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Computational Materials Science 43 (2008) 119–126

Effect of aggregate kind and graining on modelling ofplain concrete under compression

T. Sadowski a,*, G. Golewski b

a Faculty of Civil and Sanitary Engineering, Department of Solid Mechanics, Lublin University of Technology, Nadbystrzycka 40 Str., 20-618 Lublin, Polandb Faculty of Civil and Sanitary Engineering, Department of Civil Engineering Structures, Lublin University of Technology,

Nadbystrzycka 40 Str., 20-618 Lublin, Poland

Available online 14 September 2007

Abstract

Hardened concrete is a brittle composite material characterised by non-homogeneous internal structure. Under the influence of exter-nal loads composites destruction is caused by: discontinuities created by technological defects or local differences of mechanical prop-erties of concrete components. In the vicinity of discontinuities local concentration of stresses occurs. They can cause (undercompressive loading) defects initiation and their stable propagation, which leads to degradation of the whole material sample or struc-tural element. The brittle fracture process occurs when the large defects start to growth in the unstable manner.

The internal structure of the concrete is very complex due to changes of the number of defects and their shapes during the wholeloading process up to the final failure. Cracks, which arose in the interfacial aggregate–mortar transition zone (ITZ) propagate mainlyin cement matrix in the surrounding area of the concrete aggregate. Grain shapes cause: bending of cracks, wing initiation at tips ofstraight cracks and creation of defects with more complex shapes.

The paper presents the method of mesomechanical modelling of compressed concrete. The subject of investigation is cement compos-ites which are made on the basis of limestone (L) and gravel aggregates (G). The paper is particularly focused on an effect of grain diam-eters on the macroscopic response of the concrete. Crack dimensions, shapes and numbers are strictly related to the aggregate diametersof the concrete composite. The aggregate material strongly influences the mechanical properties of the ITZ and further overall mechan-ical response as well as crack propagation conditions. The presented theoretical predictions of different concretes and experimental inves-tigations lead to the conclusion that the appropriate selection of grain diameter can improve structural properties of the concretecomposites and enlarge their applications.� 2007 Elsevier B.V. All rights reserved.

PACS: 81.05.Rm; 61.43.�j; 61.72.Mn; 62.20.�x; 62.20.mj; 62.20.mt

Keywords: Concrete; Aggregate grading; Mesomechanical modelling; Compression loading

1. Introduction

Concrete is a non-homogeneous and multiphase com-posite. In its internal structure one can distinguish twobasic components: cement matrix and inclusions (grainaggregates). The remaining concrete components are: min-eral and chemical additives.

0927-0256/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2007.07.037

* Corresponding author. Tel.: +48 81 538 13 86; fax: +48 81 525 69 48.E-mail address: t.sadowski@pollub.pl (T. Sadowski).

Brittleness of concrete is responsible for durability andsafety of concrete and reinforced concrete structures [1–4]. This feature is significantly important in these areas ofconcrete structure, where small defects exist. In such brittlecomposites local failure frequently occurs under loading atlocal material discontinuities and sudden differencesof mechanical properties of the material, e.g. [5]. In thesurrounding regions of defects such as: microcracks,microvoids, discontinuities of crystal lattice and materialcorrosion, local stress concentrations take place under

120 T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126

loading. These concentrations can create local damagedevelopment and its further unstable growth leads to thedynamic failure of the concrete structural element. Themost dangerous are small cracks with sharp tips, creatingthe highest stress concentrations.

The problem of damage growth in cement composites isa very complex phenomenon. There are many different rea-sons of cracks initiation and growth and it is difficult toforesee stages of that process, which lead in the end to cat-astrophic failure of the structural element. The basic factorcausing the damage growth is an initial internal structure ofthe composite.

The most important parameters describing the initialstructure in the concrete composite are: hardenedcement paste, kind of aggregates material, also distributionof: grains and grain diameters, e.g. [6,7]. The particularlysignificant for the behaviour of the compressed concreteare: aggregate–mortar ITZ and coarse grains, e.g.[5–8].

The description of the mechanical response of the con-crete made up from a set of coarse aggregates is possibleby application of the mesomechanical model. However,this model requires introduction of the necessary experi-mental data including, e.g. [9]:

• definition of internal microstructure of the material bySEM observations,

• specification of macroscopic mechanical features of thematerial,

• definition of fracture mechanics parameters.

The description of gradual degradation of mechanicalproperties in the compressed concrete was done in 2-Dproblem by taking into account development of two typesof mesocracks with contact friction: straight and wing one.Therefore in the description of cracks growth the majorrole is played by the fracture toughness tests according toI and II model. Critical values of stress intensity factors:KIc and KIIc, necessary to develop the model, were deter-mined experimentally for the concrete composites. Theseexperimental data play the significant role because crackssubjected to compressive loading propagate under mixedmode.

The whole deformation process of the analysed concretecomposites includes: (1) purely elastic deformation ofmatrix and aggregates, (2) deformations originating fromdefects: straight and wing cracks.

In order to compare the formulated model with theexperimental data, the experiments were carried out deter-mining development of deformations in cylindrical samples(300 mm height and 150 mm diameter) subjected to theaxial compression. The tests were carried out for the com-posites in which two different types of coarse aggregatewere used: (1) crushed limestone aggregate (L) and (2) nat-ural gravel aggregate (G). The experimental data obtainedare in good conformity with the theoretical modelpredictions.

2. The role of the initial concrete microstructure

As it has been pointed out above, the initial concretecomposite microstructure is essential for the behaviour ofthe material under compressive loading. Particularly inter-faces between grains and hardened cement paste as well asshape of aggregates have a crucial importance in under-standing of microdamage process of concrete materialunder compression e.g. [5–9].

Due to technological process (curing and air drying) avirgin concrete has initial microcracks, which are mainlydistributed along aggregate–paste interface. The thicknessof ITZ is in a range 10–50 lm for normal concrete, [8],and it was found that the cement density between aggre-gates is a function of: water–cement ratio, maximumcement diameter and the distance from the aggregate sur-face. The distance between two adjacent grains can bedivided into three regions: two ITZ and central zone. Thedensity of cement particles in the central zone is constant,whereas this density significantly decreases in the neigh-bourhood of the surface walls of grains.

It was also shown that the increase of water–cement ratiocaused decrease of density of particles in the hardenedcement paste. Moreover, ITZ are sites of pre-existing ofthe interfacial micropores and microcracks. Most of thesemicrocracks occur with higher density at interfaces betweencement paste and coarse aggregates. It means that ITZ arethe weakest parts of the cement composite structure regard-ing concrete degradation. When the concrete is subjected tothe compressive loading, concentration of shear and tensilestresses occurs at interfaces due to differences in elasticmodulus of the concrete components. Similar effects wereobserved for the ceramic matrix composites [10,11].

As a result of stress concentrations, concrete microcrackswill develop in the interface zone. When the aggregates stiff-ness is distinctively higher than the hardened cement paste,microcracks tend to propagate through ITZ around aggre-gate particles, which constitute the path of weak resistance.If the stiffness of composite grains is slightly higher or equalto the stiffness of the matrix, microcracks tend to propagatethrough grains, creating shorter fracture paths. Before thepeak of the compressive stress a system of longer meso-cracks occur and link together creating unstable macro-cracks and failure of the concrete composite.

3. Mesomechanical model of the compressed concrete

3.1. Structure and growth of cracks in the

deformed concrete

The reduction of strength of materials like concreteresults from initial structural defects existing inside mate-rial. It has been proved experimentally that the first defectsinitiate, under loading, in the surrounding areas of thelarge grains. Therefore in the present model it has beenassumed that damage grows along the grain boundarieswith loading increase. Figs. 1 and 2 show consecutive

1 l 1 2 l 4

1 3

I2222 σσ ≥

I2222 σσ

1x

2x

σσ≥

I2222 σσ >

I2222 >

II2222 σσ ≥

II2222 σσ ≥

2 1 3

4

Fig. 1. Growth process of straight microcracks at ITZ: 1 – grain, 2 – matrix, 3 – ITZ, 4 – meso- and wing cracks.

l 6

l 5

13 13

II2222

2

1

σσ >

II2222 σσ >

III2222 σσ ≈

III2222 σσ ≈

3

4

Fig. 2. Development of wing cracks: 1 – grain, 2 – matrix, 3 – ITZ,4 – meso- and wing cracks.

T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126 121

phases of microcrack development, whereas Fig. 3 presentscharacteristic level of stress related to them. The stress lev-els rI

22 and rII22 (in Fig. 3) corresponding to mesocracks and

wings initiation were estimated experimentally for analysedconcretes during uniaxial compression by application ofthe acoustic emission method.

σ22

ε22

σ22II

σ22I

II - wings initiation

I - mesocracks initiation

III - failure

ε22IIε22

I

fc

Fig. 3. Stress levels corresponding to initiation of microcracks rI22 and

wing cracks rII22.

One can distinguish the following stages of this process:

• initiation of the straight cracks from micropores distrib-uted along aggregate grains, when the level of stress sat-isfies condition r22 P rI

22. The length of microcracks isdistinctly less than the grain size (Fig. 1b),

• growth of microcrack into mesocrack to the length cor-responding to straight segment of the aggregate bound-ary (Fig. 1c),

• wing cracks initiation from mesocracks, when the loadlevel corresponds to r22 P rII

22, (Fig. 1d),• growth of the wing cracks under mixed mode or mode I

in dependence on: (1) level of applied load, (2) directionof wing development, (3) properties of composite com-ponents. Wing propagation takes place through: matrix(Fig. 2a) or matrix and large aggregate (Fig. 2b).

Theoretical description of crack models, which approx-imate different sets of cracks existing in the real engineeringmaterials one can find in [12–19]. Depending on the type ofthe composite material and loading material history it ispossible to approximate exact set of cracks developmentin any structural element.

3.2. Strain component in the concrete composite undercompression

The total macroscopic strain eij in concrete compositematerial can be expressed as a sum of the following straincomponents:

• purely elastic strains (e0ij),

• strains due to initial porosity (ePij),

• strains due to cracks presence (eDij ),

which gives an expression:

eijðrklÞ ¼ e0ijðrklÞ þ eP

ijðrklÞ þ eDij ðrklÞ: ð1Þ

The idea of mesomechanical modelling is based on theanalysis of the representative volume element (RVE), hav-ing enough large volume V (Fig. 4). In order to get the

22

x2

x1

1

a bσ

A 2

3 4

5

Fig. 4. Internal structure of composite concrete: (a) representative surfaceelement (RSE), (b) concrete sample: 1 – matrix, 2 – sand grains, 3 – grainsof aggregates, 4 – pores, 5 – cracks.

122 T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126

strain components e0ij and eD

ij one can consider sets of poresand cracks existing or initiating in RVE under appliedloading. Macroscopic components of the strain state areobtained by averaging procedure over the RSE surface A

of all defects and discontinuities creating current internalmaterial structure.

Let us limit our consideration to the 2-D case, i.e.i, j = 1,2. Applying mesomechanical approach, e.g. [12–19], one can analyse all phenomena developing inside theRSE. At the beginning of the deformation process theRSE has internal microstructure created by the initialporosity, grains distribution and grain boundary distribu-tion. For advanced state of deformation, the RSE containsalso defects like straight cracks and deflected cracks (whichare approximated by wing cracks). It is worth pointing out,that each flaw generates a local stress and strain field fluctu-ation in the surrounding area. In order to get stress–strainrelations under uniaxial compression (Fig. 4), we apply anaveraging procedure over the area of the RSE. Followingthis procedure one can evaluate the particular strain com-ponents specified in the Eq. (1):

• purely elastic deformation of the material

e0ijðr22Þ ¼

1

A

ZAe

e0eijðr22ÞdA; ð2Þ

where Ae denotes the elastic part of RSE,• deformation of the material due to the initial porosity

ePijðr22Þ ¼

1

A

XNPo

s¼1

ZAðsÞ

Po

e0PoðsÞij ðr22ÞdA; ð3Þ

where NPo is the number of pores inside RSE, AðsÞPo is thearea of a single pore ‘‘s’’,

• deformation of the material due to existence and growthof the set of cracks

eDij ðr22Þ ¼

1

A

XN cr

s¼1

ZAðsÞcr

e0DðsÞij ðr22ÞdA; ð4Þ

The integral Eq. (4) is taken over the part of the RSEAAcr ¼

PN cr

s¼1AðsÞcr , which denotes the total area of cracksexisting in the RSE. AðsÞcr is the area of a single crack ‘‘s’’.Ncr is the number of cracks inside the RSE. One can pointout that the total number of cracks Ncr consists of: numberof microcracks N ðmiÞ

cr , number of straight mesocracks Nm

and number of wing cracks Nw, Figs. 1 and 2. Thereforeassuming that the defects do not interact, integral Eq. (4)can be used for calculation of all kinds of defects. Symbol‘‘s’’ in Eqs. (3) and (4) is related to a single pore or singlecrack in the RSE. The description of calculation of theappropriate components ecr

ij corresponding to differentkinds of cracks is given in [15,16,18].

Having the specified form of Eqs. (2)–(4) for particularcrack growth in any composite material it is possible toestimate averaged strain and formulate the constitutiverelation Eq. (1) for a given load path.

In the analysed problem of uniaxial compression in theconcrete, the strain components eD

ij result from presence ofdifferent shaped cracks in ITZ of coarse aggregates. Themicrocracks of the length l2 (Fig. 1b) have the infinitesi-mally small influence on the material response. Thereforein the analysis we can take into account two kinds ofstrains created by:

a) mesocracks l3, Fig. 5a, which were denoted by eDmðsÞij ,

b) set of wings (deflected part of the wing crack model ofthe length l5), Fig. 5b. This strain component for onewing was denoted by eDwðsÞ

ij .

According to [15,16] the total strain created by a singlewing crack Fig. 5b is composed of strain describing theinfluence of: the central part PP’ (eDmðsÞ

ij ) and two wings(PQ and P’Q’):

eDwðsÞij ¼ eDmðsÞ

ij þ 2eDwðsÞij ¼ eDmðsÞ

ij þ 2 eDwoðsÞij þ eDwzðsÞ

ij

� �: ð5Þ

One can notice that the strain component eDwðsÞij consists

of two parts. The first one, eDwoðsÞij , describes opening of the

wing at points P and P’ due to sliding of the central partPP’ of the wing crack model. The second one, eDwzðsÞ

ij ,describes wing closing process due to compressive externalloading.

3.3. Growth of the straight mesocracks

Initiation of mesocracks in the concrete starts when thelevel of compression stress is equal to r22 ¼ rI

22, Fig. 3.Usually this level corresponds to 0.3–0.45 of the concretestrength r22 ¼ fc. The value of rI

22 is assessed during uniax-ial compression test of cylindrical specimens by analysis oflongitudinal and transversal stains, e.g. [6,20,21]. Under thelevel rI

22 the concrete behaviour is purely elastic, describedby Hooke’s law. The level of stress rI

22 estimated from

2x1x(s)n

(s)s

1x

2x

(s)

22σ

(s)nσ

α

(s)nτ (s)

ml

1x

1x

(w)s

1x

2x

2x

Q

Q'

P'

P

2x(w)n

(w)θ

(w)

l

(s)

2c

(s)α

Fig. 5. Mesocrack and wing crack.

σ 12

22

σ22

Fig. 6. Wing crack growing along ITZ of large aggregate: 1 – RSE,2 – wing crack.

T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126 123

experiments, strongly depends on initial microstructure ofthe analysed material.

When the condition r22 P rI22 is satisfied, mesocracks

begin to initiate inside the material. These cracks have anature of cracks with contact friction (Fig. 5a) due to thecompressive loading. Their presence are described by:space orientation (vector~nðsÞ), its length (lðsÞm ) and an angleof inclination a(s). Their further sliding and growth is con-trolled by effective shear stress �ss

w acting along its directionand equal to:

�sðsÞw aðsÞ; lðsÞ; r22

� �¼ �sðsÞn

�� ��� lðsÞ �rðsÞn

�� ��¼ r22 �

1

2sin 2aðsÞ þ lðsÞ cos2 aðsÞ

� �; ð6Þ

where �s sð Þn and �r sð Þ

n are shear and normal stresses to the mes-ocrack ‘‘s’’ direction; related to the coordinate system(�x1;�x2) associated with the mesocrack. l(s) is the sliding fric-tion along the crack surface, which value should be esti-mated from experiments.

According to Coulomb’s friction law, micro- or meso-cracks start to slide when �sðsÞw > 0. These cracks can furthergrow under mode II when the crack propagation criterionis satisfied:

�sðsÞw aðsÞ; lðsÞm ; lðsÞ; r22

� �¼

ffiffiffiffiffiffiffiffiffi2

plðsÞm

sKITZ

IIcr ; ð7Þ

where lðsÞm is the current microcrack length and KITZIIcr is the

crack resistance of the ITZ. KITZIIcr should be estimated from

experiments for the II mode crack propagation.The most preferential direction of the propagation of

cracks depends on the maximum value of �sðsÞw and is ori-ented at the angle a0:

a0 ¼ arctg lðsÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½lðsÞ�2 þ 1

q� �; ð8Þ

when it is assumed that coefficient of sliding friction all ac-tive cracks is constant.

Denoting by rI22 the load level at which the first meso-

cracks start to propagate at angle a0 and using (6) as wellas (7), one can obtain the fan of inclination of active mes-ocracks am

1 6 aðsÞm 6 am2

� �which propagate to the character-

istic dimension of the grain facet, Fig. 1c:

tgam1;2 ¼ p �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip22lðsÞp � 1

q; ð9Þ

where p ¼ ��sðsÞw = 2rI22

� �.

3.4. Growth of the wing cracks

In the proposed theory, we assume that mesocracksoccupy the straight or slightly curved grain boundary facet(Fig. 1c). Their further growth depends on the internalstructure of the concrete composite. However, SEM obser-vations indicate that mesocracks deflect or change directionof propagation (kink) depending on free space of thematrix phase, Fig. 6. The existence of such zig-zag crackin the material can by approximated by application of wingcrack model, Fig. 5b.

In the general case kinks of the considered mesocrack‘‘s’’ are formed under mixed mode and can further grow,when the respective energy release rate criterion:

GðsÞ r22; aðsÞ; lðsÞ; lðsÞm ; h

ðsÞ; lðsÞw

� �¼ Gcr ð10Þ

Table 2Strength characteristics of the analysed concretes

Type ofconcrete

Concrete strength characteristics

fcm #15

[MPa]fcm Ø15/30

[MPa]fct

[MPa]Ecm

[MPa]

L1 39.2 35.7 2.57 31,325G1 35.2 32.7 2.44 30,169L2 45.1 39.1 3.17 32,645G2 38.9 35.9 2.96 31,828

d

h

σ22

Dmax

124 T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126

reaches a critical value Gcr. Thus, the directional conditionand the failure condition are

dG sð Þ

dh sð Þ ¼ 0 and G sð Þ h sð Þ� �

¼ Gcr; ð11Þ

where Gcr ¼ KITZIc

� �2 þ KITZIIc

� �2h i

=E0cm for 2-D case. Here

E0cm ¼ Ecm= 1� m2cm

� �for plane strain and E0cm ¼ Ecm for

plane stress. Ecm and mcm are initial Young’s modulusand Poisson’s ratio of the concrete composite material ob-tained from experiment.

There are various a physically motivated fracturehypotheses proposed for different materials. For the con-crete composite considered here we adopt a simple follow-ing approach:

KðsÞI

KITZIcr

!aI

þ KðsÞII

KITZIIcr

!aII

¼ 1; ð12Þ

where the four parameters: KITZIcr ;K

ITZIIcr were determined in

experiments, whereas aI and aII were assumed to be equalto 1. The fracture criterion (10) allows for estimation ofthe wing length l sð Þ

w for the given: state of stress r22, the

wing crack geometrical configuration aðsÞ; lðsÞm ; hðsÞ

� �and

coefficient of sliding friction l(s). Additionally, criterion(10) can be used for calculation of the mesocracks fan,which are able to create further two wings – aw

1 6 aðsÞw 6 aw2 .

Fig. 7. Representative cylinder section.

4. Description of deformation process in the concrete

with a changeable internal structure

Two kinds of the structural concrete made of gravelaggregate (G) and limestone aggregate (L) were analysed.In order to determine the influence of coarse aggregategrain size distribution, for each type of concrete, two typesof optimal proportions of different-sized aggregates wereused, with the maximum grain diameter up to 8 and upto 16 mm. The water/cement ratio w/c = 0.4 was the con-stant parameter in all the cement mixtures. Compositionsof concrete mixtures are summarized in Table 1 andstrength characteristics of the concrete are specified inTable 2.

Since the cracks occurring in the coarse aggregate inclu-sion area have decisive influence on the values of the finaldeformations in the concrete, the so-called representative

Table 1Weight composition of the analysed concretes

Component of concrete mix

Portland cement CEM I class 42.5 R from O _zarow Cement PlantPit sand, fraction 0–2 from Markuszow n. LublinGravel, fraction 2–8 (G1 and G2)Gravel, fraction 8–16 (G2)Limestone grit, fraction 2–8 (L1 and L2)Limestone grit, fraction 8–16 mm (L2)Water from municipal water supply systemSuperplasticizer Arpoment P (1.5% of cement mass)

cylinder sector (RCS) was assumed to determine the num-ber of the filling material grains in the most precise way,Fig. 7. It was assumed that, depending on the maximumgrain size distribution of the aggregate, the width of RCSwould be 8 or 16 mm. It was also assumed that up to thestress level of rI

22 i.e. the initiation of cracks in the compos-ite, the deformations would be linear:

e11 ¼�r22

Ecm

mcm; e22 ¼r22

Ecm

; ð13Þ

where mcm – Poisson’s ratio and Ecm – Young’s modulusobtained in the experiment.

When the level of stresses exceeds rI22, the values of

deformation components are calculated on the basis of

Quantity [kg/m3]

352676

From the deposits in Sobolewo n. Suwałki 1207 (G1), 482 (G2)725 (G2)

From the deposits in Trzuskawica n. Kielce 1207 (L1), 482 (L2)725 (L2)1415,28

0

5

10

15

20

25

30

-1500 -500 500 1500 2500

[MPa]σ22

x2

x1

0

experimenttheory

0

5

10

15

20

25

30

-1500 -500 500 1500 2500

11, 22 [1εε 0-6]

x2

x1

0

experimenttheory

[MPa]σ22

11, 22 [1εε 0-6]

b

a

Fig. 8. Deformation process of concrete with the maximum graindiameter up to 8 mm: (a) for specimens L1, (b) for specimens G1.

T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126 125

formulas (1)–(4) taking into account the occurrence ofstraight as well as wing-type defects. Taking experimentaldata into account, it was assumed that the inclination angleof the straight cracks – a(s) would be within the range from40� to 50�, where a half of the defects occurring in the con-crete would be assigned to 45� angle while the two remain-ing angles would be attributed of defects each. Whenr22 > rII

22 wing initiates at angle – h(w) = 60�. Coefficientof sliding friction l(s) was assumed to be 0.3.

It was also assumed that the defects (Nm and Nw) in thecontact layer of the coarse aggregate will be initiatedgradually. An increased number of cracks will occur inthe material at each successive load level. Consequently,the number of the coarse aggregate grains occurring inRCS was divided in proportion to the number of occurringload levels. In the final stage of deformation it wasassumed, that one wing crack corresponds with one coarsegrain.

5. Analysis of the obtained results

Fig. 8 shows examples of the function diagrams of thedeformations occurring in one of the concrete specimensof L1 and G1 series, whereas Fig. 9 presents results forspecimens L2 and G2. The deformations obtained in thetests for the cylinders having the diameter of 150 mm andthe height of 300 mm were taken into account during theanalysis (readouts on the extensometer bridge – ebm

1 ; ebm2 )

and were compared to the deformation values whichresulted from the calculations based on the mesomechani-cal model – eom

1 ; eom2 .

Table 3 summarizes final average values of the deforma-tions for specific concrete series obtained in the teststogether with the results obtained on the basis of the model.Comparison of the final deformation values obtained duringthe tests with the values calculated on the basis of the algo-rithm prepared in the mesomechanical model shows rela-tively good conformity of the obtained results. In the firstseries of the concrete specimens, the smallest differences inthe deformation values occurred: in comparison ofebm

1 to eom1 and ebm

2 to eom2 in G1 concrete (18% and 14%).

The biggest differences occurred in comparison ofebm

1 to eom1 and ebm

2 to eom2 in L1 concrete (32% and 24%).

With regard to the composites with the maximum grain-size distribution up to 16 mm, the results were even morecompatible. The smallest differences occurred in compari-son of ebm

1 to eom1 in L2 concrete (9%) and ebm

2 to eom2 in

G2 concrete (12%). The biggest differences occurred incomparison of ebm

1 to eom1 in G2 concrete (21%) and

ebm2 to eom

2 in L2 concrete (19%).

6. Final remarks

The examples of application of the mesomechanicalmodel to evaluate the deformations of the concrete com-posites have confirmed the usefulness of the model. Dueto the fact that the occurrence of the internal structural

defects was taken into account in the analysis of the con-crete deformations, it was possible to calculate eom

1 andeom

2 which are very similar to the average values ebm1 and

ebm2 obtained in the experiment.

However, the above planar model takes into consider-ation only the defects which develop in the area of contactlayers of the coarse aggregate (straight and wing-typecracks). Consequently, we can assume that the remainingdifference concerning the deformations obtained in theresearch and the deformations calculated using thedescribed model may result from the development of mic-rodefects in the area of the fine aggregate grains and thecement matrix, as well as defect interactions. The impor-tant question in mesomechanical model concerns the num-ber of different defects. Up till now it is impossible to assesexactly this data in experiments.

Presented problem requires further research aimedmainly at developing the model which will take into

0

5

10

15

20

25

30

-1500 -500 500 1500 2500

[Mσ22 Pa]

x2

x1

0

experimenttheory

0

5

10

15

20

25

30

-1500 -500 500 1500 2500

11, 22 [1εε 0-6]

x2

x1

0

experimenttheory

[Mσ22 Pa]

11, 22 [1εε 0-6]

a

b

Fig. 9. Deformation process of concrete with the maximum graindiameter up to 16 mm: (a) for specimens L2, (b) for specimens G2.

Table 3Maximum strains of the analysed concretes

Kind of concrete Mean values of maximum strains [&]

Experiments Mesomechanical model

ebm1 ebm

2 eom1 eom

2

L1 0.98 2.04 0.67 1.54G1 0.83 1.86 0.68 1.60L2 0.86 1.75 0.78 1.42G2 1.05 1.81 0.82 1.59

126 T. Sadowski, G. Golewski / Computational Materials Science 43 (2008) 119–126

consideration the spatial character of defect developmentin the concrete. It is expected that application of such amodel would contribute to even higher conformity of theobtained results.

Acknowledgement

The authors are supported by Polish Ministry of Scienceand Higher Education – grant No 65/6.PR UE/2005-2008/7.

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