Research ArticleMesoscopic Finite Element Modeling of Concrete ConsideringGeometric Boundaries of Actual Aggregates

Hao Jin Yuliang Zhou Binglong Wang and Shunhua Zhou

Key Laboratory of Road and Traffic Engineering of Ministry of Education Tongji University Shanghai 201804 China

Correspondence should be addressed to Yuliang Zhou zyl2831tongjieducn

Received 16 April 2018 Accepted 4 June 2018 Published 2 July 2018

Academic Editor Charles C Sorrell

Copyright copy 2018 Hao Jin et al+is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Concrete is nonhomogeneous and comprises aggregate mortar and interfacial transition zones at the mesoscopic scale +eaggregate shapes significantly affect the development of microcracks To deal with the problem of imprecise description of actualaggregate an innovative method of modeling concrete is proposed in this study considering geometric boundaries of actualaggregate First the geometric feature points of the actual gravel aggregates that is the shape of the actual aggregate are obtainedby laser scanning +e geometric feature points are then moved randomly in the plane Using this method an aggregate library isestablished based on the actual aggregates Finally the front polygons-rear circumcircle conflict and overlap criteria are proposedwhich can achieve a rapid placing process of the multicontrol point aggregate Using this method numerical uniaxial tensile andthree-point bending beam tests are conducted and the results are compared with the round aggregate model +e results indicatethat the geometric properties of aggregates have both blocking and guiding effects on crack development+erefore the proposedmodeling method is better suited for analyzing crack development

1 Introduction

Traditional macroscopic models assume that concrete isisotropic based on continuum mechanics However themodels cannot explain the crack propagation routes underexternal loads Furthermore it is also difficult to describe theconcrete damage caused by stress concentrations [1]+ereforethemacroscopicmodel is not able to determine the relationshipbetween the internal structure of concrete and the macroscopicmechanical properties

With the advances in computing power differentmodeling methods of concrete in the mesoscopic scale wereproposed [2ndash9] Among these methods the random ag-gregate model assumes that concrete comprises aggregatemortar and interfacial transition zones [10ndash12] It highlightsthe random distribution of the aggregate and gives an ex-plicit characterization of the aggregate shape [13]

Pebble aggregate was typically simulated by the circle andthe ellipse in two dimensions [14ndash17] or the sphere and the ovalin three dimensions [18 19] For the gravel aggregate thescholars typically reduced it to polygons or polyhedrons Forinstance Gao and Liu [20] used triangles and quadrilaterals as

an aggregate base and new aggregates were generated byadding new vertices and controlling the side lengths Sun et al[21] used random circles as the aggregate size to generatetriangular aggregate bases and then grouped all the bases si-multaneously and extended them randomly thereby generatinghigh aggregate content models Ma et al [22] used theWallavinformula to determine the distribution of two-dimensional (2D)concrete specimens and then extended the internal polygonbase until its area was the same as the circle aggregate In theabovementioned studies all the aggregate models were basedon regular geometry and specific mathematical algorithmswere applied to structure the shape However this does notreflect the characteristics of actual aggregates

Simulations based on actual aggregate have becomeincreasingly popular [23]+e primary method is to build anaggregate model using computed tomography (CT) sliceimages +ese can better characterize the actual aggregategeometry Ren et al [24] used X-ray CT images to obtain theshape and location information of aggregates and from thisthey proposed the mesoscopic finite element fracture modelQin and Du [25] used the bottom-hat transform methodsto convert CT slices into three-dimensional (3D) images

HindawiAdvances in Materials Science and EngineeringVolume 2018 Article ID 7816502 10 pageshttpsdoiorg10115520187816502

He reconstructed concrete 3D models based on the volumedata method the model took porosity into consideration Panet al [26] analyzed the size distribution of 2D aggregate basedon the 3D sections and proposed a numerical method Ap-plying this method the 2D mass-distribution function witharbitrary gradation can be determined Fu et al [27] used thepolygons to approximately describe clear outlines on the CTslices +e aggregate sizes were normalized and the doublecriterion was used to judge the aggregate intrusion Li andWang [28] established a relatively complete system forconcrete mesoscopic mechanics analysis to simulate theprocess of early-age shrinkage cracking in high-performanceconcrete based on CT image reconstruction Although the CTreconstruction technology can characterize the shape of theactual aggregate satisfactorily the aggregate shape and gra-dation are limited by the specific slice A significant number ofexperiments are needed to satisfy varying gradations there-fore this method has a low simulation efficiency

In order to improve the aggregate modeling we pro-posed an innovative method+is study uses 2Dmodeling asan example and is organized as follows Section 2 introducesthe method of aggregate generation including scanning ofactual aggregate and the establishment of an aggregate li-brary In Section 3 the aggregate placement basic principleand overlap criteria are discussed Two numerical me-chanical tests are conducted and discussed in Section 4 andthe advantages of the proposed model are analyzed Finallythe conclusions are presented in Section 5

2 Aggregate Generation

21 Scanning of Actual Aggregate For actual aggregate theoutline shape of the single aggregate is complex andconcave-convex A Creaform handheld laser scanner wasused to scan aggregate particles for the 3D shapes +ereconstructed aggregate images were obtained by scanningthe actual aggregate as shown in Figure 1

+e numerous contour points of the reconstructed ag-gregate can describe the actual aggregate accurately as shown inFigure 2(a) However this makes the meshing and calculationin finite element models highly complex +erefore it is nec-essary to deal with the contour points of the actual aggregate

To simplify the contour points the key control pointsshould be extracted from the original contour points +esepoints can describe the basic shape of the actual aggregateusing as few as possible contour points +e aggregate shapeis simplified as shown in Figure 2(b) and is known as theaggregate base of the actual aggregate

In this study the control of the curvature and theminimum distance between two points are used to reducethe number of contour points and the simplified aggregateshape can be generated by geometric control points andstraight lines Before determining the minimum distancebetween two points the average particle size of each ag-gregate is calculated as follows

Rave 1113944N

i1

Ri

N (1)

where Rave is the average particle size of the aggregate Ri isthe distance from the contour control point to the geometriccenter and N is the number of contour control points

Assuming any point on the contour as the starting pointthe distance between two adjacent control points satisfies

dis ni ni+11113868111386811138681113868

1113868111386811138681113868le αRave (2)

(a)

(b)

(c)

Figure 1 Actual aggregate laser scanning (a) Creaform laserscanner (b) Actual aggregate (c) Reconstructed aggregate

2 Advances in Materials Science and Engineering

where α is the distance control coefficient+e control conditions for curvature control using the

three-point method are as follows find the circumscribedcircle of the triangle formed by three adjacent points and thecurvature radius of the circumscribed circle can be obtainedIf the curvature radius is greater than a given value themiddle point is removed +e radius of curvature for anythree consecutive points is calculated as follows

(1) Calculate the length of each side of the triangleaccording to the coordinates of three points nini+1ni+1ni+2 and ni+2ni

(2) Calculate the angle of the triangle vertex opposite tothe circumscribed circle θ angnini+1ni+2 accordingto the cosine theorem

(3) Use the sine theorem to find the radius of curvatureR ((ni+1ni+2sin(θ2))( sin(π minus θ)))

(4) +e curvature control radius can be taken as 015ndash02times the average particle size

22 Establishment of Aggregate Library Because of thestatistical similarities among different aggregates the keycontrol points of the actual aggregates can be used as anaggregate base to generate new aggregates +e method ofgenerating new aggregates randomly is as follows

(1) Two circles are drawn with radii α1Rave and(1 + α2)Rave and the centroid of the aggregate is taken

as the center of the circle As shown in Figure 3(a) thecontrol points move between the annular bandcomprising the two circles and α1 and α2 can bedetermined according to the shape of the newaggregate

(2) Select a control point in all control points randomlyand allow it tomove randomly along the double arrowline in Figure 3(a)+e distancemoved is controlled asnRave(0lt nlt 1) and the number of random movesldquonumrdquo is selected according to the deviation from thebase shape of the original aggregate

Figures 3(b)ndash3(d) show an example of the new aggre-gates generated after random movement and the contoursof the aggregates had been determined However the par-ticle size and the location of the aggregate should be takeninto consideration when it is placed into the designated area+erefore we need ways to determine the local coordinatesof the aggregate particles and to adjust the particle size inaccordance with the particle size distribution to treat theaggregate generated above

+e local coordinates of the aggregate particles are de-termined by subtracting the geometric center coordinatesfrom the outline coordinates of the aggregate so that thegeometric center point is at the origin +erefore this pointis easily located when the aggregate is placed

Adjusting the particle size in accordance with the particlesize distribution allows the determination of the circumscribed

(a) (b)

Figure 2 Contour pointsrsquo simplification (a) Original contour points (b) Key control points

(a) (b) (c) (d)

Figure 3 Control points random movement (a) Annular band (b) New aggregate 1 (c) New aggregate 2 (d) New aggregate 3

Advances in Materials Science and Engineering 3

circle diameter of each aggregate in the aggregate library withaggregate coordinates as follows

Xi xi

Dwj

Yi yi

Dwj

(3)

+e aggregate sizes have been treated after the appli-cation of (3) and when placing aggregates the aggregatecoordinates are enlarged and located based on the gradationdetermined by the Walraven formula

After scanning the different shapes of actual aggregatethe number of aggregate bases is determined On this basisthe above methods are used to generate new aggregates anda statistically significant actual aggregate library is formed

3 Aggregate Placement

31 Basic Principle Aggregate placement requires that ag-gregates do not overlap and achieve a certain aggregate contentPreviously the bulk of random aggregate models were simpleconvex polygons In the process of placing scholars establishedthe criterion of aggregate intrusion in order to avoid theoverlapping of aggregates including the arc discriminationmethod [29] the area discriminationmethod [20] the angle sumtestmethod [21] the statematrixmethod [30] and the aggregatedetermination method based on a background grid [31] Inactual aggregates the contour shape is more complex and hasa greater number of control points than the simple convexpolygons In addition earlier discriminant methods for theplacement were limited by computational inefficiencies+erefore it is necessary to establish an aggregate placementalgorithm that adapts to the actual aggregatemulticontrol points

In this study we propose an aggregate placement al-gorithm based on the intrusion judgment of front polygons-rear circumcircle +e algorithm is proposed as follows

Determine the circumcircles of each aggregate in theactual aggregate library and then place the aggregates fromlarge to small according to gradation When placing a newaggregate determine if the new aggregate circumcircle andthe placed aggregate contours overlap the new aggregate iscorrectly located when they do not overlap +e algorithmeffectively solves the problem of low discriminating effi-ciency between polygons with significant numbers of controlpoints while simultaneously satisfying the gradation varia-tion of the aggregates without creating large gaps betweenthe aggregates +is method is a novel way to place actualaggregates with complex geometric boundaries

+e Walraven formula is used to calculate the areaoccupied by each gradation+eWalraven formula [32] is asfollows

Pc DltD0( 1113857 Pk11138741065D050 Dminus05max minus 0053D

40Dminus4max

minus 0012D60Dminus6max minus 00045D

80Dminus8max

minus 00025D100 Dminus10max1113875

(4)

An aggregate is randomly selected from the aggregatelibrary and the circumcenter circle is determined In orderto ensure the randomness of aggregates the aggregates weresubjected to aggregate rotation and particle size gradationbefore placing

Aggregate rotation is performed by randomly rotatinga selected aggregate around its circumscribed circle center

x1 y0 cos(360 times ran1)minus x0 sin(360 times ran1)

y1 y0 sin(360 times ran2)minusx0 cos(360 times ran2)(5)

where x0 and y0 are the coordinates of the control pointbefore rotation x1 and y1 are the coordinates of the controlpoint after rotation and ran1 and ran2 are the randomnumbers from 0 to 1

Particle size grading is performed by randomly expandingaggregates within a range according to the graded distributionrange of the aggregates generating aggregates meeting thespecified gradation requirements

x2 D1 + D2 minusD1( 1113857 times ran31113858 1113859 times x1

y2 D1 + D2 minusD1( 1113857 times ran31113858 1113859 times y1(6)

where x2 andy2 are the coordinates of the control point afterexpansion D1 and D2 are the maximum and minimumparticle sizes in the grading range respectively and ran3 isa random number from 0 to 1

+e aggregates are placed from large to small until thecumulative area satisfies the gradation requirements Oncethis gradation placement is completed the next gradationplacement will be done

32 Aggregate Overlap Criteria An aggregate placementprocess is complete when the circumcircle of the aggregate isplaced into the designated area +erefore it is necessary tojudge whether the circumcircles of the existing aggregatesand the new aggregate overlap when placing and this can beregarded as the intrusion relationship between polygons andcircles +e method simplifies the judgment required be-tween traditional polygons Simultaneously the principle ofplacing aggregates from large to small circumvents the problemof low aggregate content because of gaps between the cir-cumscribed circles and the polygons

When judging the location of the relationship betweencircles and polygons it can be divided into three cases (1)control point inside circle (2) polygonal edges intersect withcircles and (3) circle inside the polygon as shown in Fig-ure 4 For the first case it can be immediately determined ifthe aggregate control point is inside the circumcircle for thesecond case it is required to judge whether the distance fromthe circumcircle center to the straight line is smaller than thecircumcircle radius and whether the vertical point falls onthe line segment formed by the adjacent control point forthe third case calculate each angle between the circumcirclecenter and two adjacent vertices and then determine if thesum of these angles is equal to 360deg

4 Advances in Materials Science and Engineering

33 Aggregate Generation by Gradation +e coarse aggre-gates of concrete are divided into four gradations accordingto the particle size small stone (5ndash20mm) stone (20ndash40mm) large stone (40ndash80mm) and extra-large stone(80ndash150mm) [33] Using the method proposed in this studyto generate two three and four-gradation concrete speci-mens as shown in Figure 5 the size of the two-gradationaggregate model is 150times150mm and the aggregate contentis 4542 +e size of the three-gradation aggregate model is300times 300mm and the aggregate content is 5217 +e sizeof the four-gradation aggregate model is 450times 450mm andthe aggregate content is 5400

4 Numerical Test and Discussion

41 Concrete Uniaxial Tension Test +e 2D aggregate modelestablished in this study is based on the actual aggregateboundary features and has a high degree of similarity withthe actual aggregate In order to study the effect of aggregateshape on the macroscopic mechanical properties and failurecharacteristics of concrete numerical simulations of theuniaxial tensile test and three-point bending test of circleaggregate model and actual aggregate model were conducted

+e aim was to determine the aggregate shape influence onmacroscopic mechanical behavior

According to the above algorithms for generation andplacement an actual aggregate model and a circular ag-gregate model are generated the geometric model is shownin Figure 6 +e specimen size is 100times100mm and theaggregate size is 2ndash20mm +e aggregate contents of theactual aggregate model and the circular aggregate model are4909 and 4911 respectively and geometric shrinkagereduces the interface thickness by 05mm to form an in-terface between aggregate and mortar

+e geometric model of the aggregate is imported intothe ABAQUS finite element software+emesh is divided bya quadrilateral-based freeform grid Taking the actual ag-gregate model as an example the model grid is divided intoa grid as shown in Figure 7 and the total number of elementsis 50717 +e total number of nodes is 50149

+e ABAQUS damage plasticity model is used to describethe fracture behavior of concrete materials +is model can beused to characterize the inelastic behavior of concrete com-bined with isotropic elastic damage and isotropic tensile andcompressive plasticity theory In order to simplify the calcu-lation the evolution model of the double-fold damage variable

Contour controlpoint

(a)

Contour controlpoint

(b)

Contour controlpoint

(c)

Figure 4 Polygons-circumcircle conflict and overlap criteria (a) Control point inside circle (b) Polygonal edges intersect with circle(c) Circle inside the polygon

(a) (b) (c)

Figure 5 Aggregate generation by gradation (a) Two-gradation aggregate content 4542 (b)+ree-gradation aggregate content 5217(c) Four-gradation aggregate content 5400

Advances in Materials Science and Engineering 5

[34] is adopted as the constitutive of each phase material asshown in Figure 8

+e damage variables can be expressed as

D

0 εmax lt ε0

1minusηminus ληminus 1

ε0εmax

+1minus ληminus 1

ε0 lt εmax le εr

1minus λε0εmax

εr lt εmax le εu

1 εmax gt εu

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where ft is the tensile strength ftr is the residual tensilestrength ε0 is the principal tensile strain corresponding to ftεr is the residual strain corresponding to the tensile residualstrength ftr εu is the ultimate tensile strain and εmax is thelargest primary tensile strain value of the loading history+e relationship between parameters is ftr λfr(0lt λlt 1)εr ε0 and εu ξε0(ξ gt η)

+e concrete phase material mechanical parameters arepresented in Table 1 where λ was taken as 01 η was taken as10 and the aggregate mortar and interface ξ were taken 5 4and 4 respectively +e concrete uniaxial tensile specimencalculation diagram is shown in Figure 9 where the leftboundary is the applied horizontal constraint the left middleis the applied vertical constraint and the right boundary isthe applied displacement load

+e macroscopic stress-strain curve of the concretetensile specimen is shown in Figure 10 In the elastic phasethe gradients of the stress-strain curve of the circularaggregate model and the actual aggregate model are es-sentially identical which indicates that the stress char-acteristics of the macrospecimens in the elastic phase arethe same At the end of the elastic phase the strain of thespecimen approaches 7 times 10minus5 numerous cracks are ini-tiated at the interface in the concrete specimens andmicrocracks are formed

After entering the softening stage the damage areaexpands as the strain increases In order to observe thedevelopment of macroscopic cracks the strain of the testspecimens was increased to greater than 2 times 10minus5 where

(a) (b)

Figure 6 Specimen geometry model (a) Actual random aggregate (b) Circle random aggregate

(a) (b) (c) (d)

Figure 7 Uniaxial tensile specimen model mesh (a) Geometric model (b) Aggregate mesh (c) Interface mesh (d) Mortar mesh

6 Advances in Materials Science and Engineering

macroscopic cracks greater than 002mm became visibleApart from the small cracks that developed on the left side ofthe circular aggregate model macroscopic cracks continuedto develop along the aggregate interface and the mortar inthe middle left of the specimen and finally formed a mac-roscopic crack perpendicular to the tensile direction of thespecimen as shown in Figure 11(a) For the actual aggregatemodel the crack developed on the right of the specimenperpendicular to the direction of stretching Unlike for thecircular aggregate the macroscopic cracks in the actualaggregate model developed simultaneously at the two in-terfaces a phenomenon known as crack bifurcation andfinally met at the bottom as shown in Figure 11(b) From thestress-strain curve as the aggregate shape and aggregate

distribution of the circular aggregate model and the actualaggregate model are different the stress peak and the cor-responding strain of the actual aggregate model are mar-ginally greater than that of the circular aggregate which ledto single and bifurcation cracks +is phenomenon reflectsthe significant influence of aggregate geometry on themacroscopic mechanical properties of the specimens

In the descending section of the stress-strain curve theaggregate cracks continue to develop +e stress-straincurves of the actual aggregate model and the circular ag-gregate model decrease at a similar rate and the softenedend of the curves are coincident At this point the crack in

Table 1 Concrete phase material mechanical parameters

Material type Elastic modulus (GPa) Tensile strength (MPa) Poissonrsquos ratioAggregate 50 6 02Mortar 26 32 02Interface 25 3 02Concrete 32 02

Nonuniformregion

100 mm

100 m

m

F

Figure 9 Uniaxial tensile specimen calculation diagram

f

ft

ftr

ε0 εr εu

Figure 8 Double-fold line damage constitutive relation

000 002 004 006 008 010 01200

05

10

15

20

25

30

Stre

ss (M

Pa)

Strain (10ndash3)

Circle aggregate modelActual aggregate model

Figure 10 Macroscopic stress-strain curve

Advances in Materials Science and Engineering 7

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

where α is the distance control coefficient+e control conditions for curvature control using the

three-point method are as follows find the circumscribedcircle of the triangle formed by three adjacent points and thecurvature radius of the circumscribed circle can be obtainedIf the curvature radius is greater than a given value themiddle point is removed +e radius of curvature for anythree consecutive points is calculated as follows

(1) Calculate the length of each side of the triangleaccording to the coordinates of three points nini+1ni+1ni+2 and ni+2ni

(2) Calculate the angle of the triangle vertex opposite tothe circumscribed circle θ angnini+1ni+2 accordingto the cosine theorem

(3) Use the sine theorem to find the radius of curvatureR ((ni+1ni+2sin(θ2))( sin(π minus θ)))

(4) +e curvature control radius can be taken as 015ndash02times the average particle size

22 Establishment of Aggregate Library Because of thestatistical similarities among different aggregates the keycontrol points of the actual aggregates can be used as anaggregate base to generate new aggregates +e method ofgenerating new aggregates randomly is as follows

(1) Two circles are drawn with radii α1Rave and(1 + α2)Rave and the centroid of the aggregate is taken

as the center of the circle As shown in Figure 3(a) thecontrol points move between the annular bandcomprising the two circles and α1 and α2 can bedetermined according to the shape of the newaggregate

(2) Select a control point in all control points randomlyand allow it tomove randomly along the double arrowline in Figure 3(a)+e distancemoved is controlled asnRave(0lt nlt 1) and the number of random movesldquonumrdquo is selected according to the deviation from thebase shape of the original aggregate

Figures 3(b)ndash3(d) show an example of the new aggre-gates generated after random movement and the contoursof the aggregates had been determined However the par-ticle size and the location of the aggregate should be takeninto consideration when it is placed into the designated area+erefore we need ways to determine the local coordinatesof the aggregate particles and to adjust the particle size inaccordance with the particle size distribution to treat theaggregate generated above

+e local coordinates of the aggregate particles are de-termined by subtracting the geometric center coordinatesfrom the outline coordinates of the aggregate so that thegeometric center point is at the origin +erefore this pointis easily located when the aggregate is placed

Adjusting the particle size in accordance with the particlesize distribution allows the determination of the circumscribed

(a) (b)

Figure 2 Contour pointsrsquo simplification (a) Original contour points (b) Key control points

(a) (b) (c) (d)

Figure 3 Control points random movement (a) Annular band (b) New aggregate 1 (c) New aggregate 2 (d) New aggregate 3

Advances in Materials Science and Engineering 3

circle diameter of each aggregate in the aggregate library withaggregate coordinates as follows

Xi xi

Dwj

Yi yi

Dwj

(3)

+e aggregate sizes have been treated after the appli-cation of (3) and when placing aggregates the aggregatecoordinates are enlarged and located based on the gradationdetermined by the Walraven formula

After scanning the different shapes of actual aggregatethe number of aggregate bases is determined On this basisthe above methods are used to generate new aggregates anda statistically significant actual aggregate library is formed

3 Aggregate Placement

31 Basic Principle Aggregate placement requires that ag-gregates do not overlap and achieve a certain aggregate contentPreviously the bulk of random aggregate models were simpleconvex polygons In the process of placing scholars establishedthe criterion of aggregate intrusion in order to avoid theoverlapping of aggregates including the arc discriminationmethod [29] the area discriminationmethod [20] the angle sumtestmethod [21] the statematrixmethod [30] and the aggregatedetermination method based on a background grid [31] Inactual aggregates the contour shape is more complex and hasa greater number of control points than the simple convexpolygons In addition earlier discriminant methods for theplacement were limited by computational inefficiencies+erefore it is necessary to establish an aggregate placementalgorithm that adapts to the actual aggregatemulticontrol points

In this study we propose an aggregate placement al-gorithm based on the intrusion judgment of front polygons-rear circumcircle +e algorithm is proposed as follows

Determine the circumcircles of each aggregate in theactual aggregate library and then place the aggregates fromlarge to small according to gradation When placing a newaggregate determine if the new aggregate circumcircle andthe placed aggregate contours overlap the new aggregate iscorrectly located when they do not overlap +e algorithmeffectively solves the problem of low discriminating effi-ciency between polygons with significant numbers of controlpoints while simultaneously satisfying the gradation varia-tion of the aggregates without creating large gaps betweenthe aggregates +is method is a novel way to place actualaggregates with complex geometric boundaries

+e Walraven formula is used to calculate the areaoccupied by each gradation+eWalraven formula [32] is asfollows

Pc DltD0( 1113857 Pk11138741065D050 Dminus05max minus 0053D

40Dminus4max

minus 0012D60Dminus6max minus 00045D

80Dminus8max

minus 00025D100 Dminus10max1113875

(4)

An aggregate is randomly selected from the aggregatelibrary and the circumcenter circle is determined In orderto ensure the randomness of aggregates the aggregates weresubjected to aggregate rotation and particle size gradationbefore placing

Aggregate rotation is performed by randomly rotatinga selected aggregate around its circumscribed circle center

x1 y0 cos(360 times ran1)minus x0 sin(360 times ran1)

y1 y0 sin(360 times ran2)minusx0 cos(360 times ran2)(5)

where x0 and y0 are the coordinates of the control pointbefore rotation x1 and y1 are the coordinates of the controlpoint after rotation and ran1 and ran2 are the randomnumbers from 0 to 1

Particle size grading is performed by randomly expandingaggregates within a range according to the graded distributionrange of the aggregates generating aggregates meeting thespecified gradation requirements

x2 D1 + D2 minusD1( 1113857 times ran31113858 1113859 times x1

y2 D1 + D2 minusD1( 1113857 times ran31113858 1113859 times y1(6)

where x2 andy2 are the coordinates of the control point afterexpansion D1 and D2 are the maximum and minimumparticle sizes in the grading range respectively and ran3 isa random number from 0 to 1

+e aggregates are placed from large to small until thecumulative area satisfies the gradation requirements Oncethis gradation placement is completed the next gradationplacement will be done

32 Aggregate Overlap Criteria An aggregate placementprocess is complete when the circumcircle of the aggregate isplaced into the designated area +erefore it is necessary tojudge whether the circumcircles of the existing aggregatesand the new aggregate overlap when placing and this can beregarded as the intrusion relationship between polygons andcircles +e method simplifies the judgment required be-tween traditional polygons Simultaneously the principle ofplacing aggregates from large to small circumvents the problemof low aggregate content because of gaps between the cir-cumscribed circles and the polygons

When judging the location of the relationship betweencircles and polygons it can be divided into three cases (1)control point inside circle (2) polygonal edges intersect withcircles and (3) circle inside the polygon as shown in Fig-ure 4 For the first case it can be immediately determined ifthe aggregate control point is inside the circumcircle for thesecond case it is required to judge whether the distance fromthe circumcircle center to the straight line is smaller than thecircumcircle radius and whether the vertical point falls onthe line segment formed by the adjacent control point forthe third case calculate each angle between the circumcirclecenter and two adjacent vertices and then determine if thesum of these angles is equal to 360deg

4 Advances in Materials Science and Engineering

33 Aggregate Generation by Gradation +e coarse aggre-gates of concrete are divided into four gradations accordingto the particle size small stone (5ndash20mm) stone (20ndash40mm) large stone (40ndash80mm) and extra-large stone(80ndash150mm) [33] Using the method proposed in this studyto generate two three and four-gradation concrete speci-mens as shown in Figure 5 the size of the two-gradationaggregate model is 150times150mm and the aggregate contentis 4542 +e size of the three-gradation aggregate model is300times 300mm and the aggregate content is 5217 +e sizeof the four-gradation aggregate model is 450times 450mm andthe aggregate content is 5400

4 Numerical Test and Discussion

41 Concrete Uniaxial Tension Test +e 2D aggregate modelestablished in this study is based on the actual aggregateboundary features and has a high degree of similarity withthe actual aggregate In order to study the effect of aggregateshape on the macroscopic mechanical properties and failurecharacteristics of concrete numerical simulations of theuniaxial tensile test and three-point bending test of circleaggregate model and actual aggregate model were conducted

+e aim was to determine the aggregate shape influence onmacroscopic mechanical behavior

According to the above algorithms for generation andplacement an actual aggregate model and a circular ag-gregate model are generated the geometric model is shownin Figure 6 +e specimen size is 100times100mm and theaggregate size is 2ndash20mm +e aggregate contents of theactual aggregate model and the circular aggregate model are4909 and 4911 respectively and geometric shrinkagereduces the interface thickness by 05mm to form an in-terface between aggregate and mortar

+e geometric model of the aggregate is imported intothe ABAQUS finite element software+emesh is divided bya quadrilateral-based freeform grid Taking the actual ag-gregate model as an example the model grid is divided intoa grid as shown in Figure 7 and the total number of elementsis 50717 +e total number of nodes is 50149

+e ABAQUS damage plasticity model is used to describethe fracture behavior of concrete materials +is model can beused to characterize the inelastic behavior of concrete com-bined with isotropic elastic damage and isotropic tensile andcompressive plasticity theory In order to simplify the calcu-lation the evolution model of the double-fold damage variable

Contour controlpoint

(a)

Contour controlpoint

(b)

Contour controlpoint

(c)

Figure 4 Polygons-circumcircle conflict and overlap criteria (a) Control point inside circle (b) Polygonal edges intersect with circle(c) Circle inside the polygon

(a) (b) (c)

Figure 5 Aggregate generation by gradation (a) Two-gradation aggregate content 4542 (b)+ree-gradation aggregate content 5217(c) Four-gradation aggregate content 5400

Advances in Materials Science and Engineering 5

[34] is adopted as the constitutive of each phase material asshown in Figure 8

+e damage variables can be expressed as

D

0 εmax lt ε0

1minusηminus ληminus 1

ε0εmax

+1minus ληminus 1

ε0 lt εmax le εr

1minus λε0εmax

εr lt εmax le εu

1 εmax gt εu

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where ft is the tensile strength ftr is the residual tensilestrength ε0 is the principal tensile strain corresponding to ftεr is the residual strain corresponding to the tensile residualstrength ftr εu is the ultimate tensile strain and εmax is thelargest primary tensile strain value of the loading history+e relationship between parameters is ftr λfr(0lt λlt 1)εr ε0 and εu ξε0(ξ gt η)

+e concrete phase material mechanical parameters arepresented in Table 1 where λ was taken as 01 η was taken as10 and the aggregate mortar and interface ξ were taken 5 4and 4 respectively +e concrete uniaxial tensile specimencalculation diagram is shown in Figure 9 where the leftboundary is the applied horizontal constraint the left middleis the applied vertical constraint and the right boundary isthe applied displacement load

+e macroscopic stress-strain curve of the concretetensile specimen is shown in Figure 10 In the elastic phasethe gradients of the stress-strain curve of the circularaggregate model and the actual aggregate model are es-sentially identical which indicates that the stress char-acteristics of the macrospecimens in the elastic phase arethe same At the end of the elastic phase the strain of thespecimen approaches 7 times 10minus5 numerous cracks are ini-tiated at the interface in the concrete specimens andmicrocracks are formed

After entering the softening stage the damage areaexpands as the strain increases In order to observe thedevelopment of macroscopic cracks the strain of the testspecimens was increased to greater than 2 times 10minus5 where

(a) (b)

Figure 6 Specimen geometry model (a) Actual random aggregate (b) Circle random aggregate

(a) (b) (c) (d)

Figure 7 Uniaxial tensile specimen model mesh (a) Geometric model (b) Aggregate mesh (c) Interface mesh (d) Mortar mesh

6 Advances in Materials Science and Engineering

macroscopic cracks greater than 002mm became visibleApart from the small cracks that developed on the left side ofthe circular aggregate model macroscopic cracks continuedto develop along the aggregate interface and the mortar inthe middle left of the specimen and finally formed a mac-roscopic crack perpendicular to the tensile direction of thespecimen as shown in Figure 11(a) For the actual aggregatemodel the crack developed on the right of the specimenperpendicular to the direction of stretching Unlike for thecircular aggregate the macroscopic cracks in the actualaggregate model developed simultaneously at the two in-terfaces a phenomenon known as crack bifurcation andfinally met at the bottom as shown in Figure 11(b) From thestress-strain curve as the aggregate shape and aggregate

distribution of the circular aggregate model and the actualaggregate model are different the stress peak and the cor-responding strain of the actual aggregate model are mar-ginally greater than that of the circular aggregate which ledto single and bifurcation cracks +is phenomenon reflectsthe significant influence of aggregate geometry on themacroscopic mechanical properties of the specimens

In the descending section of the stress-strain curve theaggregate cracks continue to develop +e stress-straincurves of the actual aggregate model and the circular ag-gregate model decrease at a similar rate and the softenedend of the curves are coincident At this point the crack in

Table 1 Concrete phase material mechanical parameters

Material type Elastic modulus (GPa) Tensile strength (MPa) Poissonrsquos ratioAggregate 50 6 02Mortar 26 32 02Interface 25 3 02Concrete 32 02

Nonuniformregion

100 mm

100 m

m

F

Figure 9 Uniaxial tensile specimen calculation diagram

f

ft

ftr

ε0 εr εu

Figure 8 Double-fold line damage constitutive relation

000 002 004 006 008 010 01200

05

10

15

20

25

30

Stre

ss (M

Pa)

Strain (10ndash3)

Circle aggregate modelActual aggregate model

Figure 10 Macroscopic stress-strain curve

Advances in Materials Science and Engineering 7

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

circle diameter of each aggregate in the aggregate library withaggregate coordinates as follows

Xi xi

Dwj

Yi yi

Dwj

(3)

+e aggregate sizes have been treated after the appli-cation of (3) and when placing aggregates the aggregatecoordinates are enlarged and located based on the gradationdetermined by the Walraven formula

After scanning the different shapes of actual aggregatethe number of aggregate bases is determined On this basisthe above methods are used to generate new aggregates anda statistically significant actual aggregate library is formed

3 Aggregate Placement

31 Basic Principle Aggregate placement requires that ag-gregates do not overlap and achieve a certain aggregate contentPreviously the bulk of random aggregate models were simpleconvex polygons In the process of placing scholars establishedthe criterion of aggregate intrusion in order to avoid theoverlapping of aggregates including the arc discriminationmethod [29] the area discriminationmethod [20] the angle sumtestmethod [21] the statematrixmethod [30] and the aggregatedetermination method based on a background grid [31] Inactual aggregates the contour shape is more complex and hasa greater number of control points than the simple convexpolygons In addition earlier discriminant methods for theplacement were limited by computational inefficiencies+erefore it is necessary to establish an aggregate placementalgorithm that adapts to the actual aggregatemulticontrol points

In this study we propose an aggregate placement al-gorithm based on the intrusion judgment of front polygons-rear circumcircle +e algorithm is proposed as follows

Determine the circumcircles of each aggregate in theactual aggregate library and then place the aggregates fromlarge to small according to gradation When placing a newaggregate determine if the new aggregate circumcircle andthe placed aggregate contours overlap the new aggregate iscorrectly located when they do not overlap +e algorithmeffectively solves the problem of low discriminating effi-ciency between polygons with significant numbers of controlpoints while simultaneously satisfying the gradation varia-tion of the aggregates without creating large gaps betweenthe aggregates +is method is a novel way to place actualaggregates with complex geometric boundaries

+e Walraven formula is used to calculate the areaoccupied by each gradation+eWalraven formula [32] is asfollows

Pc DltD0( 1113857 Pk11138741065D050 Dminus05max minus 0053D

40Dminus4max

minus 0012D60Dminus6max minus 00045D

80Dminus8max

minus 00025D100 Dminus10max1113875

(4)

An aggregate is randomly selected from the aggregatelibrary and the circumcenter circle is determined In orderto ensure the randomness of aggregates the aggregates weresubjected to aggregate rotation and particle size gradationbefore placing

Aggregate rotation is performed by randomly rotatinga selected aggregate around its circumscribed circle center

x1 y0 cos(360 times ran1)minus x0 sin(360 times ran1)

y1 y0 sin(360 times ran2)minusx0 cos(360 times ran2)(5)

where x0 and y0 are the coordinates of the control pointbefore rotation x1 and y1 are the coordinates of the controlpoint after rotation and ran1 and ran2 are the randomnumbers from 0 to 1

Particle size grading is performed by randomly expandingaggregates within a range according to the graded distributionrange of the aggregates generating aggregates meeting thespecified gradation requirements

x2 D1 + D2 minusD1( 1113857 times ran31113858 1113859 times x1

y2 D1 + D2 minusD1( 1113857 times ran31113858 1113859 times y1(6)

where x2 andy2 are the coordinates of the control point afterexpansion D1 and D2 are the maximum and minimumparticle sizes in the grading range respectively and ran3 isa random number from 0 to 1

+e aggregates are placed from large to small until thecumulative area satisfies the gradation requirements Oncethis gradation placement is completed the next gradationplacement will be done

32 Aggregate Overlap Criteria An aggregate placementprocess is complete when the circumcircle of the aggregate isplaced into the designated area +erefore it is necessary tojudge whether the circumcircles of the existing aggregatesand the new aggregate overlap when placing and this can beregarded as the intrusion relationship between polygons andcircles +e method simplifies the judgment required be-tween traditional polygons Simultaneously the principle ofplacing aggregates from large to small circumvents the problemof low aggregate content because of gaps between the cir-cumscribed circles and the polygons

When judging the location of the relationship betweencircles and polygons it can be divided into three cases (1)control point inside circle (2) polygonal edges intersect withcircles and (3) circle inside the polygon as shown in Fig-ure 4 For the first case it can be immediately determined ifthe aggregate control point is inside the circumcircle for thesecond case it is required to judge whether the distance fromthe circumcircle center to the straight line is smaller than thecircumcircle radius and whether the vertical point falls onthe line segment formed by the adjacent control point forthe third case calculate each angle between the circumcirclecenter and two adjacent vertices and then determine if thesum of these angles is equal to 360deg

4 Advances in Materials Science and Engineering

33 Aggregate Generation by Gradation +e coarse aggre-gates of concrete are divided into four gradations accordingto the particle size small stone (5ndash20mm) stone (20ndash40mm) large stone (40ndash80mm) and extra-large stone(80ndash150mm) [33] Using the method proposed in this studyto generate two three and four-gradation concrete speci-mens as shown in Figure 5 the size of the two-gradationaggregate model is 150times150mm and the aggregate contentis 4542 +e size of the three-gradation aggregate model is300times 300mm and the aggregate content is 5217 +e sizeof the four-gradation aggregate model is 450times 450mm andthe aggregate content is 5400

4 Numerical Test and Discussion

41 Concrete Uniaxial Tension Test +e 2D aggregate modelestablished in this study is based on the actual aggregateboundary features and has a high degree of similarity withthe actual aggregate In order to study the effect of aggregateshape on the macroscopic mechanical properties and failurecharacteristics of concrete numerical simulations of theuniaxial tensile test and three-point bending test of circleaggregate model and actual aggregate model were conducted

+e aim was to determine the aggregate shape influence onmacroscopic mechanical behavior

According to the above algorithms for generation andplacement an actual aggregate model and a circular ag-gregate model are generated the geometric model is shownin Figure 6 +e specimen size is 100times100mm and theaggregate size is 2ndash20mm +e aggregate contents of theactual aggregate model and the circular aggregate model are4909 and 4911 respectively and geometric shrinkagereduces the interface thickness by 05mm to form an in-terface between aggregate and mortar

+e geometric model of the aggregate is imported intothe ABAQUS finite element software+emesh is divided bya quadrilateral-based freeform grid Taking the actual ag-gregate model as an example the model grid is divided intoa grid as shown in Figure 7 and the total number of elementsis 50717 +e total number of nodes is 50149

+e ABAQUS damage plasticity model is used to describethe fracture behavior of concrete materials +is model can beused to characterize the inelastic behavior of concrete com-bined with isotropic elastic damage and isotropic tensile andcompressive plasticity theory In order to simplify the calcu-lation the evolution model of the double-fold damage variable

Contour controlpoint

(a)

Contour controlpoint

(b)

Contour controlpoint

(c)

Figure 4 Polygons-circumcircle conflict and overlap criteria (a) Control point inside circle (b) Polygonal edges intersect with circle(c) Circle inside the polygon

(a) (b) (c)

Figure 5 Aggregate generation by gradation (a) Two-gradation aggregate content 4542 (b)+ree-gradation aggregate content 5217(c) Four-gradation aggregate content 5400

Advances in Materials Science and Engineering 5

[34] is adopted as the constitutive of each phase material asshown in Figure 8

+e damage variables can be expressed as

D

0 εmax lt ε0

1minusηminus ληminus 1

ε0εmax

+1minus ληminus 1

ε0 lt εmax le εr

1minus λε0εmax

εr lt εmax le εu

1 εmax gt εu

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where ft is the tensile strength ftr is the residual tensilestrength ε0 is the principal tensile strain corresponding to ftεr is the residual strain corresponding to the tensile residualstrength ftr εu is the ultimate tensile strain and εmax is thelargest primary tensile strain value of the loading history+e relationship between parameters is ftr λfr(0lt λlt 1)εr ε0 and εu ξε0(ξ gt η)

+e concrete phase material mechanical parameters arepresented in Table 1 where λ was taken as 01 η was taken as10 and the aggregate mortar and interface ξ were taken 5 4and 4 respectively +e concrete uniaxial tensile specimencalculation diagram is shown in Figure 9 where the leftboundary is the applied horizontal constraint the left middleis the applied vertical constraint and the right boundary isthe applied displacement load

+e macroscopic stress-strain curve of the concretetensile specimen is shown in Figure 10 In the elastic phasethe gradients of the stress-strain curve of the circularaggregate model and the actual aggregate model are es-sentially identical which indicates that the stress char-acteristics of the macrospecimens in the elastic phase arethe same At the end of the elastic phase the strain of thespecimen approaches 7 times 10minus5 numerous cracks are ini-tiated at the interface in the concrete specimens andmicrocracks are formed

After entering the softening stage the damage areaexpands as the strain increases In order to observe thedevelopment of macroscopic cracks the strain of the testspecimens was increased to greater than 2 times 10minus5 where

(a) (b)

Figure 6 Specimen geometry model (a) Actual random aggregate (b) Circle random aggregate

(a) (b) (c) (d)

Figure 7 Uniaxial tensile specimen model mesh (a) Geometric model (b) Aggregate mesh (c) Interface mesh (d) Mortar mesh

6 Advances in Materials Science and Engineering

macroscopic cracks greater than 002mm became visibleApart from the small cracks that developed on the left side ofthe circular aggregate model macroscopic cracks continuedto develop along the aggregate interface and the mortar inthe middle left of the specimen and finally formed a mac-roscopic crack perpendicular to the tensile direction of thespecimen as shown in Figure 11(a) For the actual aggregatemodel the crack developed on the right of the specimenperpendicular to the direction of stretching Unlike for thecircular aggregate the macroscopic cracks in the actualaggregate model developed simultaneously at the two in-terfaces a phenomenon known as crack bifurcation andfinally met at the bottom as shown in Figure 11(b) From thestress-strain curve as the aggregate shape and aggregate

distribution of the circular aggregate model and the actualaggregate model are different the stress peak and the cor-responding strain of the actual aggregate model are mar-ginally greater than that of the circular aggregate which ledto single and bifurcation cracks +is phenomenon reflectsthe significant influence of aggregate geometry on themacroscopic mechanical properties of the specimens

In the descending section of the stress-strain curve theaggregate cracks continue to develop +e stress-straincurves of the actual aggregate model and the circular ag-gregate model decrease at a similar rate and the softenedend of the curves are coincident At this point the crack in

Table 1 Concrete phase material mechanical parameters

Material type Elastic modulus (GPa) Tensile strength (MPa) Poissonrsquos ratioAggregate 50 6 02Mortar 26 32 02Interface 25 3 02Concrete 32 02

Nonuniformregion

100 mm

100 m

m

F

Figure 9 Uniaxial tensile specimen calculation diagram

f

ft

ftr

ε0 εr εu

Figure 8 Double-fold line damage constitutive relation

000 002 004 006 008 010 01200

05

10

15

20

25

30

Stre

ss (M

Pa)

Strain (10ndash3)

Circle aggregate modelActual aggregate model

Figure 10 Macroscopic stress-strain curve

Advances in Materials Science and Engineering 7

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

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Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

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Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

TribologyAdvances in

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ChemistryAdvances in

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BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

33 Aggregate Generation by Gradation +e coarse aggre-gates of concrete are divided into four gradations accordingto the particle size small stone (5ndash20mm) stone (20ndash40mm) large stone (40ndash80mm) and extra-large stone(80ndash150mm) [33] Using the method proposed in this studyto generate two three and four-gradation concrete speci-mens as shown in Figure 5 the size of the two-gradationaggregate model is 150times150mm and the aggregate contentis 4542 +e size of the three-gradation aggregate model is300times 300mm and the aggregate content is 5217 +e sizeof the four-gradation aggregate model is 450times 450mm andthe aggregate content is 5400

4 Numerical Test and Discussion

41 Concrete Uniaxial Tension Test +e 2D aggregate modelestablished in this study is based on the actual aggregateboundary features and has a high degree of similarity withthe actual aggregate In order to study the effect of aggregateshape on the macroscopic mechanical properties and failurecharacteristics of concrete numerical simulations of theuniaxial tensile test and three-point bending test of circleaggregate model and actual aggregate model were conducted

+e aim was to determine the aggregate shape influence onmacroscopic mechanical behavior

According to the above algorithms for generation andplacement an actual aggregate model and a circular ag-gregate model are generated the geometric model is shownin Figure 6 +e specimen size is 100times100mm and theaggregate size is 2ndash20mm +e aggregate contents of theactual aggregate model and the circular aggregate model are4909 and 4911 respectively and geometric shrinkagereduces the interface thickness by 05mm to form an in-terface between aggregate and mortar

+e geometric model of the aggregate is imported intothe ABAQUS finite element software+emesh is divided bya quadrilateral-based freeform grid Taking the actual ag-gregate model as an example the model grid is divided intoa grid as shown in Figure 7 and the total number of elementsis 50717 +e total number of nodes is 50149

+e ABAQUS damage plasticity model is used to describethe fracture behavior of concrete materials +is model can beused to characterize the inelastic behavior of concrete com-bined with isotropic elastic damage and isotropic tensile andcompressive plasticity theory In order to simplify the calcu-lation the evolution model of the double-fold damage variable

Contour controlpoint

(a)

Contour controlpoint

(b)

Contour controlpoint

(c)

Figure 4 Polygons-circumcircle conflict and overlap criteria (a) Control point inside circle (b) Polygonal edges intersect with circle(c) Circle inside the polygon

(a) (b) (c)

Figure 5 Aggregate generation by gradation (a) Two-gradation aggregate content 4542 (b)+ree-gradation aggregate content 5217(c) Four-gradation aggregate content 5400

Advances in Materials Science and Engineering 5

[34] is adopted as the constitutive of each phase material asshown in Figure 8

+e damage variables can be expressed as

D

0 εmax lt ε0

1minusηminus ληminus 1

ε0εmax

+1minus ληminus 1

ε0 lt εmax le εr

1minus λε0εmax

εr lt εmax le εu

1 εmax gt εu

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where ft is the tensile strength ftr is the residual tensilestrength ε0 is the principal tensile strain corresponding to ftεr is the residual strain corresponding to the tensile residualstrength ftr εu is the ultimate tensile strain and εmax is thelargest primary tensile strain value of the loading history+e relationship between parameters is ftr λfr(0lt λlt 1)εr ε0 and εu ξε0(ξ gt η)

+e concrete phase material mechanical parameters arepresented in Table 1 where λ was taken as 01 η was taken as10 and the aggregate mortar and interface ξ were taken 5 4and 4 respectively +e concrete uniaxial tensile specimencalculation diagram is shown in Figure 9 where the leftboundary is the applied horizontal constraint the left middleis the applied vertical constraint and the right boundary isthe applied displacement load

+e macroscopic stress-strain curve of the concretetensile specimen is shown in Figure 10 In the elastic phasethe gradients of the stress-strain curve of the circularaggregate model and the actual aggregate model are es-sentially identical which indicates that the stress char-acteristics of the macrospecimens in the elastic phase arethe same At the end of the elastic phase the strain of thespecimen approaches 7 times 10minus5 numerous cracks are ini-tiated at the interface in the concrete specimens andmicrocracks are formed

After entering the softening stage the damage areaexpands as the strain increases In order to observe thedevelopment of macroscopic cracks the strain of the testspecimens was increased to greater than 2 times 10minus5 where

(a) (b)

Figure 6 Specimen geometry model (a) Actual random aggregate (b) Circle random aggregate

(a) (b) (c) (d)

Figure 7 Uniaxial tensile specimen model mesh (a) Geometric model (b) Aggregate mesh (c) Interface mesh (d) Mortar mesh

6 Advances in Materials Science and Engineering

macroscopic cracks greater than 002mm became visibleApart from the small cracks that developed on the left side ofthe circular aggregate model macroscopic cracks continuedto develop along the aggregate interface and the mortar inthe middle left of the specimen and finally formed a mac-roscopic crack perpendicular to the tensile direction of thespecimen as shown in Figure 11(a) For the actual aggregatemodel the crack developed on the right of the specimenperpendicular to the direction of stretching Unlike for thecircular aggregate the macroscopic cracks in the actualaggregate model developed simultaneously at the two in-terfaces a phenomenon known as crack bifurcation andfinally met at the bottom as shown in Figure 11(b) From thestress-strain curve as the aggregate shape and aggregate

distribution of the circular aggregate model and the actualaggregate model are different the stress peak and the cor-responding strain of the actual aggregate model are mar-ginally greater than that of the circular aggregate which ledto single and bifurcation cracks +is phenomenon reflectsthe significant influence of aggregate geometry on themacroscopic mechanical properties of the specimens

In the descending section of the stress-strain curve theaggregate cracks continue to develop +e stress-straincurves of the actual aggregate model and the circular ag-gregate model decrease at a similar rate and the softenedend of the curves are coincident At this point the crack in

Table 1 Concrete phase material mechanical parameters

Material type Elastic modulus (GPa) Tensile strength (MPa) Poissonrsquos ratioAggregate 50 6 02Mortar 26 32 02Interface 25 3 02Concrete 32 02

Nonuniformregion

100 mm

100 m

m

F

Figure 9 Uniaxial tensile specimen calculation diagram

f

ft

ftr

ε0 εr εu

Figure 8 Double-fold line damage constitutive relation

000 002 004 006 008 010 01200

05

10

15

20

25

30

Stre

ss (M

Pa)

Strain (10ndash3)

Circle aggregate modelActual aggregate model

Figure 10 Macroscopic stress-strain curve

Advances in Materials Science and Engineering 7

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

[34] is adopted as the constitutive of each phase material asshown in Figure 8

+e damage variables can be expressed as

D

0 εmax lt ε0

1minusηminus ληminus 1

ε0εmax

+1minus ληminus 1

ε0 lt εmax le εr

1minus λε0εmax

εr lt εmax le εu

1 εmax gt εu

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

where ft is the tensile strength ftr is the residual tensilestrength ε0 is the principal tensile strain corresponding to ftεr is the residual strain corresponding to the tensile residualstrength ftr εu is the ultimate tensile strain and εmax is thelargest primary tensile strain value of the loading history+e relationship between parameters is ftr λfr(0lt λlt 1)εr ε0 and εu ξε0(ξ gt η)

+e concrete phase material mechanical parameters arepresented in Table 1 where λ was taken as 01 η was taken as10 and the aggregate mortar and interface ξ were taken 5 4and 4 respectively +e concrete uniaxial tensile specimencalculation diagram is shown in Figure 9 where the leftboundary is the applied horizontal constraint the left middleis the applied vertical constraint and the right boundary isthe applied displacement load

+e macroscopic stress-strain curve of the concretetensile specimen is shown in Figure 10 In the elastic phasethe gradients of the stress-strain curve of the circularaggregate model and the actual aggregate model are es-sentially identical which indicates that the stress char-acteristics of the macrospecimens in the elastic phase arethe same At the end of the elastic phase the strain of thespecimen approaches 7 times 10minus5 numerous cracks are ini-tiated at the interface in the concrete specimens andmicrocracks are formed

After entering the softening stage the damage areaexpands as the strain increases In order to observe thedevelopment of macroscopic cracks the strain of the testspecimens was increased to greater than 2 times 10minus5 where

(a) (b)

Figure 6 Specimen geometry model (a) Actual random aggregate (b) Circle random aggregate

(a) (b) (c) (d)

Figure 7 Uniaxial tensile specimen model mesh (a) Geometric model (b) Aggregate mesh (c) Interface mesh (d) Mortar mesh

6 Advances in Materials Science and Engineering

macroscopic cracks greater than 002mm became visibleApart from the small cracks that developed on the left side ofthe circular aggregate model macroscopic cracks continuedto develop along the aggregate interface and the mortar inthe middle left of the specimen and finally formed a mac-roscopic crack perpendicular to the tensile direction of thespecimen as shown in Figure 11(a) For the actual aggregatemodel the crack developed on the right of the specimenperpendicular to the direction of stretching Unlike for thecircular aggregate the macroscopic cracks in the actualaggregate model developed simultaneously at the two in-terfaces a phenomenon known as crack bifurcation andfinally met at the bottom as shown in Figure 11(b) From thestress-strain curve as the aggregate shape and aggregate

distribution of the circular aggregate model and the actualaggregate model are different the stress peak and the cor-responding strain of the actual aggregate model are mar-ginally greater than that of the circular aggregate which ledto single and bifurcation cracks +is phenomenon reflectsthe significant influence of aggregate geometry on themacroscopic mechanical properties of the specimens

In the descending section of the stress-strain curve theaggregate cracks continue to develop +e stress-straincurves of the actual aggregate model and the circular ag-gregate model decrease at a similar rate and the softenedend of the curves are coincident At this point the crack in

Table 1 Concrete phase material mechanical parameters

Material type Elastic modulus (GPa) Tensile strength (MPa) Poissonrsquos ratioAggregate 50 6 02Mortar 26 32 02Interface 25 3 02Concrete 32 02

Nonuniformregion

100 mm

100 m

m

F

Figure 9 Uniaxial tensile specimen calculation diagram

f

ft

ftr

ε0 εr εu

Figure 8 Double-fold line damage constitutive relation

000 002 004 006 008 010 01200

05

10

15

20

25

30

Stre

ss (M

Pa)

Strain (10ndash3)

Circle aggregate modelActual aggregate model

Figure 10 Macroscopic stress-strain curve

Advances in Materials Science and Engineering 7

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

macroscopic cracks greater than 002mm became visibleApart from the small cracks that developed on the left side ofthe circular aggregate model macroscopic cracks continuedto develop along the aggregate interface and the mortar inthe middle left of the specimen and finally formed a mac-roscopic crack perpendicular to the tensile direction of thespecimen as shown in Figure 11(a) For the actual aggregatemodel the crack developed on the right of the specimenperpendicular to the direction of stretching Unlike for thecircular aggregate the macroscopic cracks in the actualaggregate model developed simultaneously at the two in-terfaces a phenomenon known as crack bifurcation andfinally met at the bottom as shown in Figure 11(b) From thestress-strain curve as the aggregate shape and aggregate

distribution of the circular aggregate model and the actualaggregate model are different the stress peak and the cor-responding strain of the actual aggregate model are mar-ginally greater than that of the circular aggregate which ledto single and bifurcation cracks +is phenomenon reflectsthe significant influence of aggregate geometry on themacroscopic mechanical properties of the specimens

In the descending section of the stress-strain curve theaggregate cracks continue to develop +e stress-straincurves of the actual aggregate model and the circular ag-gregate model decrease at a similar rate and the softenedend of the curves are coincident At this point the crack in

Table 1 Concrete phase material mechanical parameters

Material type Elastic modulus (GPa) Tensile strength (MPa) Poissonrsquos ratioAggregate 50 6 02Mortar 26 32 02Interface 25 3 02Concrete 32 02

Nonuniformregion

100 mm

100 m

m

F

Figure 9 Uniaxial tensile specimen calculation diagram

f

ft

ftr

ε0 εr εu

Figure 8 Double-fold line damage constitutive relation

000 002 004 006 008 010 01200

05

10

15

20

25

30

Stre

ss (M

Pa)

Strain (10ndash3)

Circle aggregate modelActual aggregate model

Figure 10 Macroscopic stress-strain curve

Advances in Materials Science and Engineering 7

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

the specimen has approximately penetrated the specimenindicating that the geometric characteristics of the aggre-gate will result in different curves in the softening sec-tion +e ultimate stresses of the specimen were essentiallyidentical

425ree-PointBendingConcreteBeamTest +e calculationdiagram of the concrete three-point bending beam specimenis shown in Figure 12+ematerial parameters of each phaseare the same as before and the specimen size is taken as400times100mm To reduce the difficulty of meshing andcalculation the middle of the specimen is taken as a non-uniform area and both sides of the specimen are taken asa uniform area +e model mesh is shown in Figure 13 witha total number of 62988 units and 62707 nodes

+e reaction force and the deflection of the mid-bottompoint of the beam are extracted from the simulation results+e load-deflection curve is obtained as shown in Figure 14where it can be seen that the gradient of the circular ag-gregate model in the elastic phase is slightly greater than thatof the actual aggregate model and the circular aggregate hasa greater peak stress than the actual aggregate Whencompared with the uniaxial tensile stress-strain curve thesoftening section of the load-deflection curve is relatively flatwhile the curve of the circular aggregate model decreasesfaster than the actual aggregate model However theeventual trends of the two models are similar

From the final crack distribution and crack developingprocess as shown in Figure 15 it can be seen that under theinfluence of the loading and the boundary conditions themacroscopic crack initiation zone is formed in the right-bottom area and points to the loading point +e crack of thecircular aggregate model begins developing in small ag-gregates and mortar and reaches aggregate 1 as shown inFigure 15(a) where the development of cracks is blockedand continues to develop upward along the boundary ofaggregate 1 +e same phenomenon known as the aggregateblocking effect is seen for aggregates 2 3 and 4 For theactual aggregate model the cracks also occur in the interface

(a) (b)

Figure 11 Uniaxial tensile specimen crack distribution (a) Circle aggregate model (b) Actual aggregate model

Nonuniformregion

Uniformregion

100 mm300 mm50 mm 50 mm

100 m

m

F

Uniformregion

Figure 12 +ree-point bending concrete beam calculation diagram

Figure 13 +ree-point bending concrete beam model mesh

000 001 002 003 004 00500

10

20

30

40

50

60

70

80

Load

(kN

)

Deflection (mm)

Circle aggregate modelActual aggregate model

Figure 14 Load-deflection curve

8 Advances in Materials Science and Engineering

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

and mortar and the crack development changes the directionalong the boundary of the aggregate depending on the shapeof the aggregate as can be seen for aggregates 1 2 3 and 4 inFigure 15(b) +is was especially noticeable in aggregate 1where the crack clearly changed the direction indicating thatthe shape of the aggregate directs the crack development

+erefore it can be concluded that the aggregate ge-ometry has the effect of blocking and guiding the crack in themesoscale model while the blocking effect tends to occurin the circle aggregate model and the guiding effect in theactual aggregate model However the blocking can also beobserved in the actual aggregate model as shown inFigure 15(b) and for aggregate 5 the effect of blockingcracking is more obvious when the aggregate length di-rection is perpendicular to the fracture direction It can beseen that the softening end of the actual aggregate modelload-deflection curve gradually flattens

5 Conclusions

A model of concrete micromechanics based on the geo-metric boundaries of actual aggregate was proposed As

opposed to the polygon model it uses the boundaries ofactual aggregate as a base to generate new aggregates Inaddition the model could place the aggregates according totheir gradation To solve the problem of low discriminatingefficiency of polygons with a great number of vertices thefront polygons-rear circumcircle conflict and overlap cri-teria were proposed

Based on these proposed solutions numerical tests ofuniaxial tension and three-point moment beams were con-ducted +e results indicated that the geometric properties ofaggregates have both blocking and guiding effects on crackdevelopment As opposed to traditional models the simulationof crack development in the model proposed in this studyapproximates actual crack development

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] X Du and L Jin ldquoA review on meso-mechanical method forstudying the static-mechanical properties of concreterdquo Ad-vances in Mechanics vol 41 no 4 pp 411ndash426 2011

[2] E Schlangen and E J Garbociz ldquoNew method for simulatingfracture using an elastically uniform random geometry lat-ticerdquo International Journal of Engineering Science vol 34no 10 pp 1131ndash1144 1996

[3] B Chiaia A Vervuurt and J G M van Mier ldquoLattice modelevaluation of progressive failure in disordered particlecompositesrdquo Engineering Fracture Mechanics vol 57 no 2-3pp 301ndash313 1997

[4] P A Cundall and O D L Strack ldquoA discrete numericalmodel for granular assembliesrdquo Geotechnique vol 29 no 1pp 47ndash65 1979

[5] A Cundall and R D Hart ldquoNumerical modeling of dis-continuardquo in Proceedings of the 1st US Conference on DiscreteElement Methods Golden CO USA 1989

[6] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part I model de-velopment and validationrdquo ACI Materials Journal vol 96no 2 pp 196ndash203 1999

[7] A R Mohamed andW Hansen ldquoMicromechanical modelingof concrete response under static loading part II modelprediction for shear and compressive loadingrdquo ACI MaterialsJournal vol 96 no 3 pp 354ndash358 1999

[8] W Zhu and C Tang ldquoNumerical simulation on shear fractureprocess of concrete using mesoscopic mechanical modelrdquoConstructions and Building Materials vol 16 no 8 pp 453ndash463 2002

[9] C Tang and W Zhu Concrete Damage and FracturendashNumerical Test Science Press Beijing China 2003

[10] G Liu and Z Wang ldquoNumerical simulation study of fractureof concrete materials using random aggregate modelrdquo Journalof Tsinghua University vol 36 no 1 pp 84ndash89 1996

[11] Z Wang A K H Kwan and H C Chan ldquoMesoscopic studyof concrete I generation of random aggregate structure andfinite element meshrdquo Computers and Structures vol 70 no 5pp 533ndash544 1999

[12] A K H Kwan Z M Wang and H C Chan ldquoMesoscopicstudy of concrete II nonlinear finite element analysisrdquoComputers and Structures vol 70 no 5 pp 545ndash556 1999

1

2

3

4

(a)

5

4

3

2

1

(b)

Figure 15 +ree-point bending concrete beam crack distribution(a) Circle aggregate model (b) Actual aggregate model

Advances in Materials Science and Engineering 9

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

[13] C ZhangX Tang et al ldquoState-of-the-art literature review onconcrete meso-scale mechanicsrdquo Journal of HydroelectricEngineering vol 34 no 12 pp 1ndash18 2015

[14] Z P Bazant M R Tabbara M T Kazemi et al ldquoRandomparticle model for fracture of aggregate or fiber compositesrdquoJournal of Engineering Mechanics vol 116 no 8 1990

[15] X Ruan and Z Pan ldquoMesoscopic simulation method ofconcrete carbonation processrdquo Structure and InfrastructureEngineering vol 8 no 2 pp 99ndash110 2013

[16] Q Huang Z Jiang W Zhang et al ldquoNumerical analysis of theeffect of coarse aggregate distribution on concrete carbonationrdquoConstruction andBuildingMaterials vol 37 no 3 pp 27ndash35 2012

[17] Z H Xie Y C Guo Q Z Yuan et al ldquoMesoscopic numericalcomputation of compressive strength and damagemechanismof rubber concreterdquo Advances in Materials Science and En-gineering vol 2015 Article ID 279584 10 pages 2015

[18] Z Pan X Ruan and A Chen ldquoChloride diffusivity of con-crete probabilistic characteristics at meso-scalerdquo Computersand Concrete vol 13 no 2 pp 187ndash207 2014

[19] J P B Leite V Slowik and H Mihashi ldquoComputer Simu-lation of fracture processes of concrete using mesolevelmodels of lattice structuresrdquo Cement and Concrete Researchvol 34 no 6 pp 1025ndash1033 2004

[20] Z Gao and G Liu ldquoTwo-dimensional random aggregatestructure for concreterdquo Tsinghua Science and Technologyvol 43 no 5 pp 710ndash714 2003

[21] L Sun C Du and C Dai ldquoNumerical simulation of randomaggregate model for mass concreterdquo Journal of Hehai Uni-versity vol 33 no 3 article 291295 2005

[22] H Ma S Mi and H Chen ldquoA generating approach ofrandom convex polygon aggregate modelrdquo Journal of ChinaInstitute of Water Resources and Hydropower Research vol 4no 3 pp 196ndash201 2006

[23] A Chen Z Pan R Ma et al ldquoNew development of meso-scopic research on durability performance of structuralconcrete in bridgesrdquo China Journal of Highway and Transportvol 29 no 11 pp 42ndash48 2016

[24] W Ren Z Yang and Y Huang ldquoMeso-scale fracturemodelling of concrete based on X-ray computed tomographyimagesrdquo Journal of Hydraulic Engineering vol 46 no 4pp 452ndash459 2015

[25] W Qin and C Du ldquoMeso-level model of three-dimensionalconcrete based on the CT slicesrdquo Engineering Mechanicsvol 7 pp 186ndash193 2012

[26] Z Pan X Ruan and A Chen ldquoSimulation method of randomaggregate in two dimension based on arbitrary gradationrdquoJournal of Tongji University vol 41 no 5 pp 759ndash764 2013

[27] B Fu J Li and G Lin ldquoMesoscopic numerical model ofconcrete based on data base of real aggregate shapesrdquo Journalof Architecture and Civil Engineering vol 27 no 2 pp 10ndash172010

[28] G Li and Z Wang ldquoA mesoscopic simulation for the early-age shrinkage cracking process of high performance concretein Bridge Engineeringrdquo Advances in Materials Science andEngineering vol 2017 Article ID 9504945 12 pages 2017

[29] Z Wang Crack growth computer strength and deformation ofnonhomogeneous materials (concrete) PhD thesis TsinghuaUniversity Beijing China PhD thesis 1996

[30] X Tang and C Zhang ldquoLayering disposition and FE co-ordinate generation for random aggregate arrangementsrdquoTsinghua Science and Technology vol 48 no 12 pp 2048ndash2052 2008

[31] C Qin C Guo and C Zhang ldquoA pre-processing schemebased on background grid approach for meso-concrete

mechanicsrdquo Journal of Hydraulic Engineering vol 42 no 8pp 941ndash948 2011

[32] J C Walraven and H W Reinhardt5eory and Experimentson the Mechanical Behaviour of Cracks in Plain and ReinforcedConcrete Subjected to Shear Loading Delft University ofTechnology Delft Netherlands 1981

[33] H Ma and H Chen Study on the Mechanism of DynamicDamage Damage of Concrete Dam and Its Microscopic Me-chanics Analysis Method China Water and Power PressBeijing China 2008

[34] H Ma H Chen and B Li ldquoMeso-structure numericalsimulation of concrete specimensrdquo Journal of HydraulicEngineering vol 35 no 10 pp 27ndash35 2004

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

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High Energy PhysicsAdvances in

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Volume 2018

TribologyAdvances in

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ChemistryAdvances in

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Hindawiwwwhindawicom Volume 2018

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CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom