research article on the shape simulation of aggregate and...

Research ArticleOn the Shape Simulation of Aggregate andCement Particles in a DEM System

Huan He12 Piet Stroeven2 Eric Pirard3 and Luc Courard3

1College of Architecture and Civil Engineering Beijing University of Technology Beijing 100122 China2Faculty of Civil Engineering and Geosciences Delft University of Technology Stevinweg 1 2628 CN Delft Netherlands3GeMMe (Minerals Engineering Materials and Environment) University of Liege Chemin des Chevreuils 1 4000 Liege Belgium

Correspondence should be addressed to Huan He hhelivecom

Received 30 April 2015 Revised 6 July 2015 Accepted 7 July 2015

Academic Editor Ana S Guimaraes

Copyright copy 2015 Huan He et alThis 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

Aggregate occupies at least three-quarters of the volume of concrete so its impact on concretersquos properties is significant Bothsize and shape of aggregate influence workability mechanical properties and durability of concrete On the other hand the shapeof cement particles plays also an important role in the hydration process due to surface dissolution in the hardening processAdditionally grain dispersion shape and size govern the pore percolation process that is of crucial importance for concretedurability Discrete element modeling (DEM) is commonly employed for simulation of concrete structure To be able to do sothe assessed grain shape should be implemented The approaches for aggregate and cement structure simulation by a concurrentalgorithm-based DEM system are discussed in this paper Both aggregate and cement grains were experimentally analyzed byX-ray tomography method recently The results provide a real experimental database for example surface area versus volumedistribution for simulation of particles in concrete technology Optimum solutions are obtained by different simplified shapesproposed for aggregate and cement respectively In this way more reliable concepts for aggregate structure and fresh cement pastecan be simulated by a DEM system

1 Introduction

A particle is defined as the smallest discrete unit of a powdermass that cannot be easily subdivided [1] The shapes of realparticles are irregular from coarse aggregate to a mineraladmixture in concrete Aggregate occupies at least three-quarters of the volume of concrete so its impact on concretersquosproperties is significant The sieve curve traditionally definesthe aggregate size range Another essential property is grainshape Both size and shape influence compactability andworkability in the fresh state and themechanical and durabil-ity properties of the hardened concrete The definition of theactual grain shape as well as the representation in a discreteelement modeling (DEM) system is complicated howeverFurthermore the real shape of aggregate or binder particlescan vary widely The sphere is therefore generally adoptedin conventional simulation systems in concrete technologythat are mostly of random sequential addition (RSA) naturedespite imposing serious limitations and biases

The shape of cement particles plays also an importantrole in the hydration process due to surface dissolution inthe hardening process Additionally grain dispersion shapeand size govern the pore percolation process that is of crucialimportance for concrete durability [2] Cement hydrationin a DEM approach renders the possibility of investigatingmicrostructure development and resulting properties of con-crete A digital image-based model [3] as well as continuummodels [4ndash6] has been developed for this purpose Apartfrom the digital image-based model particle shape is gener-ally assumed to be spherical because of inherent simplifica-tion in algorithm formulation however at the cost of biasesin the simulation results Recently some reference cementand aggregate materials were analyzed by microtomography(120583CT) and X-ray tomography (CT) respectively [7 8] Theresults provide real experimental databases of aggregate andcement grains providing valuable parameters for the DEMsimulation This is even more so since traditional shapeanalysis methods for example manual measurement or

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2015 Article ID 692768 7 pageshttpdxdoiorg1011552015692768

2 Advances in Materials Science and Engineering

Gravel Ellipsoids Aggregate

Crushed rock Polyhedra

Figure 1 Simulation strategy of arbitrary shaped aggregate (top) differently shaped ellipsoids represent river gravel (bottom) differentlyshaped polyhedrons represent crushed rock

experimental techniques in building codes such as ASTMD-3398-97 cannot provide sufficient information for particlereconstruction in theDEMsystemsThe involved experimen-tal difficulties would be even greater for very fine particlessuch as cement grains Simulation of concrete by a DEMsystem requires a large number of particles with differentsizes to guarantee the RVE demand They will inevitably alsohave different shapes However giving all particles individualshapes in DEM would be too expensive and (computer)time-consuming for solving practical engineering problems[9] Therefore a practical strategy is to select a single shapeparameter or eventually a limited number of typical shapesto represent a particles mixture

In this paper simulation strategies are presented for therepresentation of real aggregate and cement grains in con-crete technologyThis simulation approach is incorporated ina physical concurrent algorithm-based DEM system that isHADES The system is developed for making more realisticpacking simulations by using nonspherical particles It isbased on a contact mechanism that evaluates the interactionforces exerted between segmented surfaces of neighboringparticles It can provide packing densities up to the denserandompacking state relevant for the present purposesMoredetailed information about the system can be found in [10]A practical shape analysis study is conducted on the basis ofCT and 120583CT results available in the international literature[7 8] Specific simulation strategies could be proposed forboth aggregate and cement particle simulation

2 Shape Simulation of Aggregate Grains

Aggregate in concrete can be divided into two groups thatis gravel of fluvial origin and crushed rock representingrounded and angular-shaped particles respectively as shownin Figure 1 The ellipsoid was selected to represent theriver gravel particles whereas the polyhedron was used forsimulation of crushed rock grainsThe surface of thesemodelparticles is tessellated by a mesh positioned on the grainsrsquosurfaces The mesh should be fine enough for a proper

generation of the particles and interparticle computationalrequirement [10]

As a practical example a series of regular polyhedronswith 4 to 8 facetted surfaces were employed to representcrushed rock aggregate as used in railroad beds [11] Thisgreatly simplifies the simulation since only the two param-eters should be specified that is the sieve size and themaximum size of the surface element Figure 2 presents themeshed particles of the regular polyhedrons employed in thisstudyThevolume fraction of each type of polyhedron is takensimilar to that found in the field study of Guo [11]

Recently in a study of Erdogan et al [8] the shapes of fourdifferent types of aggregate denoted by GR LS IN and AZwere investigated by the CTmethod and reconstructed by thespherical harmonic (SH) technique [8] The most interestinginformation coming from this shape analysis is the globalsurface area (119878) to volume (119881) curve of each type of aggregateA function 119878 = 119886119881119887was used to represent the 119878 versus119881 curvefor each type of aggregate [8] The coefficients 119886 and 119887 wereassessed for all cases and are listed inTable 1 [8]The variancesof the regression approximations were exceeding 098 in allcases

The regression function 119878 = 119886101584011988123 was employed tofit the different types of aggregate Next 1198861015840 is determined(volumelt 4000mm3) with a variation lower than 5 for eachtype of aggregate as given in Table 1 A common shape indexlike sphericity is finally determined for each type of aggregateby

Sphericity =119878eqsphere

119878particle=4120587 (31198814120587)23

119886101584011988123= 4836 (1198861015840)

minus1(1)

inwhich 1198861015840 is the coefficient in the above-mentioned relation-ship of 119878 versus 11988123 Sphericity is defined as the surface arearatio of the equivalent sphere and the real particle so bothhaving equal volume The sphericity values are also listed inTable 1 Obviously the crushed rock types GR and LS cannotbe distinguished by means of sphericity The natural rivergravel types IN and AZ have a relatively larger sphericity as

Advances in Materials Science and Engineering 3

Tetrahedron Pentahedron Hexahedron

Heptahedron Octahedron

I II III

IIIIII

Figure 2 Nine regular polyhedra with facet numbers 4sim8

Table 1 119878-119881 correlation coefficients found by Erdogan et al [8] for four types of aggregate 119886 and 119887 are from 119878 = 119886119881119887 while coefficient 1198861015840 isfrom 119878 = 119886101584011988123 Sphericity is calculated by (1)

Aggregate type Coefficient 119886 Coefficient 119887 Variance 1198772 Coefficient 1198861015840 SphericityGR 81 063 099 613 079LS 75 064 0997 612 079IN 86 062 099 606 080AZ 91 061 098 595 081

Table 2 Shape indices of the regular polyhedra appropriate proportions according to Guo [11] and revised proportions for better overallsimulation results

Shapes Facet number Sphericity 1198861015840 Proportion by Guo [11] Revised proportion

Tetrahedron 4 067 721 100 5Pentahedron I 5 070 695 125 5Pentahedron II 5 072 671 125 5Hexahedron I (cube) 6 081 600 175 10Hexahedron II 6 076 639 175 10Heptahedron I 7 083 583 100 155Heptahedron II 7 080 606 100 155Octahedron I 8 085 572 50 17Octahedron II 8 078 624 50 17Sphere infin 100 484 mdash mdash

could be expected But sphericity seems not very effective indistinguishing between actual shapes

The proposed proportions by Guo [11] for simulationof crushed rock in railway beds are then analyzed on thebasis of experimental results of Erdogan et al [8] Shapeindices of different regular polyhedra are listed in Table 2with the sphere as a reference It demonstrates that theshape indices of crushed rock samples GR and LS in Table 1are within the range of shape indices of these nine regularpolyhedra Guo [11] also proposed for the crushed rockproportions of each shape category as listed in Table 2 Nextthe composite polyhedramethodproposed byGuo [11] can becompared with experimental results of GR and LS [8] shown

in Figure 3 Generally speaking Guorsquos proposal fits the data ofaggregates GR and LS quite well But there is some deviationin the large particle range Therefore a better proportion ofeach category is indicated in Table 1 as well as in Figure 3 Bytheses proportions 119878-119881 information of composite polyhedracan better comply with the experimental results of GR and LStypesThis practical example on crushed rock in railway bedsshows that a proper global representation can be obtained bythe sketched simple simulation strategy

On the other hand ellipsoids are particularly interestingbecause with only 3 parameters a variety of shapes rangingfrom oblate to oblong can be described Three principalaxes (119886 represents the longest 119887 the medium size and 119888


0

500

1000

1500

2000

0 1000 2000 3000 4000

SphereGR in [8]LS in [8]

Composite polyhedra by Guo [11]Revised composite polyhedra

Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

Figure 3 119878-119881 curves of crushed rock types (GR and LS) ofcomposite polyhedra and of the sphere

the shortest axis) determine the shape of an ellipsoid Byparameter variation different ellipsoids with different 119878-119881relationships can be derived119881 and 119878 can be obtained by [12]

119881 =43120587119886119887119888

119878 = 2119887radic2int120587

0radic1198862 + 1198882 + (1198862 minus 1198882) cos (2120601) sin120601

sdot 119864(119888

119887

radic2 (1198872 minus 1198862)

1198862 + 1198882 + (1198862 minus 1198882) cos (2120601)sin120601)119889120601

(2)

in which 119864 = (1198872cos2120579 + 1198862sin2120579)sin2120601 is a coefficient of thefirst fundamental form and 120601 is a polar angle Flatness andelongation are defined as 119888119887 and 119887119886 respectively So thedistribution contours of sphericity and 11987811988123 (or 1198861015840) withdifferent elongation and flatness values can be constructedas depicted in Figure 4

Figure 4(a) reveals that if elongation and flatness are bothlarger than 05 sphericity of an ellipsoid will be close to 111987811988123-contours with elongation and flatness are plotted in

Figure 4(b) With experimental values of IN and AZ typesin Table 1 the hatched regions in Figure 4 can be selected asthe optimum solution The preferred elongation and flatnessof an ellipsoid should exceed 03 and 04 respectively Thisconclusion is comparable with the experimental results ongravel in Figure 4The selected ellipsoids are thus neither veryelongated nor flat

3 Shape Simulation of Cement Grains

Computer X-raymicrotomography offers a potential solutionfor shape assessment of cements as shown by Garboczi andBullard [7 13] The microstructures of hydrated cements

based on actual grain shape and on spherical shape werefound significantly different in the study [13] The resultsprovide an experimental database of this cement that yieldssome valuable parameters for DEM simulation of cementhydration

Using cement grains in numerical simulations similar tothe real ones would be too expensive So it is crucial findingsimpler shapes that are sufficiently representative Similaras in the case of the aggregate simulation a shape analysisstudy was therefore conducted with some simpler shapesBased on this analysis a simulation strategy is proposed forcement More realistic but still cost-effective particle shapesare therefore implemented in the DEM system

Cement is made by raw materials such as limestoneand clay after a process of calcination Raw cement clinkersconsist of hard large and roughly rounded particles ona scale of centimeters Raw cement clinkers are groundin the miller with a small amount of gypsum into smallcement grains Hence the obtained cement grains will tendto angular shapes on a scale of micrometers Polyhedra seemtherefore more appropriate for simulation of cement grainsNine polyhedra in Figure 2 are also selected for this shapeanalysis As the surface area-to-volume (119878-119881) relationshipwas the most important information obtained from theexperimental reference [7] values of these shape indicesare plotted in Figure 5 The surface to volume relationshipcan generally be expressed also by 119878 = 119886101584011988123 Within aneffective experimental volume of 10000sim150000 120583m3 [7]the coefficient 1198861015840 = 58 was found by regression analysiswith a coefficient of variation lower than 5 Proposedshapes should therefore approximate this value In this casesphericity can be calculated as 083

From Figure 5 and Table 2 it is concluded that theheptahedron 119868 and the octahedron 119868 offer themost promisingsolutionsThe octahedron is selected for the simulation studybecause of allowing an easier transformation into irregularshapes by parameter variation Three axes can be employedfor this purpose in agreement with [14] Since the 119878-119881 curveof an octahedron is close to that of the reference cement somelimited random variation can be applied accounting for thediversity of particle shape

Figure 6(a) shows the 119878-119881 distributions of randomlygenerated octahedra with an experimental reference curveThis figure reveals the cases of 1000 particles with longestaxis in the 10sim50120583msize rangeThe 119878-119881 relationships complywell with the experiments Figure 6(b) shows an example ofa loose packed structure of arbitrary octahedrons supposedlyrepresenting fresh cement

Morphological comparisons can additionally be madebetween sections of simulated particle structures and ofreal ones As an example Figure 7(a) shows a randomsection of the simulated structure shown in Figure 6(b) anda 2D section of a real cement structure obtained by X-raymicrotomography Cement particles in Figure 7(b) displayedin light grey obviously reveal the angular shape that issimilarly revealed by the section of the polyhedral cementparticles in Figure 7(a) This adds to the aforementioned


02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

Sphe

ricity

01940

02948

03955

04963

05970

06977

07900

08200

1000

(a)

02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

SV23

4800

5900

6100

7800

1260

1580

1800

2230

2480

(b)

Figure 4 Distribution contours of (a) sphericity and (b) 11987811988123 with different elongation and flatness of an ellipsoid

0

5000

10000

15000

20000

0 50000 100000 150000

SphereCCRL-133 [7]TetrahedronPentahedron IPentahedron IIHexahedron II

CubeHeptahedron IHeptahedron IIOctahedron IOctahedron II

Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

Figure 5 119878-119881 curves of reference cement and some regularpolyhedral

quantitative matching of 119878-119881 features of real and simulatedcement structures as shown by Figure 6(a) Therefore it canbe concluded that the proposed simulation method providesa significantly improved option for modelling of cementparticles as compared to the conventional approaches inwhich spheres are employed This will also have an impacton the hydration process as well The simulation of thiscomplicated interference process of hydrating particles isstill a demanding issue however A possible way out of thishydration simulation problem is to have the outer layer ofC-S-H simulated by a dynamic DEM on nanolevel as ispresently developed at Delft University of Technology in theframework of a running PhD study [15]

4 Discussion and Conclusions

Aparticulatematerial like concrete is composed of irregularlyshaped particles in the form of either aggregate cement ormineral admixtures The definition or simulation of shapeis therefore inevitably complex because it is also intimatelyconnected to that of size Reconstruction of an irregularshape is technically possible especially with the developmentof CT (or 120583CT) and SH [7 16] Even with traditional 3Dimage analysis an irregular particle shape can roughly bereconstructed by combining the vertical and horizontal pro-files with a 3D mesh program [10 17] Examples are plottedin Figure 8 Using a large amount of grains in numericalsimulations that have similar shapes as in practice would betoo expensive with the eye on equipment and computer time[9] So it is crucial finding simpler shapes that are sufficientlyrepresentative since a spherical particle shape as normallyassumed in conventional simulation systems can give rise tobiased simulation results [13]

Therefore a shape analysis study was conducted withsome simpler shapes for simulation of aggregate as well as ofcement The shape information of real aggregate and cementby CT (or 120583CT) was used as references [7 8] A simula-tion strategy based on different origins of arbitrary shapedaggregate grains is proposed A composite of nine regularpolyhedra [11] can properly represent a crushed rock forapplication in concrete This method can fully comply withthe 119878-119881 correlations extracted from experimental findings[8] Ellipsoids can be used for simulation of gravel of fluvialorigin A kind of relatively flat ellipsoids complies with the119878-119881 information available in experimental investigations [8]Furthermore surface texture simulation can also be simplyconsidered by superimposing a sine function on a regularshape [10]

For hydration simulation a study was conducted pur-suing the assessment of physically more realistic shapesthan the sphere commonly employed in simulations Thisstudy encompassed polyhedra and ellipsoids It was shownthat a limited variation of octahedra can be considered an


0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000 25000

Simulation Experiment

Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

(a)

x

y

z

(b)

Figure 6 (a) Computer simulation of 1000 particles in the 10sim50 120583m size range and experimental regression results [7] and (b) visualizedstructure of compacted grains

(a) (b)

Figure 7 (a) Section of the simulated structure (1000 octahedron grains in 10sim50 120583m size range) (b) 2D section of a real fresh cementstructure (Cement-133) obtained by X-ray microtomography wc = 035 Source httpvisiblecementnistgovcementhtml

x

z

x

z

Figure 8 Two views of a reconstructed grain with (left) fine 3D mesh and (right) coarse 3D mesh respectively


optimum solutionThey offermore realistic 119878-119881 relationshipsas spheres which is crucial for cement hydration [13] andfinally for structure formation governing the pore networksystem Morphological comparison shows the similarity ofthe packed structures of proposed polyhedra with real par-ticles Implementation of these simulation strategies in thedynamic force-based DEM approach used throughout thisstudy will lead to a more realistic structure of fresh cementpaste in concrete technology

Nowadays computational concrete also referred to asldquocompucreterdquo or virtual concrete has been developed fora wide range of research purposes It provides a properrepresentation of the heterogeneous concrete material ondifferent levels of its microstructure and renders the possi-bility of studying the effects of a wide range of technologicalparameters Many phenomena in concrete are highly relevantto particle interface properties for example interfacial tran-sition zone (ITZ) surface dissolution and porosimetry Therealistic 119878-119881 information incorporated in the DEM approachwould yield a better representation of the material and thusforms a more reliable basis for estimating relevant materialsproperties as compared to systems employing spherical par-ticles

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The first author is grateful for the continuing supportfrom the National Natural Science Foundation of China(51308013) Beijing Natural Science Foundation (8152009)Specialized Research Fund for the Doctoral Program ofHigher Education of MOE (SRFDP 20131103120022) and theGeneral Program of Science and Technology DevelopmentProject of BeijingMunicipal EducationCommission ofChina(00400054631453040) for this work

References

[1] R M German Particle Packing Characteristics Metal PowderIndustries Federation Princeton NJ USA 1989

[2] P Stroeven J Hu and D A Koleva ldquoConcrete porosimetryaspects of feasibility reliability and economyrdquo Cement andConcrete Composites vol 32 no 4 pp 291ndash299 2010

[3] D P Bentz ldquoThree-dimensional computer simulation of port-land cement hydration and microstructure developmentrdquo Jour-nal of the American Ceramic Society vol 80 no 1 pp 3ndash21 1997

[4] K van Breugel Simulation of Hydration and Formation of Struc-ture in Hardening Cement-based Materials Delft UniversityPress Delft The Netherlands 1997

[5] P Navi and C Pignat ldquoThree-dimensional characterization ofthe pore structure of a simulated cement pasterdquo Cement andConcrete Research vol 29 no 4 pp 507ndash514 1999

[6] M Stroeven Discrete numerical model for the structural assess-ment of composite materials [PhD thesis] Meinema Delft TheNetherlands 1999

[7] E J Garboczi and J W Bullard ldquoShape analysis of a referencecementrdquoCement andConcrete Research vol 34 no 10 pp 1933ndash1937 2004

[8] S T Erdogan P N Quiroga D W Fowler et al ldquoThree-dimensional shape analysis of coarse aggregates new tech-niques for and preliminary results on several different coarseaggregates and reference rocksrdquo Cement and Concrete Researchvol 36 no 9 pp 1619ndash1627 2006

[9] P A Thomas and J D Bray ldquoCapturing non-spherical shapeof granular media with disk clustersrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 125 no 2-3 pp 169ndash178 1999

[10] H He Computational modelling of particle packing in con-crete [PhD thesis] Delft University of Technology IpskampDrukkers Delft The Netherlands 2010

[11] W Guo ldquoSome material parameters on numerical statisticalcontinuum mechanics of concreterdquo TU Delft Report 25-88-38Delft University of Technology Delft The Netherlands 1988

[12] E W Weisstein Ellipsoid FromMathWorldmdashAWolframWebResource httpmathworldwolframcomEllipsoidhtml

[13] J W Bullard and E J Garboczi ldquoA model investigation ofthe influence of particle shape on portland cement hydrationrdquoCement and Concrete Research vol 36 no 6 pp 1007ndash10152006

[14] H He Z Guo P Stroeven and M Stroeven ldquoDiscrete elementapproach to packing of arbitrary shaped particles in concreterdquoInzynieria materiałowa vol 29 no 4 pp 403ndash407 2008

[15] K Li M Stroeven P Stroeven and L J Sluys ldquoC-S-H globuleclustering on nano-scale simulated by the discrete elementmethod for pore structure explorationrdquo in Proceedings of theTRANSCEND Conference Water Transport in CementitiousMaterials Guildford UK November 2013

[16] E J Garboczi ldquoThree-dimensional mathematical analysis ofparticle shape using X-ray tomography and spherical harmon-ics application to aggregates used in concreterdquo Cement andConcrete Research vol 32 no 10 pp 1621ndash1638 2002

[17] H He Z Guo P Stroeven M Stroeven and L J SluysldquoCharacterization of the packing of aggregate in concrete by adiscrete element approachrdquoMaterials Characterization vol 60no 10 pp 1082ndash1087 2009

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


Gravel Ellipsoids Aggregate

Crushed rock Polyhedra

Figure 1 Simulation strategy of arbitrary shaped aggregate (top) differently shaped ellipsoids represent river gravel (bottom) differentlyshaped polyhedrons represent crushed rock

experimental techniques in building codes such as ASTMD-3398-97 cannot provide sufficient information for particlereconstruction in theDEMsystemsThe involved experimen-tal difficulties would be even greater for very fine particlessuch as cement grains Simulation of concrete by a DEMsystem requires a large number of particles with differentsizes to guarantee the RVE demand They will inevitably alsohave different shapes However giving all particles individualshapes in DEM would be too expensive and (computer)time-consuming for solving practical engineering problems[9] Therefore a practical strategy is to select a single shapeparameter or eventually a limited number of typical shapesto represent a particles mixture

In this paper simulation strategies are presented for therepresentation of real aggregate and cement grains in con-crete technologyThis simulation approach is incorporated ina physical concurrent algorithm-based DEM system that isHADES The system is developed for making more realisticpacking simulations by using nonspherical particles It isbased on a contact mechanism that evaluates the interactionforces exerted between segmented surfaces of neighboringparticles It can provide packing densities up to the denserandompacking state relevant for the present purposesMoredetailed information about the system can be found in [10]A practical shape analysis study is conducted on the basis ofCT and 120583CT results available in the international literature[7 8] Specific simulation strategies could be proposed forboth aggregate and cement particle simulation

2 Shape Simulation of Aggregate Grains

Aggregate in concrete can be divided into two groups thatis gravel of fluvial origin and crushed rock representingrounded and angular-shaped particles respectively as shownin Figure 1 The ellipsoid was selected to represent theriver gravel particles whereas the polyhedron was used forsimulation of crushed rock grainsThe surface of thesemodelparticles is tessellated by a mesh positioned on the grainsrsquosurfaces The mesh should be fine enough for a proper

generation of the particles and interparticle computationalrequirement [10]

As a practical example a series of regular polyhedronswith 4 to 8 facetted surfaces were employed to representcrushed rock aggregate as used in railroad beds [11] Thisgreatly simplifies the simulation since only the two param-eters should be specified that is the sieve size and themaximum size of the surface element Figure 2 presents themeshed particles of the regular polyhedrons employed in thisstudyThevolume fraction of each type of polyhedron is takensimilar to that found in the field study of Guo [11]

Recently in a study of Erdogan et al [8] the shapes of fourdifferent types of aggregate denoted by GR LS IN and AZwere investigated by the CTmethod and reconstructed by thespherical harmonic (SH) technique [8] The most interestinginformation coming from this shape analysis is the globalsurface area (119878) to volume (119881) curve of each type of aggregateA function 119878 = 119886119881119887was used to represent the 119878 versus119881 curvefor each type of aggregate [8] The coefficients 119886 and 119887 wereassessed for all cases and are listed inTable 1 [8]The variancesof the regression approximations were exceeding 098 in allcases

The regression function 119878 = 119886101584011988123 was employed tofit the different types of aggregate Next 1198861015840 is determined(volumelt 4000mm3) with a variation lower than 5 for eachtype of aggregate as given in Table 1 A common shape indexlike sphericity is finally determined for each type of aggregateby

Sphericity =119878eqsphere

119878particle=4120587 (31198814120587)23

119886101584011988123= 4836 (1198861015840)

minus1(1)

inwhich 1198861015840 is the coefficient in the above-mentioned relation-ship of 119878 versus 11988123 Sphericity is defined as the surface arearatio of the equivalent sphere and the real particle so bothhaving equal volume The sphericity values are also listed inTable 1 Obviously the crushed rock types GR and LS cannotbe distinguished by means of sphericity The natural rivergravel types IN and AZ have a relatively larger sphericity as




I II III

IIIIII












0

500

1000

1500

2000

0 1000 2000 3000 4000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)



119881 =43120587119886119887119888

119878 = 2119887radic2int120587


sdot 119864(119888

119887

radic2 (1198872 minus 1198862)

1198862 + 1198882 + (1198862 minus 1198882) cos (2120601)sin120601)119889120601

(2)













02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

Sphe

ricity

01940

02948

03955

04963

05970

06977

07900

08200

1000

(a)

02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

SV23

4800

5900

6100

7800

1260

1580

1800

2230

2480

(b)


0

5000

10000

15000

20000

0 50000 100000 150000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)








0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000 25000


Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

(a)

x

y

z

(b)


(a) (b)


x

z

x

z







Acknowledgment


References

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






I II III

IIIIII












0

500

1000

1500

2000

0 1000 2000 3000 4000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)



119881 =43120587119886119887119888

119878 = 2119887radic2int120587


sdot 119864(119888

119887

radic2 (1198872 minus 1198862)

1198862 + 1198882 + (1198862 minus 1198882) cos (2120601)sin120601)119889120601

(2)













02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

Sphe

ricity

01940

02948

03955

04963

05970

06977

07900

08200

1000

(a)

02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

SV23

4800

5900

6100

7800

1260

1580

1800

2230

2480

(b)


0

5000

10000

15000

20000

0 50000 100000 150000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)








0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000 25000


Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

(a)

x

y

z

(b)


(a) (b)


x

z

x

z







Acknowledgment


References

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0

500

1000

1500

2000

0 1000 2000 3000 4000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)



119881 =43120587119886119887119888

119878 = 2119887radic2int120587


sdot 119864(119888

119887

radic2 (1198872 minus 1198862)

1198862 + 1198882 + (1198862 minus 1198882) cos (2120601)sin120601)119889120601

(2)













02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

Sphe

ricity

01940

02948

03955

04963

05970

06977

07900

08200

1000

(a)

02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

SV23

4800

5900

6100

7800

1260

1580

1800

2230

2480

(b)


0

5000

10000

15000

20000

0 50000 100000 150000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)








0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000 25000


Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

(a)

x

y

z

(b)


(a) (b)


x

z

x

z







Acknowledgment


References

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

Sphe

ricity

01940

02948

03955

04963

05970

06977

07900

08200

1000

(a)

02 04 06 08 10

Elongation

02

04

06

08

10

Flat

ness

SV23

4800

5900

6100

7800

1260

1580

1800

2230

2480

(b)


0

5000

10000

15000

20000

0 50000 100000 150000



Surfa

ce ar

ea (120583

m2)

Volume (120583m3)








0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000 25000


Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

(a)

x

y

z

(b)


(a) (b)


x

z

x

z







Acknowledgment


References

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000 25000


Surfa

ce ar

ea (120583

m2)

Volume (120583m3)

(a)

x

y

z

(b)


(a) (b)


x

z

x

z







Acknowledgment


References

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials








Acknowledgment


References

























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



research article on the shape simulation of aggregate and...

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