multiscale homogenization modeling of alkali-silica-reaction … · multiscale homogenization...

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

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

MULTISCALE HOMOGENIZATION MODELING OFALKALI-SILICA-REACTION DAMAGE IN CONCRETE

R. REZAKHANI∗, M. ALNAGGAR† AND G. CUSATIS∗

∗Northwestern UniversityEvanston, IL USA

e-mail: [email protected] - [email protected]

†Rensselaer Polytechnic InstituteTroy, NY USA

e-mail: [email protected]

Key words: Multiscale Homogenization, Asymptotic Expansion, Alkali-Silica-Reaction

Abstract. Alkali silica reaction (ASR) is one of the main reasons that cause deterioration in concretestructures, such as dams and bridges. ASR is a chemical reaction between alkali ions from cementpaste and the silica inside each aggregate piece. ASR gel, which is the product of this reaction,imbibes additional water causing swelling and cracking, which leads to degradation of concrete me-chanical properties. In this study, to model ASR-induced damage, the Lattice Discrete Particle Model(LDPM) is adopted, which is a meso-scale discrete model. LDPM simulates concrete at the levelof coarse aggregate pieces. ASR effects have been already successfully modeled by LDPM in therecent past. This paper employs a recently developed multiscale homogenization approach to derivemacroscopic constitutive equations for ASR-damaged concrete. The adopted homogenized model isused to reproduce experimental data on volumetric expansion of unrestrained concrete prisms.

1 Introduction

Alkali-Silica-Reaction (ASR) in concrete isa chemical reaction between the silica exist-ing in aggregate pieces and alkali ions con-tained in the cement paste. In the presence ofwater, ASR produces an expansive gel, regu-larly called “ASR gel”, which cause crackingand damage in concrete over time. As a re-sult of this chronic material deterioration, bothstrength and stiffness of concrete are decreased.Since presence of water is the cornerstone inASR occurrence, it is reported in concrete struc-tures exposed to high humidity environments,usually above 60%. In addition, temperaturealso plays a crucial role in driving ASR effects.

Stanton [1] studied the ASR effect for thefirst time with reference to various aspects in-

cluding: chemistry of the reaction, structuraland material consequences, development of ex-perimental techniques to assess the vulnerabil-ity of aggregate and mixes to ASR, and the for-mulation of new strategies to mitigate the ASReffects. Since then, several research efforts havebeen devoted to scrutinize different aspects ofASR.

In general, experiments on ASR effect havebeen performed on small scale accelerated ASRlaboratory specimens. Among multiple exper-imental techniques is the Accelerated MortarBar Test (AMBT) which is completed within16 days. AMBT experiments are performed onspecimens, made of crushed aggregate pieces,which are immersed in an alkali rich solution,while temperature is raised to 80◦C. ConcretePrism Test (CPT) is another testing method

1

DOI 10.21012/FC9.285

R. REZAKHANI, M. ALNAGGAR and G. CUSATIS

which provides broader insight into the ASR re-action on real structures. CPTs are usually car-ried out on specimens that are made of aggre-gates used in practice and are totally sealed tomaintain the relative humidity at high values.

As far as numerical modeling of ASR effectsare concerned, numerous research efforts havebeen carried out to explore this phenomenon atdifferent length scales and with various accu-racy levels. Bažant [2] was the first to studyASR deterioration by means of fracture me-chanics approach and to predict the pessimumaggregate size.

Multiple macroscopic models are availablein the literature to investigate ASR mechanismsand related degradation of mechanical proper-ties. The phenomenological model developedby Charlwood et al. [3] is one of the earliestones. Later, researchers formulated computa-tional models based on the ASR kinetics includ-ing more details on creep effect as well as crack-ing and implemented them into the finite ele-ment framework using the crack band approach[4]. These models allowed successfully repli-cating some experimantal data on ASR expan-sion [5]. Effect of stress state on ASR deteriora-tion mechanism were incoporated into this typeof models by Saouma and Perotti [6]. Comi etal. [7] proposed damage models which includedchemical and mechanical aspects of the ASRprocess in a consistent thermodynamic fash-ion. Humidity and temperature effects were in-tegrated into the ASR reaction kinetics law byPoyet et al. [8]. While all aformentioned mod-els were deterministic, Capra and Sellier [9]studied ASR phenomena by a probabilistic ap-proach. Detailed consideration of the chemi-cal aspect of ASR reaction were addressed bySaouma et al. [10] who derived an analyticalmodel to explain the kinetics of the alkali-silicachemical reaction.

The main drawback of all the aforemen-tioned models is their inability to simulate accu-rately crack patterns induced by ASR reaction,which hinders the prediction of ASR degrada-tion effects and requires the assumption of phe-nomenological relationships between concrete

strength and ASR gel expansion. In addition, itlimits the ability of such models to explain com-plex triaxial behavior of concrete under ASRand also forces the assumption of phenomeno-logical relationships between ASR gel expan-sion and stress state. These limitations are in-herently related to simulating concrete as anisotropic and homogenous continuum [11, 12].The ASR-LDPM model [13, 14] used in thepresent study is the first model to overcomeall aforementioned limitations. The model in-tegrates ASR effects within the Lattice Dis-crete Particle Model (LDPM) [15, 16]. LDPMsimulates the mechanical interaction of coarseaggregate pieces through a system of three-dimensional polyhedral particles, each resem-bling a spherical coarse aggregate piece with itssurrounding mortar, connected through latticestruts. LDPM has the ability of simulating theeffect of material heterogeneity on the fractureprocesses. ASR-LDPM can reproduce realisticcrack patterns under free expansion, restrainedexpansion and expansion under various loadingconditions. In addition, it can replicate the dete-rioration of concrete strength and stiffness, tem-perature effects on reaction kinetics and alkalicontent variations. It is worth noting that, ASR-LDPM can predict degradation of mechanicalproperties without the need of any phenomeno-logical relationship between such degradationand expansion.

Despite of all above mentioned capabilities,major limitation of employing ASR-LDPM inthe simulation of large concrete structural mem-bers is the immense computational cost. Thisis due to the fact that LDPM explicitly sim-ulates concrete aggregate pieces that are eachrepresented as a computational node with sixdegrees of freedom. In addition, each compu-tational node is surrounded by many triangu-lar facets, forming polyhedral cells, which arethe computational entities over which consti-tutive calculations are carried out during eachcomputational step. As a result, it is necessaryto establish a multiscale framework to increasecomputational efficiency [17]. Wu et al. [18]established a FE2 homogenization framework

2


based on averaging theorem to analyze ASReffect in concrete. A multiscale homogeniza-tion framework has been recently developed bythe authors [11, 19–21] to homogenize discretemodels to continuum, and this homogenizationscheme has been employed to evaluate elasticeffective properties of concrete as well as itsnonlinear response under different loading con-ditions.

2 ASR theory and modeling techniqueASR fundamental governing equations

along with the computational framework formodeling ASR effect in concrete is discussed inthis section.

2.1 ASR governing equationsIn the current paper, the ASR model de-

veloped in [13] is considered and incorporatedin the LDPM framework, which is employedas the meso-scale model in the homogeniza-tion scheme. The theory is established for themeso-scale, scale of coarse aggregate pieces,and lower scale mechanisms are taken into ac-count in an average sense by means of the gov-erning constitutive equations.

Gel formation. To form the ASR gel, waterand alkali ions must reach the silica present inthe aggregate pieces through a diffusion pro-cess through the aggregate volume. Therefore,gel mass Mg produced from an aggregate pieceof diameter D is obtained from the solutionof a steady state mass balance of radial diffu-sion process into the aggregate piece. Figure1a shows the diffusion process that takes placeduring ASR reaction, in which z is the diffu-sion front position measured from the aggregatecenter or the radius of the aggregate unreactedportion. Thus, one can write,

Mg = κaκgπ

6

(D3 − 8z3

)(1)

where κa = min(〈ca − ca0〉/(ca1 − ca0),1) isconsidered to account for the fact that the avail-able alkali content in the cement paste aroundthe vicinity of each aggregate piece is not nec-essarily enough for a complete ASR reaction.

ca0 is the threshold alkali content at which, noor minimal expansion is observed, and ca1 is thesaturation alkali content enough for the ASR re-action to occur completely. κg is a free param-eter. Rate of the diffusion front position can bewritten as

z = −κzwe exp

( Eag

RT0−

Eag

RT

)z(1− 2z

D

) (2)

where T0 = reference temperature; T = currenttemperature; Eag = activation energy of the dif-fusion process; and R = universal gas constant.we = density of water content in concrete sur-rounding the aggregate piece estimated as we =

(w/c− 0.188α∞c )c at saturation; in which α∞c =

(1.031w/c)/(0.194 + w/c) asymptotic hydra-tion degree, w/c = water-to-cement ratio, c =

cement content, and κz is a free parameters.

Water imbibition. The gel created throughASR reaction has the capacity to imbibe watermolecules and expand subsequently. This exertsan expansive pressure on the surrounding con-crete and result in cracking and damage. Thewater imbibition process can be formulated byrelating the rate of the imbibed water mass byASR gel Mi to the thermodynamic driving forceand a characteristic imbibition time. Consider-ing the developed equation for the gel mass Mg

given by the integration of Eq. 2, the rate ofimbibed water reads

Mi =C1

i

δ2 exp (−ηMi)[κi Mg −Mi

]exp

(Eai

RT0−

Eai

RT

)(3)

in which, at thermodynamic equilibrium, alinear relationship is considered between therate of water imbibition and the mass of formedgel through κi as the constant of proportional-ity. C1

i is the initial micro-diffusivity of the ofwater close to aggregate surface. The exponen-tially decaying factor in Eq. 3 represents thereduction of water imbibition rate as more wa-ter is imbibed into ASR gel. This is a reason-able relation, since diffusivity of ASR gel de-creases during the water imbibition process. In

3


Title Suppressed Due to Excessive Length 25

0 1 2 3 4 5 6x 10ï3

0

7

14

21

28

35

42

Strain %

Stre

ss [M

Pa]

Simulated

0 0.05 0.1 0.15 0.2 0.250

400

800

1200

1600

2000

Displacement [mm]

Load

[N]

ExperimentalSimulated

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

ga

¡imp

OldNew

0 0.2 0.4 0.6 0.8 10.2

0.56

0.92

1.28

1.64

2

Alkali Concentration [Mole/litre]Diff

usio

n Co

eff.

[m2 /d

ay×1

0ï5 ]

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

Time [Days]

Expa

nsio

n %

ExperimentalSimulated

c) b) a)

h)

et

e) f) d)

0

5

10

15

20

250 5 10 15 20 25

X-dir. [mm]

Y-di

r. [m

m]

g) 0

20

40

60

80

1000 20 40 60 80 100

X-dir. [mm]

Y-di

r. [m

m]

0

20

40

60

80

1000 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1 i) 0

20

40

60

80

1000 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

0

2

4 6 8

10

12

14 0

2

4 6 8

10

12

14

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

ga

¡imp

OldNew

Constant 0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

ga

¡imp

OldNew

Title

Supp

ress

edD

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ener

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cle

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base

don

[28]

asw

e=

(w/c

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• c)c

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=(1

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)/(0

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+w

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nt.

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bibi

tion.

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wat

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ess

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Mi

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mod

ynam

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finity

and

ach

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time.

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side

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the

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Eq.7

,the

rate

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ater

imbi

bitio

nis

give

nto

:

Mi=

C1 i d2

exp(�h

Mi)

k iM

gexp✓

Eai

RT 0

�E

ai

RT

◆�

Mi�

(8)

whe

reth

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bibe

dw

ater

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sum

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mas

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edge

lwith

k ias

the

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tant

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opor

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lity,

and

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pera

ture

-dep

ende

ntth

roug

han

Arr

heni

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peeq

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vern

edby

the

activ

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nen

-er

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the

imbi

bitio

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oces

sE

ai.d

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reff

ectiv

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ceof

wat

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oces

sfr

omth

eco

ncre

tear

ound

the

aggr

egat

ein

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eA

SRge

l.Th

em

icro

-diff

usiv

ityC

ifo

rmic

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ffus

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ofw

ater

clos

eto

the

aggr

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ew

asco

nsid

ered

tobe

ade

crea

sing

func

tion

ofM

ias

Ci=

C1 i

exp(�h

Mi)

,whe

reh

(Mi)

isan

incr

easi

ngfu

nctio

nof

Mi.

Com

paris

onbe

twee

nth

eor

igin

alA

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DPM

mod

elan

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rmul

atio

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show

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fig.1

c.W

here

the

ASR

gele

igen

stra

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for

both

cons

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riabl

ek a

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ent

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show

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alka

lico

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ts.

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ceD

iscr

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icle

Mod

elin

gof

ASR

Effe

ct.

The

rate

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ass

ofw

ater

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bed

Mi

bya

spec

ific

aggr

egat

epi

ece

isgi

ven

byEq

.8.U

sing

this

equa

tion,

the

incr

ease

inra

dius

ofea

chag

greg

ate

parti

cle

ofin

itial

radi

usr=

D/2

isgi

ven

by:

r i=

(3M

i/4p

r w+

r3 )1/

3�

r(9

)

whe

rer w

isth

em

ass

dens

ityof

wat

er.t

hera

teof

radi

usin

crea

seis

calc

ulat

edus

ing

the

chai

nru

leas

:

r i=

dri

dt=

dri

dMi

dMi

dt=

Mi

dri

dMi=

Mi

4pr w

(3M

i/4p

r w+

r3 )�2

/3(1

0)

Expa

nsio

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the

ASR

gel

can

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mm

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ithou

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ithth

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rate

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fora

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fined

as

e0 N=

⇢0.

0fo

rr i1

+r i2

2d

c(r

i1+

r i2)/`

for

r i1+

r i2>

2dc

(11)

Variable

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

ga

¡imp

OldNew

Diffusion front

z D/2

Cement Paste Reactive Aggregate

Fig. 1: a) One LDPM Cell arrond an aggregate piece. b) Idealization of gel formation in oneaggregate. c) Comparison between alkali content effect with original ASR-LDPM modeland new model. d) Nonlinear alkali diffusion coefficient change with alkali concentration.e) Simulated LDPM response of 3 point bending test compared with experimental data in[132]. f) Simulated LDPM response of Uniaxial compression test matching the compressivestrength reported in [132]. g) Alkali Concentration [mole/litre] distribution over mid sec-tion of small specimen at 0,2,4,6,8,10,12 and 14 days. h) Alkali Concentration [mole/litre]distribution over mid section of large specimen at 0,2,4,6,8,10,12 and 14 days, and i) Axialexpansion evolution for small and large specimens.

(a)

of cement, qw = 1000 kg/m3 is the mass density of water, andvair is the volume fraction of entrapped or entrained air (typi-cally 3–4%);

2. Compute the volume fraction of simulated aggregate asva0 ¼ ½1# Fðd0Þ&va ¼ ½1# ðd0=daÞnF &va;

3. Compute the total volume of simulated aggregate as Va0 = va0V;4. Compute particle diameters by sampling the cdf in Eq. (2) by a

random number generator: di ¼ d0½1# Pi 1# dq0=dq

a

! "&#1=q, where

Pi is a sequence of random numbers between 0 and 1. Fig. 1ashows a graphical representation of the particle diameter selec-tion procedure.

5. For each newly generated particle in the sequence, check that

the total volume of generated particles eV a0 ¼P

i pd3i =6

# $does

not exceed Va0. When, for the first time, eV a0 > Va0 occurs, thecurrent generated particle is discarded, and the particle gener-ation is stopped.

Fig. 1b shows the comparison between the theoretical sievecurve (solid line) and the computational sieve curve (circles), ob-tained through the procedure highlighted above for the generationof a 100-mm-side cube of concrete characterized by c = 300 kg/m3,w/c = 0.5, nF = 0.5, d0 = 4 mm, and da = 8 mm.

In order to simulate the external surfaces of the specimen vol-ume, the generated particles are augmented with zero-diameterparticles (nodes). Assuming that the external surfaces of the spec-imen volume can be described through sets of vertexes, edges, andpolyhedral faces, one node for each vertex is first added to the par-ticle list. Then, Ne = INT(Le/hs) and Np ¼ INT Ap=h2

s

# $(where the

operator INT(x) extracts the integer part of the argument x) nodesare associated with each edge e and polyhedral face p, respectively.Le is the length of a generic surface edge, Ap is the area of a genericsurface polyhedron, and the average surface mesh size hs is chosensuch that the resolution of the discretization on the surface is com-parable to the one inside the specimen. Numerical experiments

0

0.2

0.4

0.6

0.8

1

d0

di

da

Pi

0

0.2

0.4

0.6

0.8

1a

0.2

0.4

0.6

0.8

d

F(d

)

d0

b c

d

T

P3

P1

P2

E12F4

P4

E13

E24

F1

F3E14

E34

F2

e f

P1

P2

E12

d1

d2

=

=

==

P3

P1

P2

E12

g

F43*

P3

P1

P2

F4

P4

*4T

h

=

=

i

TheoreticalNumerical

Fig. 1. (a) Probability distribution function for particle size generation; (b) theoretical (solid curve) and numerical (circles) sieve curve; (c) particle system for a typical dog-bone specimen; (d) tetrahedralization for a typical dog-bone specimen; (e) tessellation of a typical LDPM tetrahedron connecting four adjacent particles; (f) edge-pointdefinition; (g) face-point definition; (h) tet-point definition; and (i) LDPM cells for two adjacent aggregate particle.

G. Cusatis et al. / Cement & Concrete Composites 33 (2011) 881–890 883

(b)

of cement, qw = 1000 kg/m3 is the mass density of water, andvair is the volume fraction of entrapped or entrained air (typi-cally 3–4%);

2. Compute the volume fraction of simulated aggregate asva0 ¼ ½1# Fðd0Þ&va ¼ ½1# ðd0=daÞnF &va;

3. Compute the total volume of simulated aggregate as Va0 = va0V;4. Compute particle diameters by sampling the cdf in Eq. (2) by a

random number generator: di ¼ d0½1# Pi 1# dq0=dq

a

! "&#1=q, where

Pi is a sequence of random numbers between 0 and 1. Fig. 1ashows a graphical representation of the particle diameter selec-tion procedure.

5. For each newly generated particle in the sequence, check that

the total volume of generated particles eV a0 ¼P

i pd3i =6

# $does

not exceed Va0. When, for the first time, eV a0 > Va0 occurs, thecurrent generated particle is discarded, and the particle gener-ation is stopped.

Fig. 1b shows the comparison between the theoretical sievecurve (solid line) and the computational sieve curve (circles), ob-tained through the procedure highlighted above for the generationof a 100-mm-side cube of concrete characterized by c = 300 kg/m3,w/c = 0.5, nF = 0.5, d0 = 4 mm, and da = 8 mm.

In order to simulate the external surfaces of the specimen vol-ume, the generated particles are augmented with zero-diameterparticles (nodes). Assuming that the external surfaces of the spec-imen volume can be described through sets of vertexes, edges, andpolyhedral faces, one node for each vertex is first added to the par-ticle list. Then, Ne = INT(Le/hs) and Np ¼ INT Ap=h2

s

# $(where the

operator INT(x) extracts the integer part of the argument x) nodesare associated with each edge e and polyhedral face p, respectively.Le is the length of a generic surface edge, Ap is the area of a genericsurface polyhedron, and the average surface mesh size hs is chosensuch that the resolution of the discretization on the surface is com-parable to the one inside the specimen. Numerical experiments

0

0.2

0.4

0.6

0.8

1

d0

di

da

Pi

0

0.2

0.4

0.6

0.8

1a

0.2

0.4

0.6

0.8

dF

(d )

d0

b c

d

T

P3

P1

P2

E12F4

P4

E13

E24

F1

F3E14

E34

F2

e f

P1

P2

E12

d1

d2

=

=

==

P3

P1

P2

E12

g

F43*

P3

P1

P2

F4

P4

*4T

h

=

=

i

TheoreticalNumerical

Fig. 1. (a) Probability distribution function for particle size generation; (b) theoretical (solid curve) and numerical (circles) sieve curve; (c) particle system for a typical dog-bone specimen; (d) tetrahedralization for a typical dog-bone specimen; (e) tessellation of a typical LDPM tetrahedron connecting four adjacent particles; (f) edge-pointdefinition; (g) face-point definition; (h) tet-point definition; and (i) LDPM cells for two adjacent aggregate particle.

G. Cusatis et al. / Cement & Concrete Composites 33 (2011) 881–890 883

(c) (d)

Figure 1: (a) ASR gel created around a generic aggregate piece. (b) LDPM polyhedral cell enclosingspherical aggregate pieces. (c) Spherical particles and (d) polyhedral cell representations of a typicaldogbone specimen.

addition, the temperature-dependence is formu-lated through an Arrhenius-type equation gov-erned by the activation energy Eai of the imbibi-tion process. δ is the average (or effective) dis-tance of water transport process from the con-crete around the aggregate into the ASR gel.

2.2 The Lattice Discrete Particle Model(LDPM)

The Lattice Discrete Particle Model (LDPM)[15,16] is a meso-scale discrete model that sim-ulates the mechanical interaction of coarse ag-gregate pieces embedded in a cementitious ma-trix (mortar). The geometrical representation ofconcrete mesostructure is constructed throughthe following steps. 1) The coarse aggregatepieces, whose shapes are assumed to be spher-ical, are introduced into the concrete volumeby a try-and-reject random procedure. 2) Zero-radius aggregate pieces (nodes) are randomlydistributed over the external surfaces to facili-tate the application of boundary conditions. 3)A three-dimensional domain tessellation, basedon the Delaunay tetrahedralization of the gener-ated aggregate centers, creates a system of poly-hedral cells (see Figure 1b) interacting throughtriangular facets and a lattice system composedby the line segments connecting the particlecenters. Figures 1c and d represent sphericalparticle and corresponding polyhedral represen-tations of a typical dogbone specimen.

In LDPM, rigid body kinematics is used to

describe the deformation of the lattice/particlesystem and the displacement jump, ~uC�, at thecentroid of each facet is used to define measuresof strain as

εN =nT~uC�

`; ε L =

lT~uC�

`; εM =

mT~uC�

`(4)

where ` = interparticle distance; and n, l, andm, are unit vectors defining a local system ofreference attached to each facet.

Next, a vectorial constitutive law governingthe behavior of the material is imposed at thecentroid of each facet. In the elastic regime,the normal and shear stresses are proportionalto the corresponding strains: tN = EN ε

∗N =

EN (εN − εξN ); tM = ET ε

∗M = ET (εM − ε

ξM ); tL =

ET ε∗L = ET (ε L − ε

ξL), where EN = E0, ET = αE0,

E0 = effective normal modulus, and α = shear-normal coupling parameter; and ε ξN , ε ξM , ε ξL aremesoscale eigenstrains that might arise from avariety of phenomena such as, but not limitedto, thermal expansion, shrinkage, and ASR ex-pansion.

For stresses and strains beyond the elas-tic limit, LDPM mesoscale nonlinear phenom-ena are characterized by three mechanisms:1) Fracture and cohesion due to tension andtension-shear, 2) Compaction and pore col-lapse from compression, and 3) Friction dueto compression-shear. One can find the detailsof the corresponding constitutive equations in[15]. Finally, the governing equations of the

4


LDPM framework are completed through theequilibrium equations of each individual parti-cle.

2.3 Modeling ASR effect within LDPMframework

The rate of imbibed water mass Mi by ageneric aggregate piece of diameter D is pre-sented in Eq. 3. In this equation, the radius of ageneric expanded aggregate piece of initial ra-dius of r = D/2 can be written as

ri = (3Mi/4πρw + r3)1/3 (5)

where ρw is the mass density of water. Usingthe chain rule along with Equation 5, the rate ofaggregate radius increase can be calculated as

ri =dri

dt=

dri

dMi

dMi

dt= Mi

dri

dMi

=Mi

4πρw(3Mi/4πρw + r3)−2/3

(6)

Therefore, for a generic facet in LDPM ge-ometry, the incompatible eigenstrain rate due toASR gel expansion εasr

N is determined as fol-lows

εasrN =

{0.0 if z = 0(ri1 + ri2)/` if z > 0 (7)

where ri1 and ri2 are the radius increase of thetwo aggregate pieces which share the consid-ered facet. It should be noted that it is hereinassumed that the shear eigenstrains due to gelexpansion imposed on each facet are approxi-mately negligible, εasr

M = εasrL ≈ 0, although this

may not be an exact assumption due to the ir-regular shape of actual aggregate pieces.

The presented formulation, is implementedinto MARS, a multi-purpose computationalcode for the explicit dynamic simulation ofstructural performance [22].

3 Multiscale homogenization theoryIn this section, fundamentals and crucial as-

pects of a general multiple scale homogeniza-tion framework proposed by Rezakhani and

Cusatis [19] based on classical asymptotic anal-ysis is discussed. In the development of the ho-mogenization theoretical framework, a partic-ulate material structure resembeling LDPM isconsidered as the underlying material domain,see Figure 3.

3.1 Fine-scale problem definitionFigure 3a shows the interaction of two neigh-

boring particles, I and J, which share a genericfacet. For the case of small strains and displace-ments, facet strains, already defined in Equation4, and curvatures can be written in terms of par-ticles displacement and rotation vectors

ε I Jα =

1r

(UJ +ΘJ × cJ −UI −ΘI × cI

)· eI Jα

(8)

χI Jα =

1r

(ΘJ −ΘI

)· eI Jα (9)

where α = N,M,L; r = |xI J |; χI Jα = facet curva-

tures; xI J = xJ − xI is the vector connecting theparticle nodes PI and PJ ; eI J

α are unit vectorsdefining a facet Cartesian system of referencesuch that eI J

N = is orthogonal to the facet andeI J

N · xI J > 0; UI , UJ = displacement vectors of

node PI and PJ ; ΘI , ΘJ = rotation vectors ofnode PI and PJ ; and cI , cJ = vectors connect-ing nodes PI and PJ to the facet centroid, seeFig. 3a. It must be noted here that displace-ments and rotations are assumed to be indepen-dent variables.

Stress traction tI J on each facet for a givenstrain vector is derived through a vectorial con-stitutive equation. In general, it can be writtenthat tI J = tα (εN ,...)eI J

α where summation ruleapplies over α. For instance, in the case of elas-tic behavior, constitutive equation can be for-mulated through; t I J

α = Eαε I Jα ; in which each

traction component and the associated strain arerelated through Eα which is fine-scale elasticconstant (summation rule does not apply). Non-linear facet constitutive equations are explainedin detail in [15], with reference to the LDPMwhich is considered in the numerical examples.

The final step in constituting the discretecomputational model is imposing the equilib-rium of each single particle under the effect of

5


all neighboring particles. Translational and ro-tational dynamic equilibrium equations of a sin-gle particle reads

MIuUI −V Ib0 =

∑FI

AtI J (10)

MIθΘ

I=

∑FI

AwI J (11)

where wI J = cI × tI J is the moment of the trac-tion tI J with respect to the particle node PI ; FIis the set of facets enclosing node PI , which isobtained by collecting all the facets associatedwith each node pair (I,J) in which J accountsfor all particles that are in contact with particleI; A = facet area; superimposed dots representtime derivatives; V I is the particle volume; b0 isthe body force vector; MI

u = mass tensor of thenode PI ; MI

θ = moment of inertia tensor.

3.2 Asymptotic expansion homogenizationFigure 3b illustrates a generic macroscopic

material domain in a global coordinate systemX. Two distinct length scales and the related lo-cal coordinate systems, x and y, are introducedat any point of the material domain to representthe local macro- and meso-scale problems, re-spectively. Material heterogeneity is neglectedin the macro-scale, in which the problem is de-fined as homogeneous continuum. On the otherhand, in the fine-scale analysis, material het-erogeneity is modeled by means of the discretemeso-scale model. As shown in Figure 3b, X isthe vector that connects the origin of the globalmacro-scale coordinate system to the mass cen-ter of a generic RVE. In Figure 3c, an enlargedrepresentation of the macro-scale material pointis depicted in the local meso-scale coordinatesystem y as a representative volume of the het-erogeneous material. It must be noted that, ac-cording to the rule of separation of scales, par-ticles I and J shown in Figure 3b in the macro-scale local coordinate system should be illus-trated mush smaller compared to their size inFigure 3c. This is ignored for the sake of clarity.According to the rule of separation of scales, themacro- and meso-scale coordinate systems are

linked as follows

x = ηy 0 < η << 1 (12)

where η is a very small positive scalar. To de-rive the governing equations that rule the prob-lem at different length scales, asymptotic def-inition of displacement and rotation fields aretaken into account. Therefore, the displacementand rotation of a generic node PI , UI = u(xI ,yI )and ΘI = θ(xI ,yI ), can be approximated byvirtue of the following asymptotic expansions

u(x,y) ≈ u0(x,y) +ηu1(x,y) (13)

θ(x,y) ≈η−1ω0(x,y) +ϕ0(x,y) +ω1(x,y)+

ηϕ1(x,y)(14)

In the above equations, terms of order O(η2)and higher are neglected. Functions u0(x,y),and u1(x,y) are continuous with respect to xand discrete (e.i. defined only at finite num-ber of points) with respect to y. It should benoted that the asymptotic expansion of rota-tion field is written considering the fact that, ingeneral, rotation vector have similar meaningof the curl of displacement vector. Therefore,if one considers a two scale dependent con-tinuous displacement-like field dη (x,y), then2ω0 = ∇y × d0; 2ϕ0 = ∇x × d0; 2ω1 = ∇y × d1;2ϕ1 = ∇x × d1; and subscripts x and y iden-tify the nabla operator in the coarse- and fine-scale, respectively. Therefore, ω0, ω1 are thefine-scale rotations, whereas ϕ0 and ϕ1 are thecorresponding coarse-scale rotations.

By substituting Equations 13 and 14 intothe expressions of facet strains and curvatures,Equations 8 and 9, along with some mathe-matical manipulation, one obtains the followingform for the multiple scale definition of facetstrains and curvatures (check Reference [19] forall the derivation details and definition of termsof different orders)

εα = η−1ε−1α + ε0

α +ηε1α (15)

η χα = η−1ψ−1α +ψ0

α +ηψ1α (16)

One can formulate the multiple scale defini-tion of facet elastic tractions as tα = η−1t−1

α +

6


of the macroscopic constitutive equations; and its implementation in computer codes is relatively simple. Within the ex-tensive literature on AEH, remarkable is the work of the following authors. Hassani and Hinton (1998a,b) investigatedformulation of homogenization theory and topology optimization and its numerical application to materials with periodicmicrostructure. Chung et al. (2001) presented detailed derivation of multiple scale formulation for elastic solids. Fish em-ployed this approach to study elastic as well as elasto-plastic composites (Fish et al., 1997). Ghosh et al. (1995) adopted MHalong with Voronoi cell finite element method (VCFEM) to study the behavior of composites with random meso-structure(Ghosh et al., 1996). More recently, Fish et al. (2007) introduced the generalized mathematical homogenization (GMH) toderive continuum constitutive equations starting from molecular dynamics (MD).

All the aforementioned work is relevant to Cauchy continuum formulations. However, homogenization schemes werealso used for the multiscale analysis of Cosserat continuum models, in which an independent rotation field appears inaddition to the displacement field. Feyel (2003) built a homogenization scheme to couple a Cauchy continuum formulationat the micro-scale giving rise to a Cosserat continuum formulation at the macro-scale. Asymptotic homogenization tech-nique was employed by Forest et al. (2001) for upscaling elastic Cosserat solids. In this work, the author studied varioustypes of asymptotic expansions for the displacement and rotation fields and investigated their effect on the resultingmacroscopic continuum behavior. Results of this investigation showed that the nature of the homogenized continuumdepends on the ratio of the Cosserat characteristic length of constituents, size of heterogeneity and typical size of thestructure.

Chan et al. (2006) derived the governing constitutive equations for strain gradient elasticity for both homogeneous andfunctionally graded materials using the strain energy density function and the related definitions of the stress fields. Theyshowed that additional terms appear in the equations that are related to the strain gradient nonlocality and the interactionbetween material nonhomogeneity. Bardenhagen and Triantafyllidis (1994) obtained a nonlinear higher order gradientcontinuum representation of discrete periodic micro-structures by means of an energy approach. The developed model wasthen employed to investigate the existence and stability of localization bands and their relationship to the model loss ofellipticity. Finally, homogenization of discrete atomic models into equivalent continuum can be found in publications wherethe authors exploited asymptotic analysis techniques (Caillerie et al., 2006) and the mathematical Γ-convergence method(Braides et al., 2006).

The present study derives a general multiscale homogenization scheme suitable for upscaling materials whose fine-scalebehavior can be successfully approximated through the use of discrete models featuring both translational and rotationaldegrees of freedom.

2. The fine-scale problem

With reference to Fig. 1a, let us consider the interaction of two adjacent particles, I and J, sharing a generic facet. If onelimits the analysis to the case of small strains and displacements – which is a reasonable assumption in the absence of largeplastic deformation prior to fracture as observed in brittle and quasi-brittle materials –meaningful measures of deformation(Cusatis et al., 2011a) can be defined as

( )Θ Θϵ = + × − − × · ( )α αrU c U c e1

1IJ J J J I I I IJ

and

Fig. 1. Geometrical explanation of the two-scale problem: (a) geometry of two neighboring particles. (b) Macro material domain. (c) Meso-scale domainwith material heterogeneity.

Please cite this article as: Rezakhani, R., Cusatis, G., Asymptotic expansion homogenization of discrete fine-scale modelswith rotational degrees of freedom for the simulation of quasi-brittle materials. J. Mech. Phys. Solids (2016), http://dx.doi.org/10.1016/j.jmps.2016.01.001i

R. Rezakhani, G. Cusatis / J. Mech. Phys. Solids ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

Figure 2: Geometrical explanation of the two-scale problem: (a) Geometry of two neighboring parti-cles. (b) Macro material domain. (c) Mesoscale domain with material heterogeneity. (d) Coordinatetransformations in the RVE analysis.

t0α + ηt1

α considering t (·)α = Eαε

(·)α and assuming

Eα ∼ O(1). It is mathematically proven that thisasymptotic form holds for the case of facet non-linear constitutive equations as well [19]. Us-ing asymptotic expansion of facet traction inequilibrium equations and some mathematicalderivation, one can derive governing equationsfor different scales O(η−2), O(η−1), and O(η0).Final equations that are obtained for each scaleis presented in the next sections.

3.2.1 Fine-scale equation: the RVE prob-lem

O(η−2) equilibrium equations refer to therigid body motion of the RVE:

u0i (X,y) = v0

i (X) + εi j k ykω0j (X) (17)

One must consider that the field variables v0

and ω0 are only dependent on the macroscopiccoordinate system X, which implies that thesequantities varies smoothly in the macro-scalematerial domain, while they are constant overthe RVE domain. Hence, v0 and ω0 correspondto the macroscopic displacement and rotationfields, respectively. Considering the fact thatω0

represents RVE rigid rotation and taking intoaccount the definition of ϕ0 and ω0, one canconclude ϕ0 = ω0 and ϕ0 is equal for all RVEparticles.

O(η−1) governing Equations are relevant tothe RVE problem, which is derived as∑FI

At0αeI J

α = 0;∑FI

A (cI × t0αeI J

α ) = 0 (18)

Equation 18 is the force and moment equi-librium equations of every single particle insidethe RVE subjected to O(1) facet traction t0

α vec-tor, which, in turn, is a function of ε0

α that iswritten as

ε0α = r−1

(u1J

i −u1Ii + εi j kω

1Jj cJ

k − εi j kω1Ij cI

k

)eI Jαi

+ Pαi j

(γi j + ε jmnκimy

cn

)(19)

where γi j = v0j,i − εi j kω

0k , κi j = ω0

j,i are themacro-scale Cosserat strain and curvature ten-sors, respectively. The vector yc is the positionvector of the facet centroid shared between par-ticles I and J in the local lower-scale coordinatesystem, shown in Figure 3a. Pα

i j = nI Ji eI J

α j is anoperator to project macroscopic strain and cur-vature tensors onto the RVE facets as normalor tangential strain components. ConsideringEquation 4, it is concluded that the first termin Equation 19 is the fine-scale definition of thefacet normal and tangential strains written interms of lower-scale displacement and rotationfields u1 and ω1. Furthermore, the second termin Equation 19, Pα

i j

(γi j + ε jmnκimy

cn

), is the pro-

jection of the macroscale Cosserat strain and

7


curvature tensors for each RVE facet. There-fore, Equation 19 states that the O(1) strains oneach RVE facet is the sum of their fine-scalestrains and the projection of the macroscopicstrain tensor onto that facet.

3.2.2 Coarse-scale equation: the macro-scopic problem

Mathematical manipulation of the O(1)terms in the multiple-scale governing equationsleads to the macroscopic translational and ro-tational equilibrium equations. The followingequations are derived for the coarse-scale equi-librium equations and their corresponding ho-mogenized stress and moment stress tensors

σ0ji,j + bi = 0 (20)

σ0i j =

12V0

∑I

∑FI

Art0αPα

i j (21)

and

εi j kσ0i j +

∂µ0ji

∂x j= 0 (22)

µ0i j =

12V0

∑I

∑FI

Art0αQα

i j (23)

where V0 is the volume of the RVE; ρu =∑I M I

u/V0 is the mass density of the macro-scopic continuum. Equation 20 correspondsto the classical differential form of equilibriumequation. In addition, Equation 21 definesmacroscopic stress tensor in terms of the so-lution of the RVE problem. Matrix Qα

i j is de-fined as Qα

i j = nI Ji ε j kl xC

k eI Jαl . µ0

i j is the macro-scopic moment stress tensor calculated usingthe results of RVE analysis, and Equation 22corresponds to the classical rotational equilib-rium equation of Cosserat continua. It is shownnumerically that the anti-symmetric part of theRVE homogenized stress tensor tends to be neg-ligible for different types of loading condition[19]. Therefore, it is rational to neglect the mo-ment stress tensor effect in the numerical exam-ples in the next section and only consider theresponse in terms of the symmetric stress ten-sor.

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]

Vol

umet

ric S

train

[%]

ExperimentsFull FineïScale Ca = 5.25

1

4.2 3.9

3.15

2.89

Figure 4: Volumetric expansion of concreteprism for different alkali contents by experi-ment and full fine-scale analysis.

4 Numerical resultsIn this section, effect of ASR expansion and

different values of alkali content are investi-gated for multiple concrete prisms that werestudied experimentally [23], by means of bothfull fine-scale analysis and the two-scale ho-mogenization framework presented in this pa-per. Free expansion of sealed concrete prismsof dimension 75mm × 75mm × 265mm arerecorded for 320 days. Aggregate size distri-bution of the concrete used in experiments is:1/3 of aggregate particles between 4.75 and 9.5mm, 1/3 between 9.5 and 12.5 mm and the lastthird between 12.5 and 19.0 mm. For the gen-eration of prisms that are completely modeledwith LDPM for full-fine scale analysis in addi-tion to the LDPM RVEs used in multiscale ho-mogenization study, the following parametersare used: minimum aggregate size, d0 = 4.75mm; maximum aggregate size, da = 19 mm;fuller curve exponent, nF = 0.55; and the fol-lowing mix composition: cement content, c =

420 kg/m3; water to cement ratio, w/c = 0.45;aggregate to cement ratio, a/c = 4.25. Figure 3shows the geometrical configuration simulatedfor the two separate analysis: Figure 3(b) showsthe polyhedral cell representation of a prismthat is fully generated by LDPM, while Fig-ure 3(a) presents the homogenization model inwhich the prism is discretized by three hexahe-

8


75 mm

265 mm

75 mm

1

Figure 3: (Left) Two-scale homogenization problem. (Right) Full mesoscale prism model.

dral finite elements with one integration point.A single RVE created by LDPM is assigned toeach integration point to calculate the effectivematerial response during the analysis, see Fig-ure 3(a).

Five levels of alkali content: ca = 2.89, 3.15,3.9, 4.2, and 5.25 kg/m3 are considered to in-vestigate its effect on ASR analysis. For eachalkali content, full fine-scale analysis is carriedout for three different LDPM mesh realizationsto examine the aggregate distribution effect, andthe average response is presented. Evolution ofvolumetric strain versus time of the full fine-scale analysis and corresponding experimentaldata are plotted in Figure 4, One can see thatthe computational results match well with theexperimental data. One can see that the rateof volumetric strain decreases and plateaus asASR reaction progresses. The ASR parame-ters are first calibrated for the 2.89 kg/m3 alkalicontent and then used to validate the responsefor other alkali content values. Calibrated pa-rameters are: ka = 1, C0

a = 2.7 kg/m3, C1a =

4.37 kg/m3, as0 = 3.45e-13 m×m/sec, C1i =

28e-11 m×m/sec, η = 439,540 kg−1, kg = 689kg/m3, Eag = Ead = 500 J/mol. Current T andreference temperature T0 are considered to beequal which means temperature effects are ne-glected. It should be noted that the specimensare sealed which maintains the relative humid-ity as high values very close to 100 %. LDPMparameters used in the simulation are: EN = 60GPa, σt = 4.75 MPa, σc0 = 150 MPa, α = 0.25,

nt = 0.2, lt = 75 mm, rst = 2.6, Hc0/E0 = 0.4,µ0 = 0.4, µ∞ = 0, kc1 = 1, kc2 = 5, σN0 = 600MPa, α = ET/EN = 0.25.

The same LDPM and ASR model parame-ters are used in the homogenization analysis.To investigate particle distribution and RVE sizeeffect, three different RVE sizes: 35, 50, 75 mmare considered, and six different mesh realiza-tions are examined for each size. The overallmultiscale numerical procedure adopted in thispaper can be summarized as follows.

• The finite element method is employedto solve the macro-scale homogeneousproblem in which external loads and es-sential BCs are applied incrementally.During each numerical step, strain incre-ments ∆γi j = ∆v0

j,i − εi j k∆ϕ0k and curva-

ture increments ∆κi j = ∆ω0j,i tensors are

calculated at each integration point basedon the nodal displacement and rotationincrements of the corresponding finite el-ement.

• Increment of ASR induced normal strain∆εasr

N , as discussed in Section 2.3, are cal-culated for all facets inside RVE.

• The macroscopic strain and curvature in-crements are projected onto the RVEfacets through the proper projection op-erators: ∆ε c

α = Pαi j

(∆γi j + ε jmn∆κimy

cn

).

These projected strains along with ASRinduced normal strain are imposed, upon

9


0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]

Vol

umet

ric S

train

[%]

35 mm RVE

2.893.153.94.25.25

2

Ca[kg/m3]

(a)

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]V

olum

etric

Stra

in [%

]

50 mm RVE

2.893.153.94.25.25

2

Ca[kg/m3]

(b)

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]

Vol

umet

ric S

train

[%]

75 mm RVE

2.893.153.94.25.25

2

Ca[kg/m3]

(c)

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]

Vol

umet

ric S

train

[%]

35 mm RVE

HomogenizationFull FineïScale

Ca = 5.25

1

4.2

3.9

3.15

2.89

(d)

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]

Vol

umet

ric S

train

[%]

50 mm RVE

HomogenizationFull FineïScale Ca = 5.25

1

4.2

3.9

3.15

2.89

(e)

0 200 400 6000

0.05

0.1

0.15

0.2

0.25

0.3

Time[Days]

Vol

umet

ric S

train

[%]

75 mm RVE

HomogenizationFull FineïScale

Ca = 5.25

1

4.2

3.9

3.15

2.89

(f)

Figure 5: Volumetric expansion of the prisms with different alkali contents obtained from homoge-nization framework using RVE size of (a) 35 mm (b) 50 mm, and (c) 75 mm. (d) to (f) Comparisonof the homogenization results using different RVE sizes with the full fine-scale ones.

sign change shown in the following equa-tion, as eigen-strains to the RVE allow-ing the calculation of the fine-scale solu-tion governed by the fine-scale constitu-tive equations.

∆ε0α = ∆ε

fα +∆ε c

α −∆εasrN δαN

= ∆εfα − (−∆ε c

α) −∆εasrN δαN

(24)

δαN is 1 for α = N and 0 for α = M or L.

• Finally, the fine-scale facet tractions areused to compute, through Equations 21,the macroscopic stresses for each Gausspoint in the FE mesh.

Variation of volumetric expansion versustime is plotted for different RVE sizes in Fig-ures 5(a) to (c). Each plot consist of six dif-ferent curves corresponding to different particledistribution for each level of alkali content. Foreach value of alkali content, one can see thatthe curves are more scattered for 35 mm RVEthan 50 mm and for 50 mm compared to 75 mmRVE size. Therefore, it is concluded that higherratio of the RVE size to the maximum particlesize results in smaller standard deviation in theresponse. In addition, since higher value of al-kali content leads to higher level of ASR dam-age and subsequently enhanced inhomogeneityof the damaged RVE, one can observe that, for

10


each RVE size, the curves are more scatteredfor higher level of alkali content. There resultsare averaged for each RVE size and different al-kali contents and compared to the full fine-scaleanalysis results in Figures 5(d) to (f). One cansee that the homogenization and the full fine-scale results match well for different RVE sizesand alkali content values. The homogenizationresults agrees more with the full fine-scale sim-ulations as the RVE size is increased. This isexpected since larger RVEs are better represen-tations of lower-scale material structure.

Finally, the significant capability of the de-veloped homogenization framework is enor-mous saving of computational cost, while it re-tains adequate precision of the results. Ho-mogenization analysis of the prism using 75mm RVE approximately takes 6 hours whichis close to full fine-scale analysis. But, anal-yses with 50 mm and 35 mm RVEs take 1.5hours and 40 minute, respectively. It shouldbe noted that only three finite elements areused in the homogenization analysis. There-fore, computational time saving in the analysisof larger structures discretized by large num-ber of finite elements will be more considerable.This is the main purpose of developing a mul-tiscale framework which is successfully accom-plished by means of the developed homogeniza-tion scheme.

5 ConclusionsA multiscale homogenization framework for

the simulation ASR effect on concrete is pre-sented. Considerable save of the computationalcost is accomplished while the results are cor-related well with the experimental data. RVEsize to maximum aggregate size ratio is the keyfactor in the scatter of the results, which de-creases as this ratio is increased. It is shownthat the homogenization framework replicatesthe full fine-scale analysis results with high ac-curacy using RVEs smaller than the volume as-sociated with the related Gauss point in the fi-nite element model. ASR degradation effectis a proper application to use homogenizationframework, since the related damage pattern is

spread throughout the specimen rather than lo-calized form which appears in fracture prob-lems.

REFERENCES[1] T. E. Stanton, Expansion of concrete

through reaction between cement and ag-gregate, Proceeding American Society ofCivil Engineers, 66:1781—1811, 1940.

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[3] R. G. Charlwood, S. V. Solymar, and D. D.Curtis, A review of alkali aggregate re-actions in hydroelectric plants and dams,in Proceedings of the International Con-ference of Alkali-Aggregate Reactions inHydroelectric Plants and Dams, volume129, Fredericton, Canada, 1992.

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