VIII International Conference on Fracture Mechanics of Concrete and Concrete StructuresFraMCoS-8
J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds)
A DISCRETE MODEL FOR ALKALI-SILICA-REACTION IN CONCRETE
M. ALNAGGAR∗, G. CUSATIS∗ AND G. DI LUZIO†
∗Northwestern University,Department of Civil and Environmental Engineering
Evanston, IL 60208, USAe-mail: [email protected],
e-mail: [email protected], web page: http://www.cusatis.us
†Politecnico di Milano,Department of Structural Engineering (DIS)
Piazza Leonardo da Vinci, 32 - 20133 Milano, Italye-mail: [email protected], web page: http://www.stru.polimi.it/people/diluzio
Key words: Cohesive Fracture, Alkali-Silica Raction, Latice Discrete Particle Model, Durability
Abstract. The safety and durability of a large number of structures, especially in high humidity envi-ronments, are endangered by Alkali-Silica Reaction (ASR). ASR is characterized by two processes:the first is the formation of gel which happens when water transmitted alkali come in contact withreactive silica in aggregates; the second is the imbibition of water into this formed basic gel and theconsequent swelling, which, in turn, causes deterioration of concrete internal structure by a diffusecracking. In this paper, the ASR effect on concrete deterioration is implemented within the frame-work of a mesoscale formulation, the Lattice Discrete Particle Model (LDPM), that simulates theheterogeneity of the concrete internal structure as well as the thermo-chemo-mechanical characteris-tics of the ASR reaction. The proposed formulation allows a precise and unique modeling of ASReffect including non-uniform expansions, expansion transfer and heterogeneous cracking. The modelcan replicate ASR cracking behavior in free and confined expansion tests. This paper presents cal-ibration and validation of the present model on the basis of experiments for unrestrained specimensunder various axial loadings undergoing ASR expansion. The results show good agreement with theexperimental data.
1 INTRODUCTION
ASR effect is reported in many concretestructures all around the world, especially thosebuilt in high humidity environments [1]. Tem-perature plays a fundamental role in the ASRrate of reaction [2] – the higher the tempera-ture the faster is the reaction. However, even incountries with average low temperatures ASRis reported to be a serious problem for the dura-bility of concrete structures. The effect of ASRis a progressive deterioration and loss of con-crete stiffness and strength that results from the
long term formation and expansion of ASR gelwhich, in turn, induce internal pressure on theconcrete meso-structure. This pressure causesnon uniform deformations leading to cracking.While the chemical representation of the reac-tion was addressed intensively in the literature,the fracture mechanics associated with the pro-gressive expansion is not as much addressed.
In this paper, the chemical model of ASRis based on the model proposed by Bazant andSteffens [3], extended hereinafter to include ad-ditional factors such as temperature effect, al-
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M. Alnaggar, G. Cusatis and G. Di Luzio
kali content in the concrete mix (Alkali con-tent effect) and gel formation effect on waterimbibition rate. The proposed model, whosepredecessors were presented in recent confer-ences [4, 5], is implemented in the Lattice dis-crete particle model (LDPM) framework [6, 7]to simulate the deterioration of concrete dueto ASR. LDPM simulates the mechanical in-teraction of coarse aggregate pieces through asystem of three-dimensional polyhedral parti-cles connected through lattice struts [6]. LDPMhas been calibrated and validated extensivelythrough the comparison to experimental dataand it has shown great capabilities in modelingconcrete damage, cracking and fracture [7].
2 ALKALI SILICA REACTION (ASR)MODELING
According to the model proposed by Bazantand Steffens [3] – adopted and extended in thissection – the overall ASR process can be ap-proximately described by considering that (1)for the ASR reaction to occur, water needs to bepresent in the pores to act as transport mediumfor hydroxyl and alkali ions required by theASR reaction; (2) the expansion of the gel is ba-sically due to water imbibition; and (3) a contin-uous supply of water is needed for the swellingto continue over time. As in the original pa-per by Bazant and Steffens [3] and consistentlywith the LDPM formulation, in this study theaggregate particles are assumed to have spher-ical shape and the whole volume of each par-ticle is assumed to be reactive with the silicaassumed to be smeared uniformly over the ag-gregate volume. This is certainly an approxi-mation compared to reality in which shape andsize of the aggregate particles may vary widely,as well as the content of reactive silica in flaws,inclusions, and veins. Under this approxima-tion, however, the dissolution of silica may beassumed to progress roughly in a uniform man-ner in the radial direction inward from the sur-face.
2.1 Gel FormationA stoichiometric relationship for the ASR re-
action is very difficult to ascertain due to thegreat variety of possible chemical equilibria fordifferent values of pH. In this model, as alsodone in Bazant and Steffens [3], the monomerH2SiO−4 is considered to be the unique formof basic gel produced by the dissolution pro-cess. In this case it can be stated that two wa-ter molecules are necessary to dissolve one sil-ica atom. The thermodynamics and kinetics ofthe ASR reaction were studied in Refs [8, 9]and it was concluded that reaction rate is fasterthan the actual rate of ASR production observedin concrete structures. This led researchers tothe conclusion that some other mechanism mustbe the dominant one and it was observed thatas the reaction progresses, the unreacted sil-ica in the interior of each of aggregate parti-cles is shielded by a spherical layer of the re-action product, the ASR gel. Through this layerfurther water molecules must diffuse in orderto reach the reacting surface of the particle anddissolve more silica. This diffusion slows downthe ASR tremendously and becomes the pro-cess governing the rate of ASR [8,9]. Thus, therate of ASR gel production can be best approx-imated by solving a diffusion problem.
A simple characterization of water diffusionacross the gel layer and towards the reactionfront at the surface of the unreacted sphericalaggregate volume can be obtained as follows.The diffusion is assumed to be governed by alinear Fick’s law. Thus, the radial flux of water,Jw, is given by
Jw = as(T )∇ · ξw (1)
where ξw = water concentration within the layerof ASR gel as a function of radial coordinate x,and as(T ) = permeability of ASR gel to water.The permeability is assumed to be temperaturedependent as
as(T ) = as0exp
(EadRT0−Ead
RT
)(2)
where Ead is the ASR activation energy, R isthe universal gas constant, T0 is a reference tem-
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M. Alnaggar, G. Cusatis and G. Di Luzio
perature at which the reference ASR gel perme-ability to water as0 is defined, and T is the cur-rent temperature. Equation 2 takes into accountthe effect of temperature on the permeability ofthe ASR gel, as proposed for the permeabil-ity of concrete by many authors [10–13]. Itshould be noted here that the time scale for tem-perature variation in real structures is the sea-son length while the ASR process takes placeon about 10 to 50 years, a period on which,the effect of change in temperature can be ne-glected. Mass conservation within infinitesimalelements of the gel requires that ξw = ∇Jw (dotdenotes derivative with respect to time t). Onehas
ξw = as(T )∇2ξw (3)
ASR gel layer
Figure 1: Idealization of gel formation in one aggregate
As discussed previously, the reaction at thesilica dissolution front may be considered to bealmost immediate compared to the duration ofwater transport to the front. Therefore, the rateof advance of the reaction front must dependsolely on the rate at which the diffusion throughthe ASR gel layer can supply water to the re-action front. The front can advance only afterenough water has been supplied to combine thesilica. Consequently, for a constant tempera-ture, the radial profile of ξw may be expectedto be the profile of a steady-state diffusion pro-cess. This profile can be expressed as in Bazantand Steffens [3] by
ξw = ws F (x), F (x) =1− 2z/Dx
1− 2z/D(4)
where ws = concentration of water in the con-crete surrounding the particle, F (x) = dimen-sionless concentration profile, x = radial coor-dinate, z = radius of the remaining unreactedparticle, D = aggregate diameter, and x = 2x/Dis the dimensionless radial coordinate (See Fig-ure 1 for details).
The condition of mass balance at the reactionfront x = z requires that
rρsdz = −as(T )∂ξw∂x
dt (5)
where r = stoichiometric ratio (mass conver-sion ratio) = 2mw/ms; mw = 18 g/mole; andms = 60.09 g/mole, and this comes from theassumption that 2 water molecules are neededto dissolve one silica molecule as stated earlierin this section. Substituting ξw = ws F (x) ac-cording to Eq. 4 and differentiating with respectto x, one obtains the following differential equa-tion for the velocity of the reaction front:
z =2wsD (dF (x)/dx)|x=z
rρsτw(6)
where τw = D2/as(T ) represents the halftimeof diffusion of water through the gel layer, andas(T ) is Fick’s permeability (having the dimen-sion of m2/s). Note that the slope of the pro-file at the reaction front becomes vertical whenz → 0.
Defining the parameters τs = 4as(T )/D2,ks = ws/rρs; and the nondimensional variablesτ = tks/τs and ζ = 2z/D, the differentialequation in Eq. 6 can be written as
dζ
dτ= −[ζ(1− ζ)]−1 (7)
In this study, this equation was solved numeri-cally with the given initial value ζ(τ = 0) = 1.The numerical solution was shown to exist be-tween τ = 0 and τ = 1/6. At τ = 1/6 theentire aggregate particle has fully reacted. Forthe sake of simplicity and for an efficient com-putational implementation, the numerical solu-tion of Eq. 7 can be well approximated by (see
3
M. Alnaggar, G. Cusatis and G. Di Luzio
figure 2)
ζ(τ) = 1− 2.22 τ 0.6 for 0 ≤ τ ≤ 1
12
ζ(τ) = 2.22
(1
6− τ)0.6
for1
12< τ ≤ 1
6(8)
0 0.03 0.06 0.09 0.12 0.15 0.180
0.2
0.4
0.6
0.8
1
τ
ζ(τ
)
NumericalApprox.
Figure 2: Numerical and approximate solutions for Eq. 7
The total mass of basic gel, ξg, produced inthe aggregate can be then determined by
ξg =4π
3
D3
8
(1− ζ3
) mg
ms
ρs (9)
where mg is the molar weights of the gel(H2SiO−4 with mg = 94.1 [g/mole]). Fur-thermore, the alkali content is not alwayspresent in enough quantity for the ASR re-action to occur. So, in order to account forthe presence of alkali, the function ρk =min(〈AC − ACth〉 /(ACreq − ACth), 1) is in-troduced such that the alkali content AC is re-lated to the required alkali content, ACreq tohave the full reaction of silica, and to an alkalicontent threshold, ACth at which, there is al-most no expansion. Eq. 9 becomes
ξg =4π
3
D3
8
(1− ζ3
) mg
ms
ρsρk (10)
2.2 Water imbibed by ASR gelThe ASR product is a solid (gel) [14], which
has the capacity – typical of colloidal systems
– to imbibe water molecules, causing extensiveswelling. Since the ASR gel is constrained inthe pores of concrete, the swelling must pro-duce pressure in the gel and the surroundingconcrete structure. This pressure induces theknown cracking and damage to the concrete.Expansion of the ASR gel can be partly accom-modated without significant pressure build upby filling the capillary pores in the hardened ce-ment paste located close to the surface of thereactive aggregate particles. This is also facili-tated by the existence of the so-called interfacialtransition zone (ITZ) that is a layer of materialwith a higher porosity in the hardened cementpaste near the aggregate surface. Similarly tothe ITZ size, the thickness, δc, of the layer inwhich the capillary pores are accessible to theswelling ASR gel may be considered constant,independent of particle size D.
As already mentioned, the physical proper-ties of the basic form of ASR gel are not yetknown. But it seems logical to assume that theformation of this basic gel causes no signifi-cant volume increase per se and that all volumechanges are caused solely by the intake of addi-tional water. Denoting by wi the mass of waterimbibed by the basic gel, the volume increaseof the gel is
∆V =wiρw
(11)
The imbibition of water from the bulk ofmortar or concrete into the basic gel cannot hap-pen instantly. It is a local micro-diffusion pro-cess that occurs with some delay. Because thegel is expelled into the pores, the geometry ofthis diffusion is no longer spherical and it ap-pears difficult to choose any particular ideal-ized geometry of the diffusion flow of waterthat should be analyzed. It seems therefore ap-propriate to conduct merely a simplified overallanalysis without reference to any particular ge-ometry. It is reasonable to assume that
wi =Aiτi
(12)
where Ai characterizes the thermodynamicaffinity and τi is a certain characteristic time.
4
M. Alnaggar, G. Cusatis and G. Di Luzio
To set up a reasonable expression for Ai, someaspects should be considered: (1) The chemi-cal potential governing water transport throughconcrete may be assumed to be roughly propor-tional to the specific water content wi; (2) Thewater imbibition rate wi, and thus Ai, should beproportional to the mass of basic gel, ξg; (3) Thewater imbibition rate wi should decrease with adecreasing relative humidity h in the pores ofconcrete near the reacting particle and shouldstop at relative humidity below which water isnot available for ASR reaction and imbibition;(4) The water imbibition rate wi should de-crease with a decreasing absolute temperatureT of concrete and vice-versa. Such dependencecan be well modeled with an Arrhenius-typeequation. According to these observations andconsidering only the case of full saturation (typ-ical, for example, for dams), the affinity (ther-modynamic driving force or chemical potentialdifference) governing the water imbibition maybe approximately expressed as
Ai = w∗i (T, h, ξg)− wi (13)
where w∗i (T, h, ξg) is the value wi at thermody-namic equilibrium, at which no water migratesinto or out of the ASR gel and can be expressedas
w∗i (T, h, ξg) = κ0i exp
(EawRT0−Eaw
RT
)ξg (14)
where Eaw is the activation energy of the imbi-bition process and κ0i represents the amount ofwater imbibed by unit mass of basic gel. Analy-sis of experimental data carried out in this studysuggests that, in absence of more precise in-formation about the water imbibition process,κ0i ≈ 1 is a reasonable assumption.
For reasons of dimensionality, the character-istic time, called the imbibition halftime, musthave the form τi = δ2/Cwiη(wi), in whichCwi is the diffusivity for microdiffusion of wa-ter close to the aggregate (surely much lowerthan the diffusivity for global water diffusionthrough a concrete structure), δ is the average(or effective) distance of water transport processfrom the concrete around the aggregate into the
ASR gel, and η(wi) is an increasing function ofwi. Roughly, for two adjacent aggregate parti-cles of diametersD1 andD2, δ can be estimatedas
δ =l −D1 −D2
2(15)
where l is the distance between the particle cen-ters. The characteristic time, τi, must be anincreasing function of wi, because the imbibi-tion of the layers of gel increases the diffusiontime of the free water to reach the unimbibedgel. Therefore, it is reasonable to assume thefunction η(wi) as η(wi) = eηwi , in which η is amaterial constant.
By considering all these effects together, onehas the following equation for water imbibitioninto the gel:
wi =(e−
EawRT ξg − wi
) Cwiδ2
e−ηwi (16)
The water imbibed into the gel is given by thefollowing differential equation:
wi =
[e−
EawRT
4π
3
D3
8
(1− ζ3
) mg
ms
ρsρk − wi]
·Cwiδ2
e−ηwi (17)
3 LATTICE DISCRETE PARTICLEMODEL
LDPM simulates concrete mesostructure bytaking into account only the coarse aggregatepieces, typically with characteristic size greaterthan about 4 mm. The mesostructure is con-structed through the following steps. 1) Thecoarse aggregate pieces, whose shapes are as-sumed to be spherical, are introduced into theconcrete volume by a try-and-reject randomprocedure. 2) Zero-radius aggregate pieces(nodes) are randomly distributed over the ex-ternal surfaces. 3) A three-dimensional domaintessellation, based on the Delaunay tetrahedral-ization of the generated aggregate centers, cre-ates a system of cells interacting through trian-gular facets, which can be represented, in a two-dimensional setting, by straight line segmentsas shown in Figure 3. A vectorial constitutivelaw governing the behavior of the model is im-posed at the centroid of the projection of each
5
M. Alnaggar, G. Cusatis and G. Di Luzio
single facet (contact point) onto a plane orthog-onal to the straight line connecting the particlecenters (edges of the tetrahedralization). Theprojections are used instead of the facets them-selves to ensure that the shear interaction be-tween adjacent particles does not depend on theshear orientation. The straight lines connectingthe particle centers define the lattice system.
Figure 3: a) Mesostructure tessellation. b) Three-dimensional discrete particle. c) Definition of nodal de-grees of freedom and contact facets in two-dimensions.
Rigid body kinematics describes the dis-placement field along the lattice struts and thedisplacement jump, JuCK, at the contact point.The strain vector is defined as the displace-ment jump at the contact point divided by theinter-particle distance, L. The components ofthe strain vector in a local system of reference,characterized by the unit vectors n, l, and m,are the normal and shear strains:
εN =nTJuCKL
; εL =lTJuCKL
;
εM =mTJuCK
L(18)
The unit vector n is orthogonal to the projectedfacet and the unit vectors l and m are mutuallyorthogonal and lie in the projected facet.
The elastic behavior is described by assum-ing that the normal and shear stresses are pro-portional to the corresponding strains:
σN = ENεN ; σM = ET εM ; σL = ET εL (19)
where EN = E0, ET = αE0, E0 = effec-tive normal modulus, and α = shear-normalcoupling parameter. E0 and α are assumedto be material properties. Detailed descrip-tion of model behavior in the nonlinear rangecan be found in Ref. [6]. Current LDPM for-mulation has been implemented into MARS, amulti-purpose computational code for the ex-plicit dynamic simulation of structural perfor-mance [15].
4 AMALGAMATION OF ASR MODELWITHIN LDPM FRAMEWORK
The radius variation of one aggregate parti-cle can be calculated from the volume variationin Equation 11 as
∆r =3
√3∆V
4π+D3
8−D/2 (20)
The calculated aggregate radius variation isused to calculate the LDPM imposed ASR gelstrain, εN0 as
εN0 = (∆r1 + ∆r2 − δc)/L (21)
where ∆r1 and ∆r2 are the radii changes of twoadjacent aggregate particles. This additionalnormal strain is imposed on the LDPM facet byadding it to the stress-dependent facet normalstrain. In the elastic regime, one can write thenormal stress as σN = EN(εN − εN0). Notethat the model formulated here approximatelyassumes that the facet shear deformations dueto gel swelling are negligible, εM0 = εL0 ≈ 0,although this might not be exactly true due tothe irregularity of actual aggregate particles.
5 NUMERICAL SIMULATIONSIn this study, simulation of experimental
results of Multon and Toutlemonde [16] wasperformed as described in the following sec-tions. These accelerated experiments have
6
M. Alnaggar, G. Cusatis and G. Di Luzio
shown good representation to the actual longterm phenomena as they could represent the ex-pansion growth the same model used by Lariveet al [17] to express dams ASR expansions. Inthe following sections, an automatic parameteridentification technique based on the nonlinearleast square method is used throughout all cali-brations to identify different model parameters.
5.1 ASR model free paramatersThe proposed model contains material pa-
rameters that need to be defined. Some param-eters, such as activation energies Eaw and Ead,reference ASR gel permeability to water as0, re-quired and threshold alkali contents ACreq andACth, and silica content ρs, can be determineddirectly if specific experimental data such asdiffusion tests and permeability tests and/or ag-gregate chemical analysis are available. Oth-ers, such as δc, Cwi, and η are very hard tomeasure and they need to be calibrated by in-verse analysis of macroscopic experimental re-sults. The remaining parameters are either welldefined or easy to assume as water content ws,current alkali content AC, T and T0. Throughout the simulations, some parameters were as-sumed as follows: ρs was assumed to be about20% of the total aggregate mass so by weightit equals 0.2 · 2200kg/m3 (silica unit weight)= 440 kg/m3. The presence of silica in ag-gregate varies based on the aggregate type andit can reach up to 50% ± 10% as reported in[18]. The free water content, ws, can be reason-ably assumed using the relation given by [19]as ws = (w/c− (0.235 + 0.19) · αc) · c, wherew/c, αc and c are the water cement ratio, reac-tion degree and cement content in kg/m3.
5.2 Calibration of LDPM concrete param-eters
To match the concrete mechanical prop-erties relevant to the analyzed experimentaldata, LDPM parameters were calibrated basedon reported values of compressive strength,f ′c, Young’s modulus, E and splitting tensilestrength, f ′t . Ten numerical specimens withdifferent meso-structures were always simu-
lated and their average response was consid-ered. The generation of the different LDPMmeso-structures was performed considering thegrain size distribution of the specimens used inthe experimental campaign as shown in Figure4. The calibration of LDPM parameters gavethe results shown in Table 1.
Agg. Diam. [mm]%
pas
sing
0 4 7 10 14 200
20
40
60
80
100
Exp.Sim.
Figure 4: Grain Size distribution for simulated and exper-imental specimens
Table 1: Mechanical properties
Property Actual Simulated Averagef ′c (MPa) 38.4 38.41f ′t (MPa) 3.2 3.19E (MPa) 37300 37695
5.3 Creep and shrinkage consideration.The experimental results show, for the con-
trol specimens, autogenous shrinkage of the or-der 0.25%. To account for the shrinkage, theCEB MC99 model was adopted for shrinkageas it divides the shrinkage into autogenous anddrying parts. Figure 5 shows the calibrationresults of the model along with experimentaldata. Also for the loaded cases in the experi-mental campaign, the results show a consider-able amount of creep. To account for creep de-formation, the B3 model was adopted. Figure6 shows the calibrated creep model as well asthe experimental data. Both creep and shrink-age strains are added to the simulated strain re-
7
M. Alnaggar, G. Cusatis and G. Di Luzio
sults and compared with the experimental re-ported measured strains.
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Figure 5: Calibration of Shrinkage Strain with CEBMC99 model.
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Figure 6: Calibration of Creep Strain with B3 model
5.4 Calibration of ASR model parameterswith free expansion results.
Since the analyzed experiments of Multonand Toutlemonde [16] are relevant to an accel-erated reaction (450 days compared to 5 to 50years in dams), and with big fraction of reactiveaggregate (all aggregates 4 mm in diameter andabove were reactive giving 64.5% by weight)and high alkali content, δc tends to be zero asalmost all pores surrounding the near aggregatesurface were filled quickly in the early 28 days
of curing. The parameters providing the tem-perature dependence (Eaw and Ead) were rea-sonably assumed on the basis of existing liter-ature as in Ref. [?] and Eaw was also assumedto be equal to Ead. Also as there was no dataabout the temperature values during the test itwas assumed to be the room temperature of 23◦C. On the contrary, the other parameters (as0,δc, Cwi and η) were calibrated using the auto-matic parameter identification procedure basedon available volumetric measured strains. Theaverage responses of volumetric strains for theten simulated specimens after adding shrinkagestrains, as well as the experimental data aver-ages, are shown in Figure 7.
0 90 180 270 360 450−0.03
0
0.03
0.06
0.09
0.12
Time [Days]
Vo
lum
etri
c S
trai
n %
Exp.
Calib.
Figure 7: Free Expansion calibration based on Volumet-ric Strain
Figure 8: Free Expansion Axial and Radial Strains
Also the model could replicate theanisotropy found in the experimental resultsof free expansion by Multon and Toutlemonde[16], which is a unique feature of the modelthat can not be done without explicitly defin-ing anisotropic properties for the model, whilehere only the different random geometry couldsimulate it. Figure 8 shows both axial and ra-dial strains for the free expansion case for bothexperimental and simulated cases showing alsothe scatter of experimental data. The values of
8
M. Alnaggar, G. Cusatis and G. Di Luzio
the calibrated, as well as the assumed parame-ters, are listed in Table 2. In these experiments,there was no study for the temperature effectsso, the activation energies have no effect onsimulations.
Table 2: Used ASR Model Parameters
Parameter Value Typews (kg/m3) 56.662 Assumedρs (kg/m3) 440 Assumedρk 1 Assumedas0(m
2/s) 9.7407 · 10−9 CalibratedEaw(Joule/mole) - AssumedEad(Joule/mole) - Assumedδc (m) 0.0 AssumedCwi(m
2/s) 3.4552 · 10−13 Calibratedη(kg−1) 36813 Calibrated
5.5 Simulation of axially loaded unre-strained specimens with 10MPa and20MPa.
After parameter calibration, all the parame-ters were kept fixed and the simulations werecarried out for the axially loaded unrestrainedspecimens. Two different axially loaded unre-strained specimens were considered: the firstwith 10 MPa and the second with 20 MPa.Ten generated specimens were used and theirresponses along with the experimental resultsfor both axial and radial strains are shown inFigures 9 and 10, which show an excellentagreement between simulations and experimen-tal data.
0 90 180 270 360 450−0.18
−0.12
−0.06
0
0.06
0.12
0.18
Time [days]
Str
ain
%
Axial
RadialMes.
Sim.
Figure 9: Unrestrained Expansion under 10 MPa Axialand Radial Strains
0 90 180 270 360 450−0.18
−0.12
−0.06
0
0.06
0.12
0.18
Time [days]
Str
ain
%
Axial
Radial
Mes.
Sim.
Figure 10: Unrestrained Expansion under 20 MPa Axialand Radial Strains
5.6 Simulated Crack patternsOne of the unique advantages of implement-
ing this ASR model within the LDPM frame-work is the capability of reproducing the veryscattered and heterogenous crack patterns thatclosely resemble crack patterns reported in ex-periments. This is illustrated in Figures 11 and12. The cracks shown represent cracks withopenings larger 0.015 mm. In Figure 11, it isclear that each simulated specimen has its owndifferent crack patterns however all the threespecimens responses showed reasonable visualagreement with typical experimental data in theliterature [2].
9
M. Alnaggar, G. Cusatis and G. Di Luzio
a) b) c)
Figure 11: Crack distribution under free expansion for:a) Specimen n.1, b) Specimen n.3, c) Specimen n.9
Also by examining the surface cracks as wellas internal cracks shown here, the very ran-dom character of cracking can be identified aswell as the characteristic of having wider crackopenings on the external surface can be shownclearly. This feature emphasizes the capabilityof the model to reproduce realistic crack propa-gation simulation.
a) b) c)
Figure 12: Crack distribution for Specimen n.1 under: a)Free Expansion, b) 10 MPa, c) 20 MPa
Moreover, the effect of axial loading on thesize and distribution of cracks is shown in Fig-ure 12 for specimen n. 1. It is clear thatthe cracks for the free expansion case are ran-domly distributed with no clear preferential di-rection, but on the contrary, cracks for both ax-ially loaded cases show redirection in the radial
direction (cracks are aligned in vertical direc-tion).
Since the gel expansion is constrained inthe concrete microstructure, the gel volume in-crease produces a pressure, which in turn causescracking in the concrete meso-structure. Thepresence of triaxial stress state reduces the ASRinduced expansion in the less compressed direc-tion leading to the so-called ASR anisotropicdevelopment which is commonly modeled us-ing the concept of ”Expansion transfer” [16,20,21]. In the current formulation, no gel redi-rection is postulated. The redirection of ASRimposed strain is interpreted as only redirec-tion of crack distribution. While some authorsin [16, 20, 21] refer to ASR induced expansionas a volumetric expansions to be calculated forthe different states of stresses, here, the effectof ASR expansion is decomposed into its twocomponents, gel volume increase due to waterimbibition and crack volume generated. Withthis unique decoupling and considering the het-erogeneity of the material by means of LDPM,deeper analysis of the effect of stress was pos-sible, leading to postulating that the real effectof stress state is the reduction of cracking inthe direction of higher compression and the in-crease of cracking in the other directions. Thisis also supported by the low likelihood for thesolid gel [14] to migrate easily under stress ef-fect through the concrete meso-structure. Byobserving both the responses shown in Figures8, 9 and 10 as well as the crack distributionredirection shown in Figure 12, it can be seenclearly that the model could capture the ”Ex-pansion transfer” effect.
6 CONCLUSIONS
A model that simulate the ASR effect hasbeen implemented within the framework of amesoscale formulation, such as the Lattice Dis-crete Particle Model. This model has been usedfor the numerical simulations of experimentaldata reported by Multon and Toutlemonde [16].Based on the observed results, the followingconclusions can be drawn:
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M. Alnaggar, G. Cusatis and G. Di Luzio
1. The model can accurately simulate freeexpansion experimental data and can re-produce realistic crack patterns.
2. Axial loading simulations show goodagreement with experimental results, es-pecially the crack patterns demonstratethe ability of the model to redirect crack-ing based on the stress state.
3. The expansion transfer concept can bemodeled by decoupling the gel expansionfrom crack volume and by this, the stressstate will only affect the crack distribu-tion and there is no need to assume thatthe ASR gel migrates under stress statechanges.
ACKNOWLEDGEMENTSThe authors gratefully acknowledge finan-
cial support under DHS grant 2010-ST-108-000016 entitled High Strain Rate Behavior ofDam Concrete: Experiments and MultiscaleModeling.
REFERENCES[1] G. Giaccio, R. Zerbino, J.M. Ponce, and
O. R. Batic, 2008. Mechanical behaviorof concretes damaged by alkali-silica re-action. Cement and Concrete Research,38:993–1004.
[2] V. Saouma and Y. Xi, 2004. Litera-ture Review of Alkali Aggregate Reac-tions in Concrete Dams. Report cu/sa-xi-2004/001, Department of Civil, Environ-mental, & Architectural Engineering Uni-versity of Colorado.
[3] Z.P. Bazant and A. Steffens, 2000. Math-ematical model for kinetics of alkali-silicareaction in concrete. Cement and ConcreteResearch, 30:419–428.
[4] G. Cusatis, G. Di Luzio, and L. Cedolin,2011. Meso-Scale Simulation of Con-crete: Blast and Penetration Effects andAAR Degradation. Applied Mechanicsand Materials, 82:75–80.
[5] G. Di Luzio, M. Alnaggar, and G. Cusatis,2012. Lattice discrete particle modelingof alkali-silica-reaction effects to concretestructures. In P. Rossi and J.L. Tailhan,editors, Numerical Modeling - Strategiesfor Sustainable Concrete Structures-SSCS2012. Aix en Provence, France.
[6] G. Cusatis, D. Pelessone, and A. Mencar-elli, 2011. Lattice Discrete Particle Model(LDPM) for Concrete failure Behavior ofConcrete. I: Theory. Cement and ConcreteComposites, 33(9):881–890.
[7] G. Cusatis, A. Mencarelli, D. Pelessone,and J.T. Baylot, 2011. Lattice DiscreteParticle Model (LDPM) for Failure Be-havior of Concrete. II: Calibration andValidation. Cement and Concrete Com-posites, 33(9):891–905.
[8] R. Dron and F. Brivot, 1992. Thermody-namic and kinetic approach to the alkali-silica reaction: Part 1. Cement and Con-crete Research, 22(5):941–948.
[9] R. Dron and F. Brivot, 1993. Thermody-namic and kinetic approach to the alkali-silica reaction: Part 2. Cement and Con-crete Research, 23(1):93–103.
[10] Z.P. Bazant and L.J. Najjar, 1972. Non-linear water diffusion in nonsaturated con-crete. Material and Structures, 5(25):3–20.
[11] M. Jooss and H.W. Reinhardt, 2002. Per-meability and diffusivity of concrete asfunction of temperature. Cement and Con-crete Research, 32(9):1497–1504.
[12] G. Di Luzio and G. Cusatis, 2009. Hygro-thermo-chemical modeling of high perfor-mance concrete. I: Theory. Cement andConcrete Composites, 31(5):301–308.
[13] G. Di Luzio and G. Cusatis, 2009. Hygro-thermo-chemical modeling of high perfor-mance concrete. II: Numerical implemen-tation, calibration, and validation. Cement
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and Concrete Composites, 31(5):309–324.
[14] T.C. Powers and H.H. Steinour, 1955.An Interpretation of Some Published Re-searches on the Alkali-Aggregate Reac-tion’. Journal of American Concrete In-stitute (Proc.), 55:497–516 and 785–812.
[15] D. Pelessone, 2009. MARS: Modeling andAnalysis of the Response of Structures -User’s Manual. ES3, Beach (CA), USA.
[16] S. Multon and F. Toutlemonde, 2006. Ef-fect of applied stresses on alkali-silicareaction-induced expansions. Cement andConcrete Research, 36:912–920.
[17] F. Ulm, O. Coussy, L. Kefei, and C. Lar-ive, 2000. Thermo-Chemo-MechanicsOf ASR Expansion In Concrete Struc-tures. Journal of Engineering Mechanics,126(3):233–242.
[18] N. Castro, B. E. Sorensen, and M. A.T.M.Broekmans, 2011. Assessment of indi-vidual ASR-Agregate particles by XRD.Proceedings of the 10th InternationalCongress for Applied Mineralogy (ICAM),pages 95–102.
[19] O. Jensen and P. Hansen, 2001. Water-entrained cement-based materials I. Prin-ciples and theoretical background. Ce-ment and Concrete Research, 31:647–654.
[20] C. Larive, A. Laplaud, and M. Joly, 1996.Behavior of AAR-affected concrete: ex-perimental data. Proc. 10th Int. Conf.AAR, Melbourne, Australia, pages 670–677.
[21] S. Multon, G. Leclainche, E. Bourdarot,and F. Toutlemonde, 2004. Alkali-silicareaction in specimens under multi-axial mechanical stresses. Proc. 4th Int.Conf. CONSEC’04, Seoul, Korea, pages2004–2011.
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