneously biaxial compression was applied inradial direction. After that, compressive forcewas released immediately by an explosive (de-crease from pre-stressing force to zero hap-pened during period of 3 · 10−5 s) and as therelaxing wave approached from both sides, itmeets in the middle where tensile failure occurs.

Another technique called Split HopkinsonPressure Bar (SHPB) [2] is based on imposingpressure on a concrete bar that finally breaksin tension after the wave is reflected at therear face into a tensile stress wave. Severaltechniques for estimation of dynamic tensile

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Josef Kveton and Jan Elias

strength by SHPB are reported in literature, e.g.using a distance where failure occurs or varioustechniques based on observation of specimenvelocity field. Simulations of SHPB test by sim-ilar discrete model with rate dependency due toviscous material model are reported in [11].

Large set of SHPB experiments is publishedin [8], including velocity of the rear face of thespecimen. These experimental data are cho-sen for comparison with the presented numer-ical model.

2 MATHEMATICAL MODEL2.1 Spatial domain discretization

Concrete fracture takes place at the scale,where one can distinguish individual aggre-gates. Material is far from homogeneous at thisscale. It is therefore convenient to use somemodel that account for material heterogeneity,such as discrete meso-scale model used here.

The domain is divided into convex polyhe-dral particles that represent larger aggregateswith surrounding cement matrix. Smaller ag-gregates are not considered explicitly, but theireffect is smeared into the constitutive model ofinterparticle interaction. Particle shape is ob-tained from Voronoi tessellation applied on a setof points randomly placed within a volume do-main with a prescribed minimum distance.

2.2 Constitutive relationsInteraction of particles is governed by the

constitutive law that is applied at contact facetsbetween neighboring particles. The contactstrains at facets are obtained from rigid bodykinematics and projected into facet normal (eN )and tangential (eM , eL) directions. Stresses(si) in corresponding directions are then calcu-lated with help two elastic material parameters,

namely meso-scale elastic modulus E0 and tan-gential to normal stiffness ratio α, and damagevariable D.

si = (1 −D)Eiei for i = N,M,L (1)EN = E0, EM,L = αE0 (2)

In linearly elastic regime when D = 0,macroscopic Young’s modulus E and Poisson’sratio ν can be approximately derived throughprinciple of virtual work [6] and then rela-tion between meso and macroscopic parametersreads

E0 =E

1 − 2να =

1 − 4ν

1 + ν(3)

Damage parameter is responsible for nonlin-ear effects. It is evaluated according to paper ofG. Cusatis [4] using additional model param-eters. We consider two of them (mesoscopicfracture energy Gf and tensile strength ft) asgoverning parameters in inelastic regime, theother material constants are derived from thesetwo. The model from [4] is further simplifiedby neglecting confinement effect. Reader inter-ested in detailed description is referred to [5].Anisotropic nature of concrete fracture is cap-tured thanks to random geometry of particlesystem.

2.3 Transient solutionSimulations of material behavior under high

strain-rates bring necessity of dynamic solu-tion. Equations of motion are solved usingan implicit time integration scheme accordingto Newmark [13], then using numerical time-derivatives of accelerations and velocities ac-cording to Eqs. (5) and (6), system of equationsstated in Eq. (4) is obtained

(K +

1

β∆t2M +

γ

β∆tC

)ut+∆t = Ft+∆t + M

(1

β∆t2ut + 1

β∆tut +

(1

2β− 1)ut

)+

+C(

γβ∆t

ut +(γβ− 1)ut + ∆t

2

(γβ− 2)ut

)(4)

ut+∆t = 1β∆t2

(ut+∆t − ut) − 1β∆t

ut −(

12β

− 1)ut (5)

ut+∆t = ut + ∆t (1 − γ) ut + γ∆tut+∆t (6)

2

Josef Kveton and Jan Elias

where M , C and K are mass, damping andstiffness matrices respectively, F and u areloading and displacement vector. Dotted sym-bol denotes time derivative. β and γ are param-eters of the Newmark method. In the presentedmodel, the system is damped only in nonlinearregime by dissipation of energy due to fracture.Additional damping by matrix C is omitted.

2.4 Rate dependencyThe model represents explicitly only the

largest mineral aggregates. It is therefore neces-sary to capture the material behavior under thisscale phenomenologically in constitutive rela-tion. To account for inertia of the interparti-cle material, the constitutive behavior of contactfacets is enriched by dependency on differencein velocities of particles it connects. Follow-ing strain rate dependent function from [3] isadopted and mechanical behavior of every con-tact is then scaled accordingly.

F (e) = 1 + c1arcsinh(el

c0

)(7)

where e is mesoscopic equivalent strain, l is dis-tance between contacting particle centers and c0

and c1 are additional material properties. Ini-tial slope of the softening curve remains un-changed, therefore more energy is dissipated.This accounts for less localized strain in mate-rial volume that is under resolution of the modelcompared to quasi-static fracture.

3 DYNAMIC TENSILE STRENGTH3.1 Split Hopkinson pressure bar

The test setup consists of long metal (usuallysteel or aluminum alloy) bar and relatively shortconcrete cylinder at its end. Metal bar is loadedby impact of a projectile or by an explosive andthe pressure wave propagates along the bar untilit reaches its end. At the contact between metaland concrete, some part of pressure wave is re-flected backwards to the metal bar as a tensilewave and the rest of it is transmitted into theconcrete specimen, where it further propagatesas a pressure wave. When it reaches the rear

face of concrete cylinder, it is reflected as a ten-sile wave and, after reaching the material tensilestrength, the specimen breaks.

To determine the dynamic tensile strengthfrom results of SHPB test, theory derived for1D longitudinal wave propagation according to[14] is usually applied. In [7], the following re-lation is stated.

ft,dyn =1

2ρ c∆Vpb (8)

whereE and ρ are macroscopic elastic modulusand density respectively and ∆Vpb is pullbackvelocity, which is difference between the maxi-mum and “residual” velocity of the rear face ofthe specimen. c is wave velocity, which can be,in case of elastic material analytically achievedfrom the following relation

c =

√E(1 + ν)

ρ(1 + ν)(1 − 2ν)(9)

3.2 Experimental data for comparisonTo validate the model, series of experimen-

tal data has been searched in literature. Test se-ries reported in [7,8] was selected because of itscomplexity and direct measurements of the rearface velocity, which is important for estimationof dynamic tensile strength.

It was a large series of specimens tested us-ing not only SHPB setup, but also tensile testperformed in conventional apparatus to deter-mine material properties under lower and quasi-static strain-rates.

Two tests were selected for comparisonwith results of the developed numerical model.Stress waves transmitted from aluminum alloybar of length 1.2 m reported in [8] are plot-ted in Fig. 1. The maximum strain-rate calcu-lated from the stress waves is 41/s and 94/s forwave 1 and 2 respectively.

Concrete specimen had length L = 140 mmand radius R = 22.5 mm. Specimens weremade of saturated (wet) concrete with the fol-lowing macroscopic parameters: elastic mod-ulus E = 42 GPa, Poisson’s ratio ν = 0.2,density ρ = 2380 kg/m3 and tensile strengthft = 3.7 MPa.

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Josef Kveton and Jan Elias

0 20 40 60 80time [µs]

0

20

40

60

Com

pres

sive

stre

ss[M

Pa]

wave 1wave 2

wave 3

Figure 1: Two stress waves reported in [8], the third waveof half intensity of wave 1 is introduced for preliminarystudy of model behavior.

4 NUMERICAL SIMULATIONS

4.1 Model setup

The geometry of the model is set accordingto experimental data from [8]. Concrete spec-imen is modeled, alloy bar is represented bya stress pressure wave as depicted in Fig. 2.Stress waves used in simulations in followingsections are plotted in Fig. 1. wave 1 and 2 cor-respond to experimental data and wave 3 withhalf intensity of wave 1 is used for preliminarystudy.

σ(t)

L

2R

Figure 2: Visualization of model geometry used in nu-merical simulation of HSB test – concrete cylinder dis-cretized into particles loaded by a stress wave.

4.2 Preliminary studyFocus of this subsection is on behavior of

the numerical model. For this purpose, influ-ence of its input parameters on results is investi-gated. Compressive stress wave imposed on thefront face of the specimen is chosen smaller (seeFig. 1) than in case of those reported in experi-mental series by Erzar & Forquin in [8] in orderto reduce inelastic behavior under compression.Note that even for this reduced pressure wave,the strain-rate reaches value 20/s.

Material parameters for this prelimi-nary study are following: E0 = 70 GPa,α = 0.237, ρ = 2340 kg/m3, ft = 8 MPaand Gf = 36.5 N/m2 (for interpretation, seesec. 2.2). Parameters of rate dependencyof constitutive law are chosen according torecommendations in [3] c0 = 10−5 s−1 andc1 = 5 · 10−2.

The reason for increasing the materialstrength and use of lower pressure intensity isto avoid material damage in stage of compres-sive wave. Even though the strength increasesdue to strain-rate dependency, tensile damageoccurs in transverse direction. This damagecauses energy dissipation leading to reductionof wave intensity and also changes material be-havior. Therefore it strongly affects obtained re-sults.

The rear face velocity in time is plotted inFig. 3 for 6 different material models. At first,elastic model response was calculated. Fromthis simulation, the value of actual wave speedwas obtained as cact = 4340 m/s. Then, theinelastic reference simulation was computedwith material parameters mentioned above. Fi-nally, four more material models were con-sidered with (i) fracture energy decreased toone half, (ii) & (iii) tensile strength decreasedto one half and one quarter and (iv) elimi-nated strain rate dependency. For rate indepen-dent constitutive law and simulation with lowertensile strength applied, significant amount ofdamage occurs during propagation of pressurewave which leads to deviation of response fromthe elastic one already before peak velocity isreached.

4

Josef Kveton and Jan Elias

0 50 100 150 200

time [µs]

0

1

2

3

4

5

6

rear

face

velo

city

[m/s

]

vp = 5.70 m/s

vr1 = 4.13 m/s

vr2 = 3.21 m/s

vr3 = 2.83 m/s

vp′ = 5.55 m/s

vr3′ = 3.49 m/s

reference sim.no rate dep.0.5×Gf

0.5× ft

0.25× ft

elastic

Figure 3: Preliminary study of model behavior – influ-ence of individual material properties.

The dynamic tensile strength is estimated us-ing Eq. (8). Looking at the rear face velocityfor reference simulation, it is unclear what valueshould be taken as so-called “pull-back” veloc-ity ∆Vpb. For this purpose, difference betweenpeak (vp) and residual (vr) velocity is calculatedin three variants as shown in Fig.3. The result-ing dynamic strengths can be found in Tab. 1.

i v [m/s] ∆Vpb [m/s] ft,dyn [MPa]vp 5.70vr1 4.13 1.57 7.97vr2 3.21 2.49 12.64vr3 2.83 2.87 14.57vp′ 5.55vr3′ 3.49 2.06 10.31

Table 1: Different ft,dyn according to three variants ofresidual velocity vr

In the first variant, vr1 is taken at the pointwhere nonlinear model response start to devi-ate from the elastic one. The value of dynamictensile strength in this case ft,dyn = 7.97 MPais close to quasi-static strength. In the remain-ing variants, vr2 and vr3 are measured at thefirst significant kink and at the first local mini-mum of pullback velocity corresponding to dy-

namic strength 12.64 and 14.57 MPa. Com-paring these values with stress profile at thetime when maximum tensile stress was reached(Fig. 4), the vr2 variant is more appropriate.

−50 0 50

x coord [mm]

0

5

10

15

σx

[MPa]

σmax = 11.80

time 84 µs

0

2

crac

kop

enin

g[µ

m]

Figure 4: Stress profile along the specimen and crack pat-tern in time of maximum tensile σx for reference simula-tion.

Focusing on response of the model with re-duced fracture energy Gf , there is only onepoint to consider. Note that the beginning ofdeviation from elastic response coincides withreference nonlinear simulation, thus this pointshould be dependent only on value of ma-terial strength used (applying the same rate-dependency parameters). However, the residualvelocity is different and so is the dynamic ten-sile strength according to Eq. (8).

Now let us focus on the responses of modelusing lower value of material tensile strength0.5 × ft and 0.25 × ft. From the graph inFig. 3, one can observe that these curves are de-viating from the elastic one already before thepeak velocity is reached. The trend is empha-sized in case of 25% reference tensile strength.What value should be taken into considerationas residual velocity? The point of deviationfrom the elastic curve does not even make sensein this case and there is no significant change inthe slope of the curve as in case of the referencesimulation, so the value vr3′ in the lowest pointis considered. The resulting dynamic strength

5

Josef Kveton and Jan Elias

is calculated in Tab. 1. Such value is howeverlargely exaggerated, because the real maximumstresses in the specimen are only about 4 Mpa,see Fig 5. We can observe from crack patternat the same figure that at the time of reachingthe maximum tensile stress there is large por-tion of volume already damaged. This dam-age happened during the pressure wave propa-gation and it caused the deviation from the elas-tic response already before the peak velocitywas reached. This corresponds with the recom-mendation reported in [9] which states that oneshould avoid pressures larger than 30% of com-pressive strength.

−50 0 50

x coord [mm]

0

2

4

6

σx

[MPa]

σmax = 3.90

time 84 µs

0.0

2.5

crac

kop

enin

g[µ

m]

Figure 5: Stress profile along the specimen and crack pat-tern in time of maximum tensile σx for simulation with0.25 × ft

4.3 Comparison to experimental dataThe experiments loaded by pressure wave 1

and 2 are simulated and results compared to theexperimental data reported in [8]. Since therelation between macro and mesoscopic elas-tic properties – Eq. (3) – is only approximate,the actual mesoscale elastic modulus was iden-tified from the wave speed and the maximumvelocity of the rear face using wave 1. Theresulting value E0 = 77 GPa is slightly higherthan 70 GPa which would be value obtainedby Eq. (3). This is in agreement [5], becauseEq. (3) underestimatesE0 for positive Poisson’sratios. For verification, results of elastic FEM

simulation with the measured material macro-scopic elastic modulus and Poisson’s ratio (seesec. 3.2) is performed. The difference betweencontinuous and discrete elastic simulation isnegligible for both waves (Fig. 6) and the differ-ence is attributed to different solution methods.The continuous model used explicit time inte-gration, while discrete model integrated in im-plicit scheme which suffers by numerical damp-ing. The remaining material parameters respon-sible for inelastic and rate dependent behaviorare listed in Tab. 2. The model response forboth loading cases along with experimental datafrom [8] is shown in Fig. 6.

Table 2: Parameter values of numerical model.

elastic modulus E0 77 GPatang./normal ratio α 0.1667density ρ 2380 kg/m3

tensile strength ft 3.7 MPafracture energy Gf 36.5 N/m2

rate parameters c0, c1 10−5 s−1, 10−1

It can be observed that the experimental peakvelocity for wave 1 corresponds to the elasticresponse of the model. However, looking at theresponse for wave 2, the model elastic responseof the same material is above the experimentalpeak velocity. It could possibly be explained byinelastic effects occurring in experiments dur-ing the pressure wave propagation, which didnot occur under lower pressure of wave 1.

The responses of nonlinear model deviatesfrom elastic response in both cases, again dueto inelastic effects during compression phase.These effects are magnified when rate depen-dency is neglected. The descending part of thesimulated pullback velocity line is not as steepas reported in experiments. There are multiplemacrocracks created in the model, shown in thebottom part Fig. 6, which corresponds to the ex-perimental evidence from [8].

5 CONCLUSIONSThe presented contribution showed numer-

ical simulations of the SHPB tests. Initially,a study of effects of the main model parameters

6

Josef Kveton and Jan Elias

50 100 150time [µs]

0.0

2.5

5.0

7.5

10.0

12.5

rear

face

velo

city

[m/s

]

wave 1

50 100 150time [µs]

0.0

2.5

5.0

7.5

10.0

12.5

rear

face

velo

city

[m/s

]

wave 2

discrete model

no rate dep.

elastic

elastic FEM

exp. Erzar

0.0

0.5

crac

kop

enin

g[m

m]

crack pattern at the end of simulations

Figure 6: Results of numerical simulations compared to the experimental data from [8].

was investigated. The large effect of inelasticmaterial behavior during pressure wave prop-agation was described. It was also discussedwhat value of residual pullback velocity shouldbe used. The most convenient is according tothis study velocity at the first kink, the first localminimum might provide exaggerated dynamicstrength. Difficulty in determining the residualvelocity is caused by inelastic material behaviorduring the pressure phase as well. We thereforesupport recommendation from [9] to avoid pres-sures larger than 30% of compressive strengthin the SHPB tests.

The comparison of the model response tothe experimental data showed differences dueto excessive fracturing during the compressionphase. The strain rate dependency of the modelconstitutive relation helps to reduce these in-elastic effect, but only partially.

Acknowledgement

Financial support provided by the Czech Sci-ence Foundation under project No. GA19-

12197S is gratefully acknowledged.

REFERENCES[1] CEB-FIP MODEL CODE, chapter 2. Ma-

terial Properties, pages 33–81. 1990.

[2] A. Brara and J.R. Klepaczko. Experi-mental characterization of concrete in dy-namic tension. Mechanics of Materials,38(3):253–267, 2006.

[3] G. Cusatis. Strain-rate effects on concretebehavior. International Journal of ImpactEngineering, 38(4):162–170, 2011.

[4] G. Cusatis and L. Cedolin. Two-scalestudy of concrete fracturing behavior. En-gineering Fracture Mechanics, 74(1-2):3–17, 2007.

[5] Jan Elias. Boundary layer effect on be-havior of discrete models. Materials,10(2):157, 2017.

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Josef Kveton and Jan Elias

[6] J. Elias. Adaptive technique for discretemodels of fracture. International Journalof Solids and Structures, 100–101:376–387, 2016.

[7] B. Erzar and P. Forquin. An experimentalmethod to determine the tensile strengthof concrete at high rates of strain. Experi-mental Mechanics, 50(7):941–955, 2010.

[8] B. Erzar and P. Forquin. Experiments andmesoscopic modelling of dynamic test-ing of concrete. Mechanics of Materials,43(9):505–527, 2011.

[9] P. Forquin, W. Riedel, and J. Weerheijm.Chapter 6 - dynamic test devices for an-alyzing the tensile properties of concrete.In Jaap Weerheijm, editor, Understand-ing the Tensile Properties of Concrete,Woodhead Publishing Series in Civil andStructural Engineering, pages 137 – 181.Woodhead Publishing, 2013.

[10] J. K. Gran, L. Seaman, and Y. M.Gupta. Application of a new technique tostudy the dynamic tensile failure of con-crete. Le Journal de Physique Colloques,46(C5):C5–617–C5–622, 1985.

[11] Young Kwang Hwang, John E. Bolan-der, and Yun Mook Lim. Simulation ofconcrete tensile failure under high load-ing rates using three-dimensional irregularlattice models. Mechanics of Materials,101:136–146, 2016.

[12] F.M. Mellinger and D.L. Birkimer. Mea-surements of stress and strain on cylindri-cal test specimens of rock and concreteunder impact loading. Technical report,Ohio River Div Labs Cincinnati, 1966.

[13] N. Newmark. A method of computationfor structural dynamics. University of Illi-nois, Urbana, 1959.

[14] S.A. Novikov, I.I. Divnov, and A.G.Ivanov. Investigation of the fracture ofsteel, aluminum and copper during explo-sive loading. Fiz Metallov Metalloved,21(4):608–615, 1966.

[15] J. Ozbolt, A. Sharma, and N. Bede. Dy-namic fracture of L and CT concrete spec-imens. Proceedings of the 9th Interna-tional Conference on Fracture Mechan-ics of Concrete and Concrete Structures,2016-5-29.

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