Delft University of Technology

Simulation of dynamic behavior of quasi-brittle materials with new rate dependent damagemodel

Magalhaes Pereira, Luis; Weerheijm, Jaap; Sluijs, Bert

DOI10.21012/FC9.036Publication date2016Document VersionFinal published versionPublished in9th International Conference on Fracture Mechanics of Concrete and Concrete Structures

Citation (APA)Magalhaes Pereira, L., Weerheijm, J., & Sluijs, B. (2016). Simulation of dynamic behavior of quasi-brittlematerials with new rate dependent damage model. In V. Saouma, J. Bolander, & E. Landis (Eds.), 9thInternational Conference on Fracture Mechanics of Concrete and Concrete Structureshttps://doi.org/10.21012/FC9.036Important noteTo cite this publication, please use the final published version (if applicable).Please check the document version above.

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9th International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-9

V. Saouma, J. Bolander, and E. Landis (Eds)

SIMULATION OF DYNAMIC BEHAVIOR OF QUASI-BRITTLEMATERIALS WITH NEW RATE DEPENDENT DAMAGE MODEL

L. Pereira∗ ‡, J. Weerheijm∗ † and L.J. Sluys∗

∗Delft University of TechnologyDelft, the Netherlands

e-mail: L.F.MagalhaesPereira@TUDelft.nl

†TNO Defence, Safety and SecurityRijswijk, the Netherlands

‡Academia da Forca Aerea PortuguesaSintra, Portugal

Key words: Stress-based nonlocal model, Damage, Rate dependency, Dynamic crack-branching

Abstract. In concrete often complex fracture and fragmentation patterns develop when subjectedto high straining loads. The proper simulation of the dynamic cracking process in concrete is crucialfor good predictions of the residual bearing capacity of structures in the risk of being exposed toextraordinary events like explosions, high velocity impacts or earthquakes.

As it is well known, concrete is a highly rate dependent material. Experimental and numericalstudies indicate that the evolution of damage is governed by complex phenomena taking place simul-taneously at different material scales, i.e. micro, meso and macro-scales. Therefore, the constitutivelaw, and its rate dependency, must be adjusted to the level of representation. For a proper phenomeno-logical (macroscopic) representation of the reality, the constitutive law has to explicitly describe allphenomena taking place at the lower material scales. Macro-scale inertia effects are implicitly simu-lated by the equation of motion.

In the current paper, dynamic crack propagation and branching is studied with a new rate-dependentstress-based nonlocal damage model. The definition of rate in the constitutive law is changed to ac-count for the inherent meso-scale structural inertia effects. This is accomplished by a new conceptof effective rate which governs the dynamic delayed response of the material to variations of the de-formation (strain) rate, usually described as micro-inertia effects. The proposed model realisticallysimulates dynamic crack propagation and crack branching phenomena in concrete.

1 Introduction

Extraordinary actions such as blast loadings,high velocity impact and earthquakes are rare,but usually have devastating effects. Thus, mak-ing critical (infra-)structures, such as power-plants, dams, bridges, hospitals, etc., more re-silient is one of the best ways to protect our-selves and our societies from these hazards.Since concrete is a very common construction

material, the development of realistic numeri-cal tools to efficiently simulate its failure be-havior under extreme loading conditions is ofparamount significance.

For example, in case of a high velocity im-pact of a rigid projectile on a concrete struc-tural element, high rate loads are induced inthe target leading to multiple damage modes,such as cratering at the impacted face, spalling

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DOI 10.21012/FC9.036

L. Pereira, J. Weerheijm and L.J. Sluys

at the opposite face and radial cracking. Mostmodels used in practical engineering applica-tion are used to predict the extension of scab-bing and spalling. However, this is clearly in-sufficient to estimate the integrity and residualbearing capacity of concrete structures wherecracking usually spreads over an area much big-ger than the crater. Thus, simulating crack ini-tiation, development and branching is essentialfor a proper representation of dynamic damagedevelopment in concrete [1].

Crack branching in brittle and quasi-brittlematerials under dynamic loading occurs whencrack propagation reaches a critical speed. Thislimit is usually considered to be material depen-dent. In brittle materials the critical crack speedis usually set between 40% and 60% of theRayleigh velocity (CR). For concrete, this limitusually drops to 20% to 40% of CR [2]. Fur-thermore, experimental data (cf. Ravi-Chandra[3]) show that under dynamic loading condi-tions the evolution and bifurcation of cracks isalso highly dependent on the load history andthe geometry of the specimen (boundary con-ditions). Different theories have been devel-oped over the years to explain dynamic frac-ture of materials, in particular the existence of acritical crack propagation speed and consequentbranch phenomena. For example, according toRavi-Chandar [3], the crack propagation speedis limited by the evolution of the dissipationprocesses that occurs over a finite fracture pro-cess zone (FPZ) around the crack tip involvingmicro-cracking and other damage mechanisms.Hence, the resistance to crack propagation re-sults in a limited crack speed dependent onthe inherent material properties that govern thefracture (damage) process ahead of the cracktip. This concept is particularly appealing forcontinuum mechanics modeling because, if thefracturing/damage processes are correctly rep-resented by the constitutive law, crack branch-ing phenomena should be automatically simu-lated without the need for any specific branch-ing criterion. In this contribution we explorethis theory and present a new damage model tosimulate dynamic fracturing in concrete.

As it is well known, quasi-brittle materialsshow a highly rate dependent behavior. Al-though the dynamic behavior of quasi-brittlematerials is far from being completely under-stood, the possible explanations for the ob-served rate effects are: i) the enhanced resis-tance of moisture in the pores [4, 5]; ii) thedynamic redistribution of stresses in the frac-tured zone associated with the material’s weak-ening by micro-cracking; iii) inertia at micro-level which retards the initiation and propaga-tion of micro-cracks [6, 7]; and iv) increase ofmicro-cracking area (damage) [8, 9]. The com-bined effect of these complex phenomena leadsto significant dynamic strength increase [10] as-sociated with a raise of the fracture energy, avisible change of the fracturing patterns and asmall (usually neglected) increase of the ma-terial stiffness. Furthermore, it has also beenobserved that the stress-strain relations in a dy-namic setting change with a variation of strainrates in time, i.e. the load history [11]. Inother words, in case of an abrupt variation ofthe strain rate, its effects are not ’felt’ instan-taneously due to micro dynamic (inertia) phe-nomena. Thus, the common phenomenologi-cal rules which consider the dynamic increasefactor (DIF) depending on an assumed con-stant strain rate history are not adequate to de-scribe the dynamic nature of the material. Newsound approaches based on physical behaviorare needed.

Different damage models have been success-fully used to represent crack branching in brittlematerials (see for example the work of Ha andBabaru [12] and Wolff et al. [13]). In this con-tribution a rate enhanced version of the latestMazars damage model [14] is used to numer-ically study dynamic crack branching in con-crete. An effective rate is defined in order toaccount for the dynamic delayed response ofthe material to variations of the deformation(strain) rate, usually ignored in continuum me-chanics. This new parameter is used to updatethe damage threshold and softening law follow-ing a simple damage delay formulation. Finally,a stress-based nonlocal regularization scheme is

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L. Pereira, J. Weerheijm and L.J. Sluys

added to the formulation to minimize spuriousmesh dependency. After a brief description ofthe material model, a set of dynamic compacttension tests conducted by Ozbolt et al. [15] isused for validation purposes.

2 Stress-based nonlocal damage model2.1 Constitutive law

Continuum damage mechanics has beenwidely and successfully used to model con-crete for a broad range of applications andloading conditions. For the present study, anadapted version of the recently proposed µ dam-age model [14] has been developed and imple-mented in LS-DYNA [16].

A classical definition of isotropic damage isconsidered. The single scalar variable ω repre-sents the material’s stiffness degradation. Thestress tensor is expressed as:

σ = (1− ω)σ with σ = C : ε , (1)

where ε is the strain tensor, C the elastic stiff-ness tensor and σ is the effective stress tensor[17]. The total amount of damage results fromthe combination of tensile (ωt) and compressive(ωc) damage variables following an adapted ver-sion of Lee and Fenves’s formulation [18].

ω = 1− (1− ωt)(1− ωc) (2)

The evolution of both damage variables isderived from two principal equivalent strainscalar quantities, εt and εc, which represent thelocal strain state in tension and compression, re-spectively

εt =0.5

1− 2νIε +

0.5

1 + ν

√3Jε (3)

εc =0.2

1− 2νIε +

1.2

1 + ν

√3Jε , (4)

where ν is the Poisson’s ratio, Iε is the first in-variant of the strain tensor and Jε the secondinvariant of the strain deviator tensor, accordingto:

Iε = ε1 + ε2 + ε3 ,

Jε =1

6[(ε1 − ε2)2 + (ε2 − ε3)2 + (ε3 − ε1)2]

The evolution of the nonlinear response isdirectly related to the growth of two mono-tonic internal variables Yt and Yc, which ac-count for the historical maximum equivalentstrain reached during loading

Yi(t) = max[ri εi, Yi(τ)] for all t ≥ τ (5)

where rt and rc are internal variables derivedfrom the triaxiality factor (r), which providesinformation on the actual loading state [14, 18].This triaxiality factor varies between 1 and 0 intension-compression situations, being 1 in ten-sile and 0 in compressive conditions

r =

∑3I=1〈σI〉∑3I=1 |σI |

, (6)

rt = rα , (7)

rc = (1− r)α (8)

where 〈σI〉 and |σI | are the positive and abso-lute values of the principal effective stresses.The parameter α controls the ”amount” of ten-sile or compressive straining that contributes tothe damage evolution in a traction-compressionstress-state (see fig. 1). This formulation is sim-pler than the one used in the original model, butwithout compromising the representativity ofconcrete’s deviatoric (mixed-mode) response.As it can be seen in fig. 2 both models pro-duce similar 2D failure envelopes, consideringα < 0.1 (black line).

r0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

rt

rc

Figure 1: Variation of rt and rc for α = 0.1

The damage evolution laws for tension (ωt)and compression (ωc) are of the same type as in

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L. Pereira, J. Weerheijm and L.J. Sluys

the original Mazars model [19].

ωi = 1− (1− Ai)Yi0Yi− Ai e−Bi(Yi−Yi0) (9)

where Ai and Bi are material parameters andY0i are the equivalent strain damage thresholds.The index i in the equations should be inter-preted as c for compression and t for tension.

σ1

-1 -0.5 0

σ2

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0FailureYieldMazarsExp.

Figure 2: Biaxial failure envelopes (α = 0.1)compared to Mazars µ damage model [14] and

experimental results by Kupfer et al. [20]

2.2 Regularization modelTo overcome the spurious mesh sensitivity

typical for standard ”local” damage models aregularization scheme is needed. In this study,the stress-based nonlocal formulation presentedby Giry et al. [21] is used. In this enhancedformulation the nonlocal characteristic length(lr) is no longer constant. It varies as a func-tion of the stress state of the neighboring el-ements. The internal length is maximum (lr)when the material is fully stressed (σI = ft)and tends to zero when the material is unloaded.To avoid numerical instabilities, a minimum in-ternal length (lmin) is introduced in the formu-lation. Consequently, the interactions betweenelements decrease close to free boundaries, ge-ometrical discontinuities in their normal direc-tions and parallel to damaged areas. This resultsin a more realistic representation of the evolu-tion of stresses in these situations. For more

information about this model and the compu-tation scheme used for this research the readeris referred to [21–23].

3 Rate-dependent damage modelIt is easily understood that simulating con-

crete as a combinations of its different phases(aggregates, cement paste and voids) some in-herent structural inertia effects being present atmeso-scale are naturally taken into account bythe equation of motion. For example, the re-tardation of micro-cracking at high deformationrates associated to micro and meso-scale iner-tia effects can be analyzed by rate-independentconstitutive theories as long as the material isdiscretized in all its phases [24]. However, ina homogeneous (macro-scale) representation ofthe material this part of the rate effects has to beexplicitly modeled [25]; i.e. the description ofrate effects within the constitutive law must beadjusted to the level of representation [6].

Several analytical, experimental and numer-ical studies demonstrated that the evolution ofmicro-cracks cannot occur arbitrarily fast dueto inertia effects [26–28]. Thus, if damage isused to describe microcracking , its evolutionhas to be retarded in case of high strain rates,i.e. the evolution of damage is a strongly timedependent phenomenon. Moreover, Eibl andSchmidt-Hurtienne [6] suggest that in case ofa sudden drop of the strain rate, the effectivestrength of the material does not decrease in-stantaneously. The material has some kind ofmemory and the effects of rate take some timeto wear-off, i.e. relax. Consequently it has beensuggested that ”the dynamic damage evolutionrequires a strain-history formulation of the dy-namic strength instead of the commonly useddependency on the current strain rate in consti-tutive modeling” (cf. Plotzitza et al. [25]). Inother words, in case of an abrupt variation ofthe strain rate, its effects are not ’felt’ instan-taneously. Consequently, considering only thecurrent strain state to define an ’instantaneous’rate (ε) and derive the corresponding rate ef-fects, as it is usually done, does not adequatelydescribe the complex evolution of stresses in

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L. Pereira, J. Weerheijm and L.J. Sluys

dynamics. Here, we propose a new formulationto incorporate in the constitutive law both therelaxation and retardation effects.

Definition of effective rate - RDifferent enhanced formulations have been

proposed to describe the dynamic damage evo-lution as a viscous and/or a retardation phe-nomenon. First, a viscous-damage formula-tion, similar to a Perzyna-type of viscoplas-ticity, was used to describe the rate of dam-age (see for example [29–31]). Plotzitza et al.[25] proposed a formulation where the evolu-tion of damage depends to some extent on theimmediately preceding strain history. A his-tory function, derived from a simple Maxwellrheological model, is used to retard the evo-lution of damage, representing the resistanceto micro-cracks growth due to inertia effects.Recently, Haussler-Combe and Kuhn [11] pro-posed a model to address both viscosity and re-tarded damage.

Independently of the approach, all thesemodels focused on directly constraining theevolution (i.e. the rate) of damage. In the for-mulation proposed hereafter, instead of chang-ing the actual damage law, an effective rate -R is derived to account for the relaxation andretardation effects. Instead of the typically con-sidered ’instantaneous’ strain rate (ε), the newparameter R is used to determine the strengthand fracture energy in the rate dependent con-stitutive law, i.e. updating Y0 and Bt in eq. 9on the basis of R. It is assumed that the ef-fects of a variation of the strain rate take sometime (λ) to be ’felt’. Simple Maxwell rheologi-cal models govern the evolution of the effectiverate in case of a decrease or increase of the de-formation rate, by the following relaxation andretardation laws, respectively:

R1 = max[ετ e

− t−τλ , εt

]for all t ≥ τ (10)

R2 = min[ετ e

t−τλ , εt

]for all t ≥ τ (11)

where λ is the user defined material’s reactiontime, t the current time and τ the time when ’in-stantaneous’ rate was previously constant (ετ 1).Please note that both mechanisms are relatedto inherent dynamic properties of the material,so λ may be considered as an intrinsic materialproperty.

To better understand this formulation, let usconsider the theoretical examples where a sud-den variation of rate takes place before and afterdamage initiation, represented in fig. 3 by a andb, respectively. As it can be seen in fig. 3 in caseof a sudden drop of ’instantaneous’ rate (ε1), thedecrease of the effective rate (R) is limited bythe relaxation (memory) law. In the completelyopposite situation (ε3), an increase of effectiverate (R) is now limited by the retardation law.This effect is only triggered after damage initia-tion because it is considered to be related to theresistance to microcracking evolution. Finally,as expected, in case of small or no variation ofstraining (ε2) these dynamic mechanisms do notaffect the effective rate (R = ε).

(a) (b)

Figure 3: Maximum allowed evolution ofeffective rate for: a) ω = 0 and b) ω > 0

Damage-delay formulationThe dynamic strength increase is simulated

by a simple damage-delay formulation, similarto the one introduced by Pontiroli et al. [32]. Inthis approach the damage threshold Yt0 is mod-ified as a function of effective rate (R). For ex-

1In our formulation, strain rate (ε) is independently defined for tension and compression as the positive value of therespective equivalent strain (εt and εc) variation in time. For example, in tension we have ε = 〈εt〉.

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L. Pereira, J. Weerheijm and L.J. Sluys

ample, for tension we have:

Y dynt0 = Yt0(1 + η R n) , (12)

where η and n are material constants.The post peak response (softening) of the

material associated with the fracturing processis also updated with rate. By changing param-eter Bt in the damage evolution law (eq. 9), itis possible to control hardening and softeningafter damage initiation.

Bdynt = Bt

[1− δ ln

(R

Y0

)], (13)

where δ is an input parameter and Y0 is the rateafter which the loading is considered dynamic.Consistently with the fib recommendation [10]Y0 = 10−6s−1 is considered.

This evolution of the damage law is meantto describe the variation of fracture energy withrate. It has been experimentally observed that inquasi-static tensile failure conditions, for exam-ple, micro-cracks form across the most stressedareas and rapidly coalesce to a main macro-crack which propagates along the weakest re-gions of the material’s matrix. With increasingdeformation rates, the concentration of micro-cracks around the main crack(s) tend to in-crease. This phenomenon contributes to anoverall increase of the fracture energy repre-sented by eq. 13. It has been experimentallyobserved that the distribution of these micro-cracks across the fracture process zone alsochanges [33]. This phenomenon is indirectlydescribed by the nonlocal length variable usedin the regularization scheme. Thus, it is not sur-prising that calibration of δ is highly dependenton the material characteristic length.

Fig. 4 shows the variation of the stress-strainrelations with rate (ε = const) for concrete withft = 3.8 MPa, as described in tab. 1, consider-ing δ = 0.0 and δ = 0.045. The respective peakstresses are presented in the DIF plot in fig. 5.Analyzing these two figures, one may noticethat a variation of Bdyn

t through δ has a directimpact on the strain-energy and dynamic max-imum strain of the model. Thus, for a proper

calibration of the model one has to consider thecombined effect of eqs. 12 and 13, i.e parame-ters η, n and δ.

ǫ ×10-30 0.5 1 1.5 2

σ1

0

5

10

15Static0.11.10.30.100.

Figure 4: σ − ε relation for different strainrates (R), considering δ = 0.0 (dashed) and

δ = 0.045 (solid).

Rate [s-1]

10-2 100 102

DIF

2

4

6

8

10

δ=0.0δ=0.45CEB

Figure 5: DIF functions.

4 Simulation of crack branching [15]Although studies on dynamic cracking (frac-

ture) are abundant in literature, few are dedi-cated to study this phenomenon in quasi-brittlematerials. Recently, Ozbolt and coworkers con-ducted two sets of studies, both numerical andexperimental, specially designed to study crackbranching in plain concrete [15,34–36]. The in-fluence of loading rate on the response of com-pact tension specimens (CT) and L-specimensof normal strength concrete was investigated.The test results of the CT tests [15] are used asa reference for our study (see fig. 6).

Plain concrete (fc = 53 MPa) CT speci-mens (200×200×25mm) were tested at differ-ent loading rates ranging from 0.06 to 4.3 m/s.The displacement controlled loads were applied

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L. Pereira, J. Weerheijm and L.J. Sluys

at the surface of the right edge of the notch,while the left edge was fixed (see fig. 6). In theexperimental study several sensors were usedto accurately record, apart of the final crackpatterns, the crack propagation speeds, reactionforces and displacement at the notch.

Figure 6: Schematic representation of CT test.

Model description and calibrationThe experiments were simulated with a La-

grangian plane-strain explicit algorithm usingthe stress-based nonlocal damage model de-scribed in section 2. Two structured meshesof quadrilateral element with 2 mm (M2) and1 mm (M1) characteristic element size wereused to model the specimens. Displacementcontrolled loads, similar to the experimentallymeasured deformation rates, were applied toone edge of the specimen’s notch, while theother edge was fixed. The quasi-static proper-ties determined during the experimental cam-paign [15] were used for the basic calibrationof the model (tab. 1).

Since there is no direct relation between frac-ture energy (GF ) and strain energy in nonlocaldamage mechanics, the shape of the softeninglaw (eq. 9) was calibrated considering the av-erage behavior of plain concrete. The parame-ters in the rate dependent law (η, n and δ) werecalibrated to fit the simulated experiments. Theuniaxial tensile stress-strain relations producedby this model in a single element test for differ-ent loading rates and the resulting dynamic in-crease factors (DIF = fdynt /fstatt ) are presentedin figs. 4 and 5, respectively. Finally, the max-

imum and minimum nonlocal intrinsic lengthswere set as lr = 12mm and lmin = 3mm, con-sidering the tensile equivalent strain as nonlocalvariable (εt). The calibration of these parame-ters is not a trivial task and is subject for a sepa-rate ongoing study, and therefore not discussedin this paper. Notwithstanding, convergent re-sults were obtained for the stress-based nonlo-cal formulation with lr ε [12, 16] mm. lmin wascalibrated as a function of the minimum meshsize to avoid numerical instabilities [21].

Finally, since an explicit computationscheme is used, a sensitivity study was con-ducted to determine the minimal time-step. Itwas determined that 30% (used in this study) orless of the critical time-step ensures numericalstability. In the following section we comparethe simulations with the experimental results.

Table 1: Constitutive law parametrization

Symbol Parameter Symbol ParameterE 36 GPa η 0.80

ρ 2400 kg/m3 n 0.30

ν 0.18 δ 0.045

ft 3.8 MPa λ 1.0 µs

Comparison of simulations with experimentsIn fig. 7 the simulated damage profiles con-

sidering the 2 mm mesh (M2) are given to-gether with the respective experimentally ob-served crack patterns. Fig. 8 shows equiva-lent results, but now considering a finer mesh(M1). The maximum reaction forces, registeredat the fixed edge of the compact tension spec-imen’s notch, are summarized in table 2. Fi-nally, the crack propagation velocities for thedisplacement rates of 4.3 m/s are presented infig. 9 alongside to the respective damage pro-file. The simulated velocities were determinedalong the center line of the damage profile (i.e.FPZ) considering a time difference (∆t) neededfor two consecutive elements (at a distance ds)to reach a specific damage level. Fig. 9 presentsthe computed velocities considering the damagelevels ω = 0.4 (left) and ω = 0.6 (right).

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L. Pereira, J. Weerheijm and L.J. Sluys

(a)

(b)

(c)

Figure 7: Comparison between final damageprofiles (M2) and experimentally observed

crack patterns for different displacement rates:a) 0.5; b) 1.4 and c) 3.3 m/s

(a) (b)

Figure 8: Final damage profiles consideringM1 for displacement rates: a) 1.4 and b) 3.3 m/s

As observed in the experiments, the pre-dicted crack patterns and reaction forces change

with loading rate. For loading with a relativelylow displacement rate (0.5 m/s) damage devel-ops almost perpendicular to the loading direc-tion at relative low propagation speeds. Withincreasing deformation rates, the damage pro-cess accelerates (i.e. crack speed increases)and starts to develop under an inclined angle.In both, simulations and experiments, crackbranching is first observed at a displacementrate of 3.3 m/s. As expected, branching occursin a region where the damage (crack) propaga-tion speed is the highest, followed by a slightdrop in the split cracks. Although the maxi-mum predicted crack speed is slightly higherthan the experimentally observed speed beforebranching, the prediction at the initial stages ofcracking are identical (see fig. 9). Overall, thesimulations are objective and agree very wellwith the experimental results. Thus, the dom-inant dynamic fracture phenomena in concreteare well represented by the proposed damagemodel.

Table 2: Summary of reaction force fordifferent loading velocities

Load Max reaction(kN )(m/s) Test M2 M10.5 4.05 4.20 4.20

1.4 4.64 - 5.76 4.42 4.28

3.3 4.59 - 6.88 5.64 5.62

4.3 5.66 5.80 5.66

(a) (b)

Figure 9: Crack speed and final damage profilefor displacement rate of 4.3 m/s

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L. Pereira, J. Weerheijm and L.J. Sluys

Critical evaluation of crack branchingIn this section the simulated results are fur-

ther studied in order to shed some light on theorigins of crack branching. Fig. 10 shows theevolution of the principal stresses for deforma-tion rates of 1.4 and 3.3 m/s, respectively, in theleft and right columns.

(a)

(b)

(c)

Figure 10: Evolution of principal stresses andcrack speed (computed for ω = 0.5)

for v = 1.4 m/s (left) and v = 3.3 m/s (right).a) t = 80µs; b) t = 100µs and c) t = 120µs

It is clear from these images that the princi-pal stress field at the crack tip evolves in time,changing from pure mode-I to mixed-mode.The rate of this change is directly related tothe displacement rate the material is subjectedto and the crack propagation speed. It is evi-dent in the higher loading rate case (right col-

umn fig. 10), that the stress field evolves from apeanut-like shape in the beginning of the failureprocess to a clover-like shape at crack branch-ing. Since cracking develops perpendicularlyto the principal stress direction, this change inthe stress field creates the necessary instabilitycondition that leads to crack branching. Thecombined effect of the specimen’s geometry(boundary conditions) with the loading historydictates the final shape of the crack. This is par-ticularly evident in the lower loading rate ex-ample (left column fig. 10) were the asymmet-ric evolution of the stress field makes the crackcurve.

It is worth noting that all these phenomenaare in good agreement with theoretical predic-tions and experimental evidences (see for exam-ple [37–39]). According to Ravi-Chandar [3],the observed evolution of the principal stressfield is a consequence of the evolution of dam-age in the FPZ. The associated degradation ofthe effective Rayleigh wave speed in the mate-rial surrounding the crack tip, naturally repre-sented by the used damage model [40], is themost probable cause for the observed evolutionof stresses. This observation supports the ideathat crack branching is governed by the evolu-tion of damage ahead of the crack tip, as sug-gested by Ravi-Chandra [3], and not by the ma-terial’s characteristic wave speeds, such as theRayleigh velocity, or structural inertia effects.

Finally, in order to study the impact of howrate effects are simulated in the overall responseof the material, we repeated the simulationsconsidering a) a rate independent constitutivelaw and b) a variation of the rate dependentlaw where the micro-inertia effects are excluded(R = ε). As we can see in figs. 11 and 12, bothmodels cannot represent the experimentally ob-served crack patterns (see fig. 7).

When rate effects are excluded (fig. 11)crack branching occurs at lower loading ratessimply because the dynamic variation ofstrength and energy are not modeled, so the ma-terial is simply weaker (see fig. 4). In the othercase an excessive damage development is ob-served. This is the consequence of ignoring

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L. Pereira, J. Weerheijm and L.J. Sluys

the dynamic resistance to micro-cracking evolu-tion, i.e. ignoring the retardation effect (eq. 11).When this effect is not explicitly simulated, af-ter damage initiation the material starts to softenrapidly leading to a localized straining and con-sequent increase of strain rate. Since in this casethe ’instantaneous’ rate (ε) is considered insteadof effective rate (R) in eqs. 12 and 13, the vari-ation of ε has a direct impact in the constitutivelaw.

(a) (b)

Figure 11: Final damage profiles excludingrate from constitutive law for the loading rates:

a) 0.5 m/s and b) 1.4 m/s

(a) (b)

Figure 12: Final damage profiles excludingmaterial’s reaction time (λ = 0), for theloading rates: a) 0.5 m/s and b) 1.4 m/s

5 ConclusionsIn this contribution the dynamic propagation

and branching of cracks in concrete were inves-tigated using a new rate-dependent stress-basednonlocal damage model. We introduced a newconcept of effective rate to explicitly simulatethe micro and meso-scale dynamic effects, such

as the retardation of microcraking due to inertiaforces and the delay response of the material incase of variations of strain rate in time (mem-ory effect). The model is validated against a setof dynamic compact tension tests conducted byOzbolt et al. [15]

The simulations are objective and agree verywell with the experimental results. All ma-jor phenomena associated with dynamic crackpropagation and branching observed in the ex-periments can be captured. Thus, the proposedmodel simulates the dynamic fracturing of con-crete realistically.

The results suggest that the crack branchingphenomenon is directly related to a distortionof the principal stress field ahead of the cracktip at a critical crack propagation speed. Thus,crack branching is governed by the evolution ofdamage ahead of the crack tip.

AcknowledgmentsThis research is supported by the grant

SFRH/BD/79451/2011 from the PortugueseFundacao para a Ciencia e Tecnologia (FCT),Lisbon, Portugal and co-funded by the Euro-pean Social Fund and by Programa OperacionalPotencial Humano (POPH). The contributionof the Portuguese Air Force Academy is alsogratefully acknowledged.

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