Analysis and modelling of the cohesion strength of ... · and Forquin, 2010, 2011). Moreover,...

International Journal of Solids and Structures 51 (2014) 2559–2574

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International Journal of Solids and Structures

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Analysis and modelling of the cohesion strength of concrete at highstrain-rates

http://dx.doi.org/10.1016/j.ijsolstr.2014.01.0230020-7683/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +33 4 76 82 52 48; fax: +33 4 76 82 70 43.E-mail address: [email protected] (P. Forquin).

B. Erzar a, P. Forquin b,⇑a CEA, DAM, GRAMAT, F-46500 Gramat, Franceb Laboratoire Sols, Solides, Structures, Risques, Univ. Grenoble Alpes, BP53, 38041 Grenoble, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 May 2013Received in revised form 18 December 2013Available online 15 February 2014

Keywords:Hopkinson barsDynamic fractureSpalling testCohesion strengthDamage model

With the exponential increase of computational power, numerical simulations are more and more used tomodel the response of concrete structures subjected to dynamic loadings such as detonation near a con-crete structural element or projectile-impact. Such loadings lead to intense damage modes resulting fromhigh strain-rate tensile loadings in the concrete structure. However, the modelling of the post-peak ten-sile response of concrete still remains difficult due to the lack of experimental data at high strain-rates.This work aims at improving the modelling of the softening behaviour of concrete based on the followingstatement: despite the propagation of unstable cracks in the tested specimen cohesion strength exists inthe vicinity of triggered cracks and is driving the whole softening behaviour of concrete. This statement isjustified in the present work by means of experiments and Monte-Carlo calculations: firstly, concretesamples have been subjected to a dynamic tensile loading by means of spalling experiments. Severalspecimens have been recovered in a damaged but unbroken state and have been subsequently loadedin quasi-static tensile experiments to characterise the residual strength and damage level in the sample.In addition, Monte-Carlo simulations have been conducted to clarify the possible influence of cohesionstrength in the vicinity of cracks. Finally, the DFH (Denoual–Forquin–Hild) anisotropic damage modelhas been adapted to take into account the cohesion strength in the damaged zone and to describe thesoftening behaviour of concrete. Numerical simulations of experiments conducted on dry and saturatedsamples at different levels of loading-rate illustrate the new capability of the model.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Concrete is the main material used in civil engineering allaround the world. However many aspects of its behaviour still be-long to the research area. The tensile behaviour is one of them.Although it is the Achilles’ heel of concrete structures loaded byexplosion or projectile-impact, few experimental works have re-ported dynamic tensile behaviour in terms of stress–strain curvesat high strain-rates. This lack of reliable test data affects the qualityof modelling and limits the accuracy of numerical predictions tocalculate the response of a concrete structure subjected to an acci-dent or an intentional attack.

At high rates of loading, the spalling technique is the mostadapted tool to characterise the tensile strength of brittle materi-als. ‘‘Spalling fracture’’ is based on the reflection of a compressivewave on a free boundary (or an impedance discontinuity). Thedynamic tensile loading induced in the core of the sample leads

to the onset, the growth and the coalescence of multiple cracks(Antoun et al., 2003; Forquin and Erzar, 2010). This kind of fracturemode can be observed during the impact of a target by a projectile(Meyers, 1994; Li et al., 2005). Nevertheless this technique ismostly used for identifying accurately the dynamic tensile strengthof materials, and a derivation of the classical split Hopkinson Pres-sure Bars device has been employed to identify the strain-rate sen-sitivity of the tensile strength of concrete (Schuler et al., 2006;Schuler and Hansson, 2006; Erzar and Forquin, 2010; Weerheijmand Van Doormaal, 2007). In parallel several works (Bazant,2002; Bazant and Planas, 1998; Zielinski, 1982; Weerheijm,1992; Cadoni et al., 2009) focused on the post-peak behaviour ofconcrete. Hillerborg et al. (1976) proposed a cohesive model to de-scribe the propagation of cracks in concrete. In this approach, it isassumed that the propagation of crack is not the only zone whereenergy is consumed: indeed, presence of inclusions induces a cohe-sive zone behind the crack front since aggregates are responsiblefor bridging phenomena locally resisting to crack opening(Fig. 1). This explanation is close to the idea of fracture processzone experimentally studied by Mihashi and Nomura (1996), Shah

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Aggregates

Crack �p

Cemen��ous matrix

Fig. 1. Bridging phenomenon in concrete.

2560 B. Erzar, P. Forquin / International Journal of Solids and Structures 51 (2014) 2559–2574

and Kishen (2010) and Hu and Duan(2004). Likewise, Zhang et al.(2010) presented 3-points bending tests of high performance con-crete. Specimens were notched and instrumented with gaugesglued at 10 mm of the expected crack path. These tests allowed fol-lowing the crack front (when tensile strain reaches its maximum)and the end of the cohesive zone (when strain drops to zero).

In previous works, Erzar and Forquin have investigated theinfluence of strain-rate and water content on the maximum tensilestrength and cracking pattern of concrete by means of spallingtests and edge-on impact tests (Erzar and Forquin, 2010, 2011).The present work aims at quantitatively evaluating the cohesion(post-peak) strength of a common concrete and a micro-concreteand proposing a modelling of their softening behaviour. To achievethis goal, several spalling tests have been performed to damagewithout breaking the specimen. Subsequently, the recovered sam-ple has been subjected to a direct tensile test to determine theirresidual strength. The second part of this work focuses on model-ling of the dynamic tensile behaviour of concrete including cohe-sion strength in obscured zones. In both experimental andmodelling parts, a particular attention is accorded to the role offree water content as recent studies have reported the influenceof this parameter on quasi-static and dynamic strength of concrete.Finally the proposed model is compared to experimental data pro-vided by spalling tests.

2. Experimental study of tensile strength of concrete at highstrain-rates

2.1. Concrete grades

Two types of concrete with very different mesostructures havebeen tested. On the one hand, the MB50 micro-concrete, providingan aggregate distribution similar to a standard concrete but at alower scale, is an interesting candidate for dynamic testing at labo-ratory scale. This material has already been characterised in directtension at low regime (10�6 to 1 s�1) by Toutlemonde (1994) andunder dynamic confined compression (Forquin et al., 2008; 2010).On the other hand, the R30A7 concrete is closer to a standardconcrete with a compressive strength of about 30 MPa and siliceous

Table 1Composition and main mechanical properties of MB50 and R30A7 concrete grades.

Composition MB50(Bernier and Dalle,1998)

R30A7(Vu et al.,2009)

Aggregates (kg/m3) 0 1008Sand (kg/m3) 1783 838Cement (kg/m3) 400 263Water (kg/m3) 200 169Admixture (kg/m3) 12 0Water/cement 0.5 0.64Maximum aggregate size

(mm)2 8

Main mechanical propertiesCompressive strength

(MPa)70 29

Tensile strength (MPa) 3 3.6

aggregates up to 8 mm diameter (Vu et al., 2009). The compositionand main mechanical properties are gathered in Table 1.

A comparison of the results obtained with both microstructuresallows assessing the influence of aggregate size on the dynamictensile response of concrete at high strain-rates.

2.2. Spalling tests performed on concrete specimens (Erzar andForquin, 2010, 2011)

Among the techniques used to investigate the tensile behaviourof concrete at strain-rates above a ten of 1/s, the spalling techniquefigures prominently. Derived from the classical SHPB technique,the concrete cylinder is placed at the end of the input bar and animpedance discontinuity is ensured at the other side (cf. Fig. 2).A small detonation (Weerheijm and Van Doormaal, 2007) or aprojectile-impact (Schuler et al., 2006; Klepaczko and Brara,2001; Erzar and Forquin, 2010) induces a compressive loading thatpropagates through the Hopkinson bar until reaching the bar-spec-imen interface. Part of this wave is reflected whereas the majorpart of the loading is transmitted to the sample. When this pulsereaches the free opposite face, it is reflected as a tensile loadingleading to damaging the material.

The instrumentation and data processing used by Erzar and For-quin are described in details in (Erzar and Forquin, 2010, 2011).Combining data gathered from the gauges glued on the specimenand laser interferometer pointed out on the rear surface of thespecimen (Fig. 2) allows identifying the wave speed, the dynamicYoung’s modulus, the strain-rate at failure and the dynamic tensilestrength (Schuler et al., 2006; Schuler and Hansson, 2006; Erzarand Forquin, 2010, 2011). Moreover, thanks to finite-element sim-ulations, spherical-cap ended projectiles have been optimised toobtain more homogeneous tensile stress and strain-rate fields inthe specimen than with flat-end projectile (Erzar and Forquin,2010). An ultra-high speed camera is used to visualise the frag-mentation of the concrete sample. More recently a new methodol-ogy was proposed by Pierron and Forquin (2012) to identify theYoung modulus and the stress field based on the use of the VirtualFields Method.

More than 40 spalling tests have been performed on wet anddry specimens of MB50 and R30A7 concretes. As shown inFig. 3(a), a difference of about 3 MPa is observed in spall strengthbetween dry and wet specimens. Fig. 3(b) presents the velocityprofile of two spalling tests conducted at the same impact velocityon the micro-concrete. Firstly, it is observed that free-water has aninfluence on the level of velocity rebound which is directly linkedto the dynamic strength of the material (Novikov et al., 1966). Thetensile strength of the concrete specimen increases due to the vis-cosity of water. Secondly, the post-rebound curve, revealing thetensile post-peak behaviour of the material (Kanel, 2010), is largelyaffected too (Fig. 3(b)). Indeed, a sharp rebound and a series ofoscillations are observed with wet samples due to the entrapmentof waves in the spall. In contrast, supposedly due to a loss of elasticenergy in phenomena of micro-cracks opening and closing, dryconcrete provides a plateau after the rebound. In consequence,the experimental data highlights to what extent the free-wateraffects the dynamic strength as well as the post-peak behaviourof concrete at high rates of strain.

To complete the analysis, post mortem investigations have beenconducted on the tested specimens. The recovered samples wereinfiltrated with a hyperfluid resin, then cut and polished to revealthe inner cracks network. This post-mortem analysis shows a largenumber of cracks perpendicular to the specimen axis which cannotbe observed with a classical observation of the external face (cf.Fig. 4).

Moreover, the fracture planes are mainly located in the cemen-titious matrix: cracks had circumvented the aggregates. Although

Compressed air chamber

Launch tubeHopkinson bar

Ø = 46 mm, L = 1200 mm

High-speed camera andintense ligh�ng device

Laser probe

Concrete specimenØ = 46 mm, L = 140 mm

Fig. 2. Experimental configuration used for spalling tests (Erzar and Forquin, 2010, 2011).

(a) (b)

6

8

10

12

14

16

18

20 50 100 200

R30A7 concrete - WetR30A7 concrete - DryMicro-concrete - WetMicro-concrete - Dry

Strain-rate [s-1]

Dyn

amic

tens

ile s

tren

gth

[MPa

]

0

2

4

6

8

10

12

14

0 50 100 150 200

15w Test - Wet MB5022d Test - Dry MB50

Free

sur

face

vel

ocity

[m/s

]

Time [µs]

Fig. 3. Experimental data of spalling tests. (a) Dynamic tensile strength of MB50 and R30A7 concretes vs strain-rate, (b) free-surface velocity profiles recorded on wet and dryconcrete specimen.

(a)

(b) Free surface

Contact surface with Hopkinson bar

Fig. 4. Post mortem analysis of (a) a MB50 wet specimen (15w test) and (b) a R30A7 dry specimen (20d test) tested at 135 s�1.

B. Erzar, P. Forquin / International Journal of Solids and Structures 51 (2014) 2559–2574 2561

very few aggregates are broken, the dynamic strength is muchhigher than the quasi-static value. In the literature, several authorshave linked strain-rate sensitivity of concrete to transgranular frac-turing of the aggregates at high strain-rates (Zielinski, 1982). How-ever this explanation appears no longer sufficient as very few

aggregates were fractured in the present case. Finally post mortemanalyses conducted on both (MB50 and R30A7) concrete samplestested at different loading-rates illustrate that a strain-rate in-crease induces a widest damaged zone and a more intense crackingdensity (Forquin and Erzar, 2010).

(a)

(b) (c)

(d) (e)

-1 10-3

-5 10-4

0

5 10-4

1 10-3

1.5 10-3

2 10-3

-100 -50 0 50 100 150 200

42d test - dry MB50

G - 100 mmG - 60 mmG - 40 mm

Stra

in [-

]

Time [µs]

-5 10-4

0

5 10-4

1 10-3

1.5 10-3

2 10-3

-100 -50 0 50 100 150 200

G - 120 mmG - 60 mmG - 40 mm

Stra

in [-

]

Time [µs]

16d Test - dry R30A7

0

1

2

3

4

5

6

-100 -50 0 50 100 150 200

42d test - dry MB50

Free

sur

face

vel

ocity

[m/s

]

Time [µs]

0

1

2

3

4

5

6

-100 -50 0 50 100 150 200

Free

sur

face

vel

ocity

[m/s

]

Time [µs]


Laser interferometer

Fig. 5. Experimental configuration of (a) the ‘‘42d’’ and the ‘‘16d’’ tests conducted on MB50 concrete, (b) and (c) corresponding strain-gauges history, (d) and (e) free-surfacevelocity recorded by a laser interferometry technique in these two experiments.


2.3. Residual strength of damaged specimens after spalling tests

2.3.1. Spalling damageDuring the experimental campaign, several initial velocities of

the projectile have been used to reach tensile strain-rates in thespecimen ranging from 20 s�1 to 150 s�1. When the projectile

struck the Hopkinson bar at low initial velocity (few m/s), the con-crete specimens were recovered in one piece without any apparentcrack. However, signals extracted from strain-gauges placed at60 mm and 40 mm from the rear face exhibit a strain peak emax fol-lowed by a residual strain eres (Fig. 5). Moreover, the velocity signalrecorded by the laser interferometer presents a pronounced

(a) (b)

0

0.2

0.4

0.6

0.8

1

1.2

0 4 10-5 8 10-5 1.2 10-4

LVDT

G - 60 mmG - 100 mm

G - 40 mm

Stre

ss [M

Pa]

Strain [-]

30 GPa

25 GPa

10 GPa


Fig. 6. (a) Experimental configuration of a quasi-static direct tensile test performedon a dynamically damaged specimen and (b) results obtained for a dry specimen ofR30A7 concrete (‘‘16d’’ test).


rebound suggesting a tensile damage (Fig. 5(d) and (e)). Thus, thestress peak has been reached and an onset of damage is expected.

2.3.2. Direct tensile tests on damaged specimensThe level of damage is difficult to quantify directly from the

damage pattern or from the experimental data. This is why addi-tional quasi-static tensile experiments have been performed onthe spalled specimens. After the spalling tests, unbroken speci-mens have been glued to a specific device (cf. Fig. 6(a)) and furtherloaded in tension at low strain-rate (10�6 s�1). The strain-gaugesused during the dynamic load have been re-used. Moreover LVDTsensors (Linear Variable Differential Transformer) have been addedfor measuring the axial displacement between the plateaus of themachine.

A series of five experiments have been conducted: two of themon wet R30A7 specimens, one on dry R30A7 sample and the otherones on dry and saturated MB50 concrete. The results gatheredfrom an experiment performed on a dry specimen are plotted inFig. 6(b). Depending on the zone where the gauge is glued, differentresponses are obtained: on the one hand, a lower Edamaged ¼ dr=de isnoted in the damaged zones. On the other hand, the signal of gaugeslocated close to the contact surface indicates a Young’s modulussimilar to the initial (undamaged) value identified experimentallyduring the spalling test (G-100 mm in Fig. 6(b)).

A summary of data obtained from these tensile experiments isgathered in Table 2 and in Fig. 7. Table 2 provides several informa-tion concerning spalling tests conducted without breaking thespecimen to pieces. First, the strain-rate of the spalling test isgiven. Only tests conducted at strain-rates up to 60 s�1 endedwithout breaking the sample. Secondly, the dynamic Young’smodulus Edyn is deduced from the wave speed C0. This last data isdirectly measured during the test by measuring the travel time

Table 2Summary of experimental data for specimens dynamically damaged and subsequently su

Spalling data

_e (s�1) Edyn (GPa) rdyn (MPa) edynmax (lst

16d – dry R30A7 33 31.5 7.5 6703w – wet R30A7 60 42.0 11.7 309021w – wet R30A7 28 43.2 10.9 60042d – dry MB50 58 30.4 8.6 159045w – wet MB50 41 39.1 12.9 954

of the compressive wave from the strain gauge to the free-surface.In each experiment reported in Table 2, a velocity rebound is mea-sured on the rear face resulting from the initiation of tensile dam-age in the concrete specimen. According to Novikov et al. (1966)the so-called pullback velocity DVpb (the difference between themaximum velocity reached during the test and the value corre-sponding to the rebound) is related to the dynamic strength ofthe material rdyn thanks to the following acoustic approximationrdyn ¼ 1

2 qC0DVpb, q being the density the tested sample. This ulti-

mate tensile stress is also reported in Table 2. Finally, edynmax and

edynres respectively the maximum and residual strains are identified

directly from a strain-gauge placed over the damaged zone (cf.Fig. 5). The damaged Young’s modulus and the quasi-static residualstrength are deduced from the direct tensile test conducted ondamaged samples.

Fig. 7 correlates the level of the strain peak with the residualstrain in spalling test and the damaged Young’s modulus measuredin quasi-static test: the higher the strain peak reached during thespalling test, the higher the residual strain and, the lower the qua-si-static strength of the damaged sample. Similarly the damagedmodulus is weaker in case of higher residual strains while it isclose to its dynamic (undamaged) value when the residual strainis low. Based on this damaged Young’s modulus the Fig. 7(c) and(d) present the damage level respectively as function of the peakstrain and strain-rate in spalling test. Again, the level of damageis directly linked to the level of peak strain reached during thespalling test. Finally, these ‘‘post-spalling quasi-static tensileexperiments’’ appear as a possible tool to evaluate the level ofdamage induced in the spalling experiments.

3. Modelling of the dynamic tensile behaviour of concrete

Even if the computational power increases exponentially, somekey aspects of the dynamic response of materials are still difficultto model. The dynamic fragmentation of brittle materials has beenwidely investigated during the last decades. Several models havebeen proposed to fill the gap in dynamic conditions. Holmquistet al. (1993) proposed a damage model including plasticity andirreversible compaction under high confining pressure. This formu-lation has been extended by Riedel et al. (1999) to model the resid-ual strength of the material. Based on the model of Mazars (1984),the PRM (Pontiroli–Rouquand–Mazars) model (Pontiroli, 1995;Pontiroli et al., 2010) allows modelling both compressive and ten-sile non-linear behaviour taking into account phenomena like hys-teresis or crack closure. Visco-plastic damage models have beendeveloped and applied to concrete (Sercombe, 1997; Pedersenet al., 2008). Lu and Xu (2004) have proposed a damage modelbased on concepts of initiation, growth and coalescence of micro-cracks. This model is fitted to describe the increase of strength withloading rate. In this work, the DFH (Denoual–Forquin–Hild) aniso-tropic damage model has been extended to concrete. Firstly,Monte-Carlo simulations have been used to assess the macroscopicinfluence of cohesive strength in obscuration volumes atmicroscopic scale. A cohesion law is proposed to describe the

bjected to direct tensile test.

Damaged Young’smodulus (GPa)

Quasi-static residualstrength (MPa)

rain) edynres (lstrain)

136 10 1.041230 1 0.26

90 22 0.98430 2.5 0.40192 9 1.68

(a) (b)

(c) (d)

0

2 10-4

4 10-4

6 10-4

8 10-4

1 10-3

1.2 10-3

1.4 10-3

0 1 10-3 2 10-3 3 10-3 4 10-3

R30A7 - DryR30A7 - WetMB50 - DryMB50 - Wet

ε res [-

]

εmax

[-]

0

10

20

30

40

50

0 1 10-3 2 10-3 3 10-3 4 10-3

R30A7 - DryR30A7 - WetMB50 - DryMB50 - WetAverage E

dyn - Dry

Average Edyn

- Wet

E dam

[GPa

]

εmax

[-]

0

0.2

0.4

0.6

0.8

1

0 1 10-3 2 10-3 3 10-3 4 10-3


1 - E

dam

/ E dy

n [-]

εmax

[-]0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70


1 - E

dam

/ E dy

n [-]

Strain-rate [s-1]

Fig. 7. Experimental results obtained on dynamically damaged sample: (a) Residual strain and (b) damaged elastic modulus function of maximal strain reached during thespalling test, (c) Damage level function of (c) maximal strain and (d) strain-rate of the preceding dynamic experiment.

Table 3DFH model (Forquin and Erzar, 2010) and parameters of the cohesive law for dry andwet microconcrete.

MB50 microconcrete Wet Dry

E (GPa) 38 31m 0.2 0.2q (kg/m3) 2320 2200

Quasi-static increase factor_e0

(s�1) 1e�5 –nQS 0.05 0

Weibull parametersm 10 12rw (MPa) 6.41 9.99Veff (mm3) 55.8 39.9

Parameters related to obscuration volumesn 3 3S 3.74 3.74k 0.38 0.38C0 (m/s) 4050 3750

Cohesive law parametersa 1 1.5rcoh

0 (MPa) 16 7.5

ecoh0

1 � 10�3 3 � 10�3

ncoh 0.6 1_ecoh

0100 100

Obscured defect

Horizon of (P,T)

Fig. 8. Obscuration of a point P at time T resulting from the activation of a criticaldefect in its horizon.


softening behaviour of concrete. It has been implemented in thecommercial finite-element code Abaqus/explicit through a VUMAT

user subroutine. Finally a comparison of the experimental andnumerical results allows evaluating the improvement of the modelapplied to concrete.

3.1. DFH model: a micromechanical approach

The Denoual–Forquin–Hild model (Denoual and Hild, 2000;Forquin and Hild, 2010) is based on the description of micromech-anisms activated during the dynamic fragmentation process. It de-picts the random distribution of defects in the microstructure, theonset, the propagation of unstable cracks and the obscuration ofcritical defects in the vicinity of cracks.

(a) (b)

= ̇ 111 = 1

1−∆ − ̇ ℎ . ∆

2,31 = 2,3

1−∆ + ̇ . ∆

kC0

1

23

1,22 = 1,2

2−∆ − ̇ ℎ . ∆

32 = 3

2−∆ + ̇ . ∆

1,2,33 = 1,2,3

3−∆ − ̇ ℎ . ∆

kC0

AB

C

AB

C

1 = 1−∆ − ̇ ℎ . ∆

,1 = ,

1−∆ + ̇ . ∆

2 = 2−∆ − ̇ ℎ . ∆

,2 = 2−∆ − ̇ ℎ . ∆

AB

3 = 3−∆ − ̇ ℎ . ∆3 = 3−∆ − ̇ ℎ . ∆3 = min ( 3−∆ , 3−∆ ) − ̇ ℎ . ∆

T1

T2

T3

T1

T2

T3

C

Fig. 9. (a) Model B and (b) model C used to assess the influence of dynamic cohesion on the macroscopic response of concrete.

(a) (b)

0

5

10

15

20

0 2 10-4 4 10-4 6 10-4 8 10-4 1 10-3

Macroscopic stress - Model B1 element - model BMacroscopic stress - Model ADifference between Models A and BAnalytical solution - DFH model

Tens

ile s

tres

s [M

Pa]

Strain [-]

Dry microconcrete

ε° = 100/s0

5

10

15

20

0 2 10-4 4 10-4 6 10-4 8 10-4 1 10-3

Dry microconcreteε° = 100/s

Macroscopic stress - Model C1 element - Model CDifference between Models A and CAnalytical solutions - DFH model

Tens

ile s

tres

s [M

Pa]

Strain [-]

Arrival of new obscuration zones

Fig. 10. Macroscopic and microscopic (for 1 element) stress–strain relationships obtained for each cohesive law: (a) comparison between models A and B and (b) betweenmodels A and C.


3.1.1. Distribution of critical defectsWeibull first assimilated the quasi-static fracture of brittle

materials to a statistical process (Weibull, 1939; 1951). In dynamicconditions, the distribution of critical defects is supposed to be thesame than the one identified in quasi-static tests for example in3-points bending experiments (Forquin and Erzar, 2010). Thedensity of critical defects kt is assumed to be a power law functionof the applied stress r:

kt ¼ k0hrir0

� �m

ð1Þ

where h�i is used to define the positive part of �, m is the Weibullmodulus (it reflects the scatter, i.e. the lower m, the higher the scat-ter) and rm

0 =k0 constitutes the Weibull scale parameter. The Weibullparameters of dry and wet MB50 and R30A7 concretes have beenset in previous works (Erzar and Forquin, 2010, 2011). These param-eters are gathered in Table 3.

Moreover, for wet concrete, an increase of tensile strength isobserved for low to intermediate strain-rates, which is theconsequence of the presence of free-water. A simple law has beenproposed (Forquin and Erzar, 2010) to take into account theinfluence of free water on the triggering of cracks:

rwet0 ¼ r0

_e_e0

� �nQS

ð2Þ

_e0 being a reference strain-rate and nQS a parameter fitted to ob-tain a tensile strength doubled over the range 10�6 to 1 s�1, accord-ingly to results of quasi-static experiments conducted byToutlemonde (1994) on concrete and microconcrete.

3.1.2. Propagation of cracksWhen the stress level reaches the critical stress of a defect an

unstable crack propagates in the material at a celerity assumedto be constant and proportional to the wave speed C0:

Vcrack ¼ kC0 ð3Þ

Experimentally, several works have been conducted to charac-terise the crack speed in brittle materials. Dynamic experimentshave reported values ranging from 0.23 to 0.59 C0 (Ravi-chandarand Knauss, 1982; 1984a,b,c,d; Sieradzki and Dienes, 1988; Sharonet al., 1995). Recently, a new testing technique named the rockingspalling test has been proposed to identify the crack speed in con-crete (Forquin and Cheriguène, 2011; Forquin, 2012). According tothis technique a value of k close to 0.32 has been evaluated.

Based on the concept of energy conservation, an analytical solu-tion of k has been reported by Broek (1982) and Kanninen andPopelar (1985). Considering the energy consumed by the growthof an elliptic crack from an initial length of 2a0 to 2a and equatingthis expression to the kinetic energy obtained from displacementfields yields to a crack speed Vcrack ¼ 0:38C0ð1� a0=aÞ. This

(a) (b)

(c) (d)

Fig. 11. Parametric study of the cohesive term influence on the dynamic tensile behaviour of concrete.

T = 30 μs T = 60 μs

T = 75 μs T = 78.45 μs

Loading pulse

Free

surf

ace

Fig. 12. Numerical simulation of 45 h spalling test. Visualisation of the axial stress field (in Pa).


0

4

8

12

16

0 1 10-3 2 10-3 3 10-3 4 10-3

ε° = 50 s-1

ε° = 100 s-1

ε° = 150 s-1

Tens

ile s

tres

s [M

Pa]

Strain [-]

Wet MB50 microconcrete

Fig. 13. Stress–strain curve of saturated MB50 microconcrete at 50, 100 and150 s�1.

(a) (

(c) (d

0

1

2

3

4

5

6

0 50 100 150 200

Experimental dataDFH modelDFH model with cohesion

Free

-sur

face

vel

ocity

[m/s

]

Time [µs]

45w test wet microconcrete

0

2

4

6

8

10

12

14

0 50 100 150 200

15w testWet microconcrete

Experimental data

DFH modelDFH model with cohesion

Free

-sur

face

vel

ocity

[m/s

]

Time [µs]

Fig. 14. Comparison between numerical results and experimental data of spalling te( _e = 130 s�1).


expression tends rapidly towards an asymptote of 0:38C0. Conse-quently, in the fragmentation model, it has been considered thatk ¼ 0:38.

3.1.3. The obscuration phenomenonWhen a crack is initiated, the stress field in the surrounding

matter is affected. The tensile stress is locally released and no crit-ical defect localised in the vicinity of this crack can be activated.The obscuration volume Z0 around the crack increases proportion-ally to the crack length at power n. It is expressed according to therelation (4) (Denoual and Hild, 2000) as illustrated in Fig. 8.

Z0½T � t� ¼ SðV crackðT � tÞÞn; ð4Þ

where T and t denote respectively the current time and the incep-tion time, Z0 denote the cracking velocity, S is a shape parameterdepending on dimension n of the considered domain, i.e.Sn¼3 ¼ 4p=3 for a sphere, Sn¼2 ¼ p for a circle or Sn¼1 ¼ 2 for aone-dimensional problem.

3.1.4. The damage modelThe description of the three main mechanisms activated during

the fragmentation of brittle materials allows defining a non-obscuration probability Pno for a point P at a time T corresponding

b)

)

-0.001

-0.0005

0

0.0005

0.001

0.0015

-60 -40 -20 0 20 40 60 80 100

Exp. data - G100

Exp. data - G60

Exp. data - G40DFH with cohesion - G40

DFH with cohesion - G100


Axi

al s

trai

n [-]

Time [µs]

45w test

-0.002

0

0.002

0.004

0.006

0.008

0.01

-60 -40 -20 0 20 40 60 80 100

Saturation

Exp. data - G60Exp. data - G60bis

Exp. data - G40DFH with cohesion - G40


Axi

al s

trai

n [-]

Time [µs]

15w test

sts performed on wet microconcrete. (a) Test 45w ( _e = 41 s�1) and (b) Test 15w

(a) 45w test - DDFH (= P0 ) (c) 15w test - DDFH (= P0 )

(b) 45w test - (ε°

exp = 41 s-1)(d) 15w test - (ε°

exp = 130 s-1)

(e)

10 mm

Free surface


Max. principal strain

D~

Simula�on of 15w testt = 125μs

Free

surf

ace

Free

surf

ace

Free

surf

ace

Free

surf

ace

Free

surf

ace

Free

surf

ace

D~ D~

Fig. 15. (a)–(c) Probabilities of obscuration and damage patterns predicted by the DFH model including the cohesive term, (e) comparison of damage patterns obtainedexperimentally and numerically (at t = 125 ls) for the 15w test (cracks observed on the sample have been superimposed in white to the numerical patterns).


to the probability of non-existence of a crack in the horizon of thepoint considered (cf. Fig. 8) (Forquin and Hild, 2010):

PnoðP; TÞ ¼Y

horizon of ðP;TÞPi9= ðx; tÞ with Pi

9= ðx; tÞ

¼ exp � @ktðx; tÞdt

dZdt� �

: ð5Þ

Considering a homogeneous stress field for any time t < T, theobscuration probability Po reads (Denoual and Hild, 2000):

P0ðTÞ ¼ 1� PnoðTÞ ¼ 1� exp �Z T

0

dktðtÞdt

Z0ðT � tÞdt� �

: ð6Þ

As new cracks are initiated outside obscured zones, the evolu-tion of the cracking density kb is linked to the density of critical de-fects kt by:

dkcrack

dt¼ ð1� P0Þ

dkt

dt: ð7Þ

According to Lemaître and Chaboche (1985), a damage variablemay be defined as the real density of discontinuities of the matter.In the fragmentation model described herein, the probability ofobscuration may be utilised as damage variable assuming thatthere is no cohesive strength in obscured zones. The growth ofthe obscuration probability Po can be deduced from Eqs. (4) and(6) as:

0

2

4

6

8

10

12

0 1 10-3 2 10-3 3 10-3 4 10-3

ε° = 50 s-1

ε° = 100 s-1

ε° = 150 s-1

Tens

ile s

tres

s [M

Pa]

Strain [-]

Dry MB50 microconcrete

Fig. 16. Stress–strain curves of dry microconcrete at 50, 100 and 150 s�1.


dn�1

dtn�1

11� DDFH

dP0

dt

� �¼ kt½rðtÞ�n!SðkCÞn when

drdt

> 0 and r > 0: ð8Þ

Assuming a constant stress-rate, characteristic parameters can beestablished (Denoual and Hild, 2000). The whole fragmentationprocess ends when the entire volume is obscured: each crack ob-scures an average volume equivalent to the total volume dividedby the total number of cracks:

ktðtcÞZ0ðtcÞ ¼ 1: ð9Þ

The characteristic time tc of the fragmentation process is deducedfrom the preceding equations. The characteristic stress rc as wellas the characteristic density of critical defects kc can be computed(Eq. (11) and (12)):

tc ¼ k�1

m0 r0

� � mmþn

_r� mmþn

1

S1nVcrack

! nmþn

; ð10Þ

rc ¼ _rtc ¼ k�1

m0 r0

� � mmþn _r

S1nVcrack

! nmþn

; ð11Þ

kc ¼ ktðtcÞ ¼ k1m

_rr0

� � mnmþn

S1nVcrack

� ��mnmþn: ð12Þ

Again, considering a constant stress-rate, one may derive an analyt-ical expression of the obscuration probability from Eqs. (6) and(8):

P0 ¼ 1� exp�n!m!

ðmþ nÞ!rrc

� �mþn� �: ð13Þ

In a case of zero cohesion strength on the crack lips a damage var-iable may be defined as the obscured fraction of domain equal toobscuration probability (Dno cohesion ¼ P0) (Denoual and Hild, 2000;Forquin and Hild, 2010). The macroscopic stress is computedaccording to:

Rno cohesion ¼ ð1� Dno cohesionÞr: ð14Þ

The tensile damage being anisotropic in brittle materials, onedamage variable is used for each principal direction:

e11

e22

e33

8><>:9>=>; ¼ 1

E

11�D1

�m �m

�m 11�D2

�m

�m �m 11�D3

26643775

R11

R22

R33

8><>:9>=>;; ð15Þ

where E is the Young’s modulus, m the Poisson’s ratio of the undam-aged material.

3.2. Modelling of cohesive behaviour of concrete

3.2.1. Monte Carlo simulationsThe bridging phenomenon is frequent in composite materials.

Concrete may be compared to composite material since large andhard inclusions are spread all over the cementitious matrix. To as-sess the influence of microscopic cohesive strength on the macro-scopic behaviour of concrete at high strain-rates, Monte-Carlosimulations have been performed using Matlab�. The routine iscomposed of 5 steps. Firstly, a two-dimensional mesh of the do-main considered is created. The second step consists in drawinglots a random failure stress for each element considering a Weibulllaw while using parameters already identified for concrete in a pre-vious study (Forquin and Erzar, 2010). Afterwards, the stress fieldincreases in the whole domain at a constant stress-rate. Whenthe stress reaches the failure stress of one element, the stress isset to zero in this element and in a spherical obscuration volumecentred on this spot. Its radius develops at a constant speed of0:38C0. The simulation is stopped when the whole domain isobscured.

This basic approach has been used to evaluate the dynamic re-sponse of three scenarios (cf. Fig. 9). In the first one, named ‘‘A’’, thestress in each obscuration volume is set to zero corresponding tothe classical assumption made in the DFH model (Forquin and Hild,2010). In the ‘‘B’’ scenario, a constant negative stress-rate is as-sumed for any point M in an obscuration zone starting from themaximum level of stress reached on this spot. In the last one(‘‘C’’), the loading in the obscuration zone is driven by a constantnegative stress-rate starting from the failure stress of the criticaldefect that obscured the point M. In case of overlapping of theobscuration areas, the stress drops to the minimum value of thetwo zones. In this last case, sudden drops of the stress level maybe observed (Fig. 9(b)).

Monte-Carlo simulations have been used to determine thestress–strain relationship in each scenario. A domain 40 �40 mm2 in size is meshed with squared elements (Le = 0.2 mm).The Fig. 10 presents the results obtained with the Weibull param-eters identified for the wet MB50 micro-concrete (Forquin andErzar, 2010) at a constant stress-rate of 100 s�1. In order tofacilitate the comparison between the 3 configurations, the samenegative stress-rate has been considered: _rcoh ¼ �1000 GPa/s.

Finally, according to Fig. 10, the use of cohesion strength in ob-scured areas has a strong influence on the post-peak behaviour atmacroscopic scale. However the number of activated defects re-mains unchanged since no additional defect is triggered in the ob-scured domain. Indeed, in any cases (A, B, C) the stress level isdecreasing in obscuration zones. Furthermore, the three scenariosproviding more or less the same peak stress, it is lastly noted thatsoftening (post-peak) behaviour is mainly driven by cohesion lawwhereas the ultimate tensile strength (peak stress) is mainly re-lated to obscuration probability. Based on these conclusions somemodifications of the DFH model are proposed.

3.2.2. Introduction of a cohesion strengthSimple approaches have been proposed in the literature to

model the post-peak behaviour of concrete. For instance, severalauthors have used a bi-linear or multi-linear softening curve

(a) (b)

(c) 42d test -

(ε°exp = 58 s-1)

(d) 24d test -

(ε°exp = 135 s-1)

0

1

2

3

4

5

6

7

0 50 100 150 200

42d testDry microconcrete

Experimental dataDFH with cohesionDFH model

Free

-sur

face

vel

ocity

[m/s

]

Time [µs]

0

2

4

6

8

10

12

0 50 100 150 200

24d testDry microconcrete

Experimental data

DFH modelDFH model with cohesion

Free

-sur

face

vel

ocity

[m/s

]

Time [µs]

Free

surf

ace

Free

surf

ace

(e)

Simula�on of 24d testt = 125μs


Free surface

Max. principal strain

D~

Free

surf

ace

Free

surf

ace

D~ D~

D~

Fig. 17. Comparison of numerical results with experimental data performed on a dry microconcrete. Velocity profiles of (a) Test 42d(e� = 58 s�1) and (b) Test 24d(e� = 135 s�1), (c) and (d) damaged patterns, (e) comparison of numerical damage pattern (at t = 125 ls) with post mortem analysis in 24d concrete sample (cracks identifiedon sample have been superimposed in white to numerical patterns).


(CEB, 1990; Petersson, 1981; Wittmann et al. 1988; Rokugo et al.,1989) to model the macroscopic response of concrete.

In the present work, cohesion strength is considered in obscura-tion zones for describing the bridging phenomenon in the

micro-cracked matrix. Based on the previous experimental andnumerical analysis an additional term is added to the classical termexpressed in Eq. (14). For each principal direction a new damage var-iable eD may be calculated and the macroscopic stress is expressed as:

0

0.2

0.4

0.6

0.8

1

0 0.001 0.002 0.003 0.004 0.005 0.006

DFH with cohesion - Wet, ε° = 45 s-1

DFH with cohesion - Dry, ε° = 45 s-1

Experimental data - cf. Fig. 7

Dam

age

leve

l [-]

Strain [-]

Fig. 18. Comparison of damage level obtained with the cohesive model withexperimental data of dynamically damaged specimens subjected to direct tensiletest.


R ¼ ð1� eDÞr ¼ ð1� P0Þrþ ðP0ÞarCOH: ð16Þ

The cohesion strength in obscured zones rCOH is expressed as:

rCOHðe; _eÞ ¼ rcoh0 � exp � e

ecoh0

� �ncoh� �

�_e

_ecoh0

� � nnþm

; ð17Þ

where rcoh0 , ecoh

0 , ncoh, _ecoh0 Þ are parameters to be fitted from experi-

mental data. This additional cohesive term rCOH is supposed to bestrain-rate sensitive similarly to the increase of tensile strengthwith strain-rate for which a power law applies (exponentn/(m + n)). The relative influence of each parameter included inthe cohesive term is plotted on Fig. 11. A constant strain-rate of100 s�1 is again assumed and the DFH parameters used for thedry microconcrete come from a previous work (Forquin and Erzar,2010). As in Monte-Carlo simulations, it is noted that additionalcohesive strength has low influence on the stress peak, especiallyfor a ¼ 1:5 or 2. This model has been implemented in Abaqus/explicit through a user material subroutine VUMAT in order toperform numerical simulations with more complex loadings.

3.3. Numerical simulations of spalling tests

3.3.1. Identification of the cohesive behaviour for saturatedmicroconcrete

A series of computations of spalling tests have been conductedwith Abaqus/explicit finite element code. A pressure pulseextracted from experimental data is applied on one end of the cyl-inder. The compressive wave, having a characteristic duration from70 to 80 ls, travels through the specimen until reaching the free-surface (Fig. 12). A tensile loading increases in the core of the spec-imen (Fig. 12, T = 75 ls). Subsequently, the tensile damage inducesa drop of axial stress in the whole sample (Fig. 12, T = 78.4 ls).

Based on the analysis of calculations of spalling tests performedat different strain-rates (40 s�1 for 45w test, 135 s�1 for 15w test),a heuristic determination of parameters for the cohesive law hasbeen carried out. The stress–strain curves of wet microconcreteare plotted in Fig. 13 considering distinct strain-rates. Theexperimental and numerical velocity profiles are compared inFig. 14 including and excluding respectively the term of cohesionstrength. The use of an additional cohesive strength allows improv-ing significantly the consistency between numerical simulationsand experimental data (Fig. 14). The strain signals picked in thesimulation are consistent with gauges measurements.

At the end of the simulation, the obscured zone (Po = 1) spreadsout a large part of concrete samples (Fig. 15(a) and (c)). The newdamage variable ~D is plotted in Fig. 15(b) and (d).

eD ¼ 1� R=r ¼ P0 � ðP0ÞarCOH=r: ð18Þ

Whereas an intense damage is noted at high strain-rate (130 s�1 inthe 15w test, Fig. 15(d)), the damage is less pronounced and less ex-tended at lower strain rate (40 s�1 in the 45w test, Fig. 15(b)).

As illustrated in Fig. 15(e), the damage model including cohe-sion strength is predicting a diffuse damage in addition to severalmajor cracking planes. Moreover, the wideness of the damage zoneis correctly predicted. Therefore, the damage law with the cohesiveterm can be used to model qualitatively (damage patterns) andquantitatively (local signals of gauges and laser interferometer)the dynamic response of the concrete.

3.3.2. Identification of cohesive behaviour for dry microconcreteIn order to identify the cohesive parameters of the dry MB50

concrete, two numerical simulations of spalling tests have beencomputed with the cohesive damage model, one at low impactvelocity (Test 42d, _e = 55 s�1) and one at a higher loading rate(Tests 24d, _e = 145 s�1). Stress–strain curves of dry microconcrete

are plotted in Fig. 16 considering distinct strain-rates. The cohesiveparameters have been iteratively adjusted in order to improve thefitting of the free-surface velocity (Table 3). According to Fig. 17,the cohesive term allows improving substantially the quality ofthe numerical predictions. In the case of the 24d experiment, thedamage is more heterogeneous and the strain is localised in severalmain damage planes. Again, the wideness of the damage zone pre-dicted by the simulation is consistent with experimental observa-tions (24d test) and the heterogeneity of damage in dry concrete ismodelled in a satisfactory way. Moreover, the cohesive strengthhas to be strain-rate sensitive in order to fit properly the free-surface velocity obtained at the highest loading rate.

The model predictions using parameters identified through aninverse approach can be compared to experimental results gath-ered on dynamically damaged specimens then subjected to directtensile tests. The theoretical evolution of the newly defined dam-age variable ~D of the DFH model with cohesion and the value of1� Edam=Edyn experimentally determined are plotted in Fig. 18.The model predictions have been plotted for wet and dry concreteconsidering a constant strain-rate of (45 s�1) corresponding to theaverage value of strain-rates obtained in spalling tests withoutbreaking the specimen. The model and experimental results areconsistent even if the damage kinetic seems to be slightly too rapidat the onset of damage. In the future, experimental data obtainedon damaged samples can be directly used to identify the post-peaktensile behaviour of concrete.

3.3.3. Sensitivity to element sizeThe low sensitivity of the original model (without cohesion

strength) has been discussed in previous works (Denoual and Hild,2000; Forquin and Hild, 2010). In the present work, a series of cal-culations of spalling tests have been carried out considering 3 ele-ment sizes to check the mesh sensitivity of the proposed model(DFH model with cohesion strength). Element sizes of 1, 2 and3 mm have been used. Larger elements have not been consideredas rougher mesh could affect the loading itself by acting as alow-pass filter. In the present case, numerical simulations havebeen performed considering the 45w test. The damage patternand the particle velocities picked on the free-surface of the numer-ical samples are compared in the Fig. 19. Even if strong differencesare observed comparing the DFH model predictions with and

(a)

(b)

0

2

4

6

8

10

12

14

0 50 100 150 200

Le = 1 mm

Le = 2 mm

Le = 3 mm

9

9.5

10

10.5

11

11.5

12

80 100 120 140 160 180 200

Velo

city

[m/s

]

Time [µs]

Le = 2 mm

Le = 3 mm

Free

surf

ace

Free

surf

ace

Le = 1 mm

Free

surf

ace

Fig. 19. Mesh size influence (a) on the free-surface velocity and (b) on the damage pattern – 15w test.


without cohesion (cf. Figs. 14, 15 and 17), mesh sensitivity appearsto play a minor role in the tested range (element size ranging fromLe = 1 mm to Le = 3 mm). Indeed, less than 10% of scatter is ob-served on the residual velocity of the spall. Moreover, the widthof the damage zone is not affected by the spatial discretization.

4. Conclusions

In the present work, the cohesion (post-peak) tensile behaviourof concrete has been investigated by means of spalling experi-ments, quasi-static experiments consecutively to spalling testsand Monte-Carlo calculations.

First, spalling experiments have been performed at strain-ratesranging from 30 to 60 s�1. Whereas one or several fracture planeswere observed at higher strain-rates no fracture plane is visible insamples tested at such low strain-rates (30–60 s�1). However,according to data from strain gauges, the strain level in the centreof the sample exceeds several times the maximum elastic straindefined as the strength to initial Young’s modulus ratio even atlow strain-rates and a residual strain is also noted. In addition,the velocity profile measured on the rear face of the sample pre-sents a rebound indicating tensile damage. Finally, despite an onsetof damage during the spalling test, a ‘‘cohesion strength’’ of theconcrete sample is expected. This statement is confirmed byquasi-static tensile tests performed with the damaged samples of


dry and wet microconcrete and common concrete. The damagedsamples present a lower but non-zero strength compared to thevirgin specimens. A lower ‘‘damaged Young’s modulus’’ (i.e. stressto strain slope) derived from the data of strain gauges is also notedcompared to the initial Young’s modulus. Based on these experi-mental observations, the following statement is proposed: despitethe propagation of cracks in the spalled samples cohesion strengthexists in the vicinity of triggered cracks resulting from bridgingphenomena.

This statement is used in the next parts in Monte-Carlo simula-tions and in an analytical and numerical modelling. Previous workshave shown the ability of the DFH (Denoual–Forquin–Hild) aniso-tropic damage model for describing the increase of spall strengthand cracking density with strain-rate in concrete and microcon-crete. However, none cohesion strength is taken into account inthe model. In the present paper, cohesion strength is assumed inobscured zones in the vicinity of cracks. This assumption is inves-tigated through Monte-Carlo simulations considering three cases:no cohesion strength, a homogeneous decreasing stress level or aconstant negative stress-rate confined in obscured zones. Finally,it is concluded that, even if additional cohesion strength in ob-scured volumes has a very small influence on the macroscopic ulti-mate strength and cracking density, it is actually driving the wholesoftening behaviour of the material.

Based on this analysis a closed form solution of the cohesionstrength is proposed as function of strain and strain-rate. It wasintegrated to the DFH obscuration model and implemented in aFE code as a Vumat subroutine. The parameters are identified bymeans of a heuristic approach based on a series of numerical sim-ulations of spalling tests. Finally, numerical simulations provide agood description of velocity profile of spalling tests performed withdry and wet samples. Moreover the theoretical evolution of dam-age with the proposed model appeared to be consistent withexperimental data extracted from direct tensile tests conductedon dynamically damaged specimens. A mesh size analysis was con-ducted to check the mesh dependency of the modelling.

In subsequent works, additional experiments might be per-formed considering intermediate saturation ratio and a widerrange of strain-rates. More complex loadings (such as impact load-ings) should be investigated to evaluate the advantages providedby the proposed modelling.

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