Research Article Combined Numerical-Statistical Analyses ...downloads.hindawi.com/journals/mpe/2015/702563.pdf · Research Article Combined Numerical-Statistical Analyses of Damage

Research ArticleCombined Numerical-Statistical Analyses of Damage andFailure of 2D and 3D Mesoscale Heterogeneous Concrete

Xiaofeng Wang and Andrey P. Jivkov

School of Mechanical, Aerospace and Civil Engineering, the University of Manchester, Manchester M13 9PL, UK

Correspondence should be addressed to Xiaofeng Wang; [email protected]

Received 6 May 2015; Accepted 16 July 2015

Academic Editor: Eugene Kogan

Copyright © 2015 X. Wang and A. P. Jivkov. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Generation and packing algorithms are developed to createmodels ofmesoscale heterogeneous concrete with randomly distributedelliptical/polygonal aggregates and circular/elliptical voids in two dimensions (2D) or ellipsoidal/polyhedral aggregates andspherical/ellipsoidal voids in three dimensions (3D). The generation process is based on the Monte Carlo simulation methodwherein the aggregates and voids are generated from prescribed distributions of their size, shape, and volume fraction. A combinednumerical-statistical method is proposed to investigate damage and failure of mesoscale heterogeneous concrete: the geometricalmodels are first generated and meshed automatically, simulated by using cohesive zone model, and then results are statisticallyanalysed. Zero-thickness cohesive elements with different traction-separation laws within the mortar, within the aggregates, andat the interfaces between these phases are preinserted inside solid element meshes to represent potential cracks. The proposedmethodology provides an effective and efficient tool for damage and failure analysis of mesoscale heterogeneous concrete, and acomprehensive study was conducted for both 2D and 3D concrete in this paper.

1. Introduction

Concrete is the most widely used construction material inthe world due to its good strength and durability relativeto its cost. Numerical simulations coupled with theory andexperiment are considered to be an important tool forexamining the mechanical behaviour through computationalmaterials science. Wittmann [1] proposed three levels ofobservation: microlevel (10−6m), mesolevel (10−3m), andmacrolevel (100m). The microlevel represents the most basiclevel, in which the internal structure of cement paste isconsidered andwhere physical and chemical forces dominate.In the mesolevel, concrete is usually represented as a materialmade up of coarse aggregates, mortar with fine aggregatesand cement paste embedded inside, and interfaces betweencoarse aggregates and mortar. At the macroscale, concrete istreated as a homogenous and continuous material in whichthe internal structure is not considered [2]. As concrete isgenerally used in large-sized structures and its dependenceof mechanical behaviours on mesostructures is significant,mesoscale modelling of concrete becomes essential and wasinvestigated in this study.

An extensive literature review recently carried out byWang et al. [3] shows that numerical image processingtechnique and parameterization modelling technique aretwo main approaches to generate mesostructure models ofconcrete. Although themesh generated from images is closestto nature, it is presently very costly and time consumingto generate sufficient scanned images and convert them tomesh for meaningful statistical analyses (e.g., [4–6]). In theparameterization modelling technique, both direct (e.g., [3,7–9]) and indirect approaches (e.g., [10–13]) to characterizematerial random heterogeneity have been used. The indirectapproach is able to generate a large number of sampleswith ease but the key multiphasic parameters which couldsignificantly influence the mechanical behaviour cannot beconsidered. The direct approach can take into account mostof the key parameters such as shape, size, gradation andspatial distribution of voids and aggregates, phase volumefractions, and aggregate-mortar interfaces. So among all theapproaches, it seems that the direct approach is particularlysuitable for statistical analysis of concrete samples and wasused in this study.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 702563, 12 pageshttp://dx.doi.org/10.1155/2015/702563

2 Mathematical Problems in Engineering

(a) Glass ball concrete (b) Rolled aggregate concrete (c) Crushed aggregate concrete

Voids

Aggregates

Mortar

Figure 1: Tomographic cross-sectional view of three concretes varying by coarse aggregate shape (after [14]).

Identification and generation of unit cell geometry are avital step in the mesoscale analysis of concrete. Both shapeand size of aggregates have significant influences on the stressdistribution, crack initiation, and damage accumulation upto the macroscopic failure within the concrete material [16].Most of the existing algorithms for mesostructure generationof concrete only consider random aggregates but neglectvoids [2, 7, 17–19]. However, the X-ray CT images [4, 14, 20]clearly show that voids exist in concrete at this scale and haveseverely adverse effects on the specimen strength [3, 9]. Soautomatic generation of morphological details of materialswith randomly distributed different shape of aggregates andvoids poses new challenges. In this paper, a computationallyefficient and versatile in-house program of heterogeneousmaterial generator (HMG) has been developed consideringboth 2D and 3D random aggregates and voids based on theprescribed parameters.

Several numerical models for crack propagation at meso-scopic level have been used to study the heterogeneity ofconcrete. Continuum based finite element models are themain approaches employed in the literature [21–24], primar-ily based on cohesive zone model [25]. Other alternativeapproaches have also been developed, such as discrete ele-mentmodel (DEM) [26] and latticemodel [11, 12].The biggestdisadvantage of DEM and lattice model is the difficultyin determining the equivalent model parameters, which isrelatively straight forward for continuumbased finite elementmodels [27].The numerical model used in this work for crackpropagation at mesoscopic level is based on the cohesivezone model [3]. It is becoming more and more popular inmodelling crack propagation due to its simple formationand easy implementation in the form of cohesive interfaceelements (CIEs).

With the preprocessing functionality in ANSYS [29] andexplicit solver in ABAQUS [30], nonlinear modelling ofmultiple crack propagation with a considerable number ofconcrete samples has been performed by high performancecomputing in this study. Statistically, analysis of mechanicalbehaviour of both 2D and 3D specimens has been conducted

Table 1: Aggregate size distribution [28].

Sieve size(mm)

Total percentageretained (%)

Total percentage passing(%)

12.70 0 1009.50 39 614.75 90 102.36 98.6 1.4

using the combined numerical-statistical method, leading tovaluable statistical results that may help improve designs ofconcrete materials.

2. 2D and 3D Heterogeneous Concrete

2.1. Aggregate and Void Size Distribution. The size distribu-tion of aggregates in concrete is often described by the Fullercurve [16], which is discretised into a certain number ofsegments. The aggregate size distributions found in Hirsch[28] and summarised in Table 1 are used in this study. In thisstudy, concrete is treated as a material consisting of coarseaggregates, mortar incorporating fine aggregates dissolved init, voids, and interfaces between aggregates and mortar. Hereonly coarse aggregates are explicitly modelled as mesoscalefeatures. For normal strength concrete cast with a mould inthe laboratory, coarse aggregates usually represent 30–50% ofthe concrete volume.

From X-ray CT images, it was found that gravel aggre-gates have circular/elliptical shapes in 2D and spheri-cal/ellipsoidal shapes in 3D, while crushed aggregates havepolygonal shapes in 2D and polyhedral shapes in 3D [14].A number of voids can be clearly observed in concrete atmesoscale from CT images (see Figure 1). Therefore, voidswith size ranging from 2 to 4mm as presented in [31] areincluded in the mesostructure of concrete in this study.Coarse aggregates are considered to have elliptical/polygonalshapes in 2D and ellipsoidal/polyhedral shapes in 3D, and

Mathematical Problems in Engineering 3

YesExit loop

No Save and go for next inclusion

Loop over all the previous generated inclusions until prescribed volume fraction is achieved

Yes

YesExit loop

Hierarchy I: radial distance check

Hierarchy II: check if the centres of any previous inclusions

filtered in Hierarchy I lie inside the new inclusion

Hierarchy III: check if any node on new inclusion lies inside

of inclusions filtered in Hierarchy II

No

NoSave and go for next inclusion

End loop

Figure 2: Flowchart to hierarchical intersection and overlapping checking.

(a) Elliptical aggregates and cir-cular voids

(b) Elliptical aggregates and ellip-tical voids

(c) Ellipsoidal aggregates and spheri-cal voids

(d) Ellipsoidal aggregates and ellip-soidal voids

Figure 3: Numerically generated 2D and 3D models for concrete with gravel aggregates (𝑃agg = 30%, 𝑃void = 2%).

the shape of voids is assumed to be circle/ellipse in 2D orsphere/ellipsoid in 3D.

2.2. Specimen Generation. The basic idea is to create theaggregates and voids in the concrete in a repeated manner,until the target area/volume is filled with aggregates andvoids. The generation starts with the information inputprocess and is followed by taking and placing voids withinsize range; the aggregates within the grading segments areproduced last. The “input” process reads the controllingparameters for generating a heterogeneous concrete; the“taking” process generates an individual void or aggregatein accordance with the random size and shape descriptions.The “placing” process positions the aggregates and voids intothe predefined area/volume in a random manner, subject tothe prescribed physical constraints. There are three physicalconditions to be satisfied simultaneously: (1) the wholeinclusion (void or aggregate) must be within the concrete, (2)there is no overlapping of inclusions, and (3) there is no inter-section between any two inclusions. A three-level hierarchicalmethod is proposed to reduce the computational cost and isoutlined in Figure 2. The procedure was programmed usingMATLAB [32] in this study, and the detailed algorithms andprocedure in generating mesostructures can be found in ourprevious work [3, 33].

2.3. Mesostructure Models. Elliptical aggregates and circu-lar/elliptical voids in 2D or ellipsoidal aggregates and spheri-cal/ellipsoidal voids in 3D were used for concrete with gravelaggregates. A series of concrete samples with dimensions of50mm × 50mm in 2D and 50mm × 50mm × 50mm in3D are generated. Figure 3 shows the numerically generatedmodels with gravel aggregates (aggregates in green and voidsin red). Aggregate gradation in Table 1, aggregate volumefraction 𝑃agg = 30%, void content 𝑃void = 2%, aspect ratiosfor elliptical/ellipsoidal aggregates and voids (the ratio of themajor axis to the minor axis) 𝑅

1

= 𝑅2

= [1, 2], minimumspace between the edge of an aggregate and the boundary ofthe concrete specimen 𝛾

1

= 0.2mm, andminimumgapwidthbetween any two aggregates 𝛾

2

= 0.2mm are adopted here[31].

Polygonal aggregates and circular/elliptical voids in 2D orpolyhedral aggregates and spherical/ellipsoidal voids in 3Dare considered for concrete with crushed aggregates. A seriesof concrete samples with the same dimensions as the previousgravel aggregate samples are generated. Figure 4 shows thenumerically generated models with crushed aggregates. Theminimum andmaximum edges of the polygons/polyhedronsare set to𝑁min = 8,𝑁max = 16. All the other parameters arethe same as the values used for gravel aggregates.


(a) Polygonal aggregates and cir-cular voids

(b) Polygonal aggregates andelliptical voids

(c) Polyhedral aggregates and spheri-cal voids

(d) Polyhedral aggregates and ellip-soidal voids

Figure 4: Numerically generated 2D and 3D models for concrete with crushed aggregates (𝑃agg = 30%, 𝑃void = 2%).

3. Combined Numerical-Statistical Method

3.1. Description of the Method. A combined numerical-statistical method is proposed in this paper to study thematerial behaviour of concrete in a statistical sense. Thedetailed procedure is as follows:

(1) Generate a model with prescribed variables, forexample, sample size, aggregate volume fraction, voidcontent, and aggregate shape.

(2) Perform a finite element simulation of the sample forgiven boundary conditions.

(3) Compute the mean value, standard deviation, andcoefficient of variation (CoV) of effective property forthe considered sample size.

(4) Repeat steps (1) to (3) for sufficient number of randomsamples to meet the required precision, and conductstatistical analysis.

This procedure is automated by running a batch file in thisstudy.

Results from all realisations are evaluated statistically.Thestandard deviation 𝑠 within a series of 𝑛 samples is given by

𝑠2=

1𝑛 − 1

𝑛

∑

𝑖=1(𝑥𝑖

−𝑥)2, (1)

where 𝑥 = (1/𝑛)∑𝑛𝑖=1

𝑥𝑖

is the series average result and 𝑥𝑖

isthe result from sample 𝑖.

To compare results obtained with different sample sizesquantitatively, we use the coefficient of variation given by

𝜀 =𝑠

𝑥. (2)

This expresses the fluctuation of measured property relativeto its average value.

3.2. Cohesive Zone Model in ABAQUS. The cohesive zonemodel developed by Barenblatt [34, 35] and Dugdale [36]enables the simulation of the energy dissipation process inthe fracture process zone (FPZ) during fracture, as illustrated

in Figure 5. A bilinear cohesive crack model is used hereto predict discrete crack propagation in concrete specimensunder tension loading. A linear ascending branch is added inthe softening curve to model the initially uncrackedmaterial.

The separation displacement is difficult to derive fromexperiments, so cohesive fracture energy and cohesivestrength are usually used. Among them, only two parametersare independent, and the fracture energy can be obtained as

𝐺 = ∫

𝛿𝑓

0𝑡 (𝛿) 𝑑𝛿 =

12𝑡0𝛿𝑓, (3)

where 𝐺 is the cohesive fracture energy, 𝑡0

is the cohesivestrength, and 𝛿

𝑓

is the separation displacement.In the 3D cohesive zone model, it is assumed that there

exist a normal traction 𝑡𝑛

and two tangential tractions (shearcohesion) 𝑡

𝑠

and 𝑡𝑛

across the crack surfaces, through mech-anisms such as material bonding, aggregate interlocking,and surface friction in the fracture process zone. Figures6(a) and 6(b) illustrate typical linear softening curves for𝑡𝑛

− 𝛿𝑛

and 𝑡𝑠

(𝑡𝑡

) − 𝛿𝑠

(𝛿𝑡

), where 𝛿𝑛𝑓

and 𝛿𝑠𝑓

(𝛿𝑡𝑓

) are thecritical relative displacements when the tractions diminishfor normal traction and tangential traction components,respectively. The unloading paths are also indicated.

It is worth noting that the initial tensile stiffness𝑘𝑛0

should be high enough to ensure displacement continuityacross the cohesive interfaces before the tensile strength 𝑡

𝑛0

isreached, but not too high to cause numerical instability dueto ill-conditioned global stiffness matrix. Reasonable initialshear stiffness, 𝑘

𝑠0

and 𝑘𝑡0

, is also needed before the shearstrength, 𝑡

𝑠0

or 𝑡𝑡0

, is reached. The effects of initial stiffnesson computational results have been investigated previously[37, 38]. The following relation is suggested in [38] as aguideline for initial stiffness selection:

𝑘𝑛0

= 𝑘𝑠0

= 𝑘𝑡0

=𝑐 (1 − V)

𝑏 (1 + V) (1 − 2V)𝐸, (4)

where 𝐸 and V are Young’s modulus and Poisson’s ratio, 𝑏 isthe characteristic size of elements, and 𝑐 is taken as 10∼100from the experience in [38].

The areas under the curves in Figures 6(a) and 6(b)(calculated by (3)) represent the normal facture energy


Aggregate

Real crack Fracture process zone (FPZ)

Concrete

Traction distribution along FPZt0

Figure 5: Characterization of cohesive zone model (after [15]).

LoadingUnloading

Atn

tn0

kn0

kn

Gnf

𝛿n0 𝛿nf 𝛿n

(a) 𝑡𝑛

− 𝛿

𝑛

curve in normal direction

LoadingUnloading

A

−𝛿sf(−𝛿tf)

ts(tt)

ts0(tt0)

ts0(tt0)

ks0(kt0)

ks(kt)

Gsf(Gtf)

𝛿s0(𝛿t0) 𝛿sf(𝛿tf)𝛿s(𝛿t)

(b) 𝑡𝑠

(𝑡

𝑡

) − 𝛿

𝑠

(𝛿

𝑡

) curve in tangential direction

Figure 6: Bilinear softening laws for cohesive elements.

𝐺𝑛𝑓

and twice the tangential fracture energy, 𝐺𝑠𝑓

and 𝐺𝑡𝑓

,respectively:

𝐺𝑛𝑓

= ∫

𝛿𝑛𝑓

0𝑡𝑛

(𝛿𝑛

) 𝑑𝛿𝑛

=12𝑡𝑛0𝛿𝑛𝑓,

𝐺𝑠𝑓

= ∫

𝛿𝑠𝑓

0𝑡𝑠

(𝛿𝑠

) 𝑑𝛿𝑠

=12𝑡𝑠0𝛿𝑠𝑓,

𝐺𝑡𝑓

= ∫

𝛿𝑡𝑓

0𝑡𝑠

(𝛿𝑠

) 𝑑𝛿𝑠

=12𝑡𝑠0𝛿𝑡𝑓.

(5)

Cohesive elements in ABAQUS based on the cohesivezone model are used here. The damage is characterised by ascalar index 𝐷 representing the overall damage of the crackcaused by all physical mechanisms. The effective relative

displacements 𝛿𝑚

combining the effects of 𝛿𝑛

, 𝛿𝑠

, and 𝛿𝑡

canbe obtained as

𝛿𝑚

= √⟨𝛿𝑛

⟩2

+ 𝛿2𝑠

+ 𝛿2

𝑡

, (6)

where ⟨ ⟩ is the Macaulay bracket and

⟨𝛿𝑛

⟩ =

{

{

{

𝛿𝑛

, 𝛿𝑛

≥ 0 (tension)

0, 𝛿𝑛

< 0 (compression).(7)

The damage evolution law is given by

𝐷 =

𝛿𝑚𝑓

(𝛿𝑚,max − 𝛿𝑚0)

𝛿𝑚,max (𝛿𝑚𝑓 − 𝛿𝑚0)

, (8)

where 𝛿𝑚,max is the maximum effective relative displacement

attained during the loading history. 𝛿𝑚0 and 𝛿𝑚𝑓 are effective


relative displacements at damage initiation and completefailure, respectively. It is obvious that the damage variable 𝐷evolves monotonically from 0 to 1 upon further loading afterthe initiation of damage.

The damage initiation and evolution will degrade theunloading and reloading stiffness coefficients 𝑘

𝑛

, 𝑘𝑠

, and 𝑘𝑡

,which can be calculated as

𝑘𝑛

= (1−𝐷) 𝑘𝑛0,

𝑘𝑠

= (1−𝐷) 𝑘𝑠0,

𝑘𝑡

= (1−𝐷) 𝑘𝑡0.

(9)

The tractions are also affected by the damage according to

𝑡𝑛

=

{

{

{

(1 − 𝐷) 𝑡𝑛

, 𝑡𝑛

≥ 0

𝑡𝑛

, 𝑡𝑛

< 0,

𝑡𝑠

= (1−𝐷) 𝑡𝑠

,

𝑡𝑠

= (1−𝐷) 𝑡𝑡

,

(10)

where 𝑡𝑛

and 𝑡𝑠

are the traction components predicted bythe elastic traction-displacement behaviour for the currentseparation without damage.

In this study, damage in the cohesive zone model isassumed to initiate when a quadratic interaction functioninvolving the nominal stress ratios reaches a value of one

{⟨𝑡𝑛

⟩

𝑡𝑛0}

2

+{𝑡𝑠

𝑡𝑠0}

2+{

𝑡𝑡

𝑡𝑡0}

2= 1. (11)

The material properties, such as density, Young’s modu-lus, Poisson’s ratio, tensile strength, and fracture energy, areset for continuum elements of aggregates and mortar, threedifferent interface cohesive elements. The material hetero-geneity is investigated by considering different phases in theconcrete specimen with corresponding material properties.

In the fracture process zone for a 2D case, tractions existin the normal direction 𝑡

𝑛

and shear direction 𝑡𝑠

across thecrack interface, and the corresponding relative displacementsare 𝛿𝑛

and 𝛿𝑠

. So the 2D constitutive law could be easilydeduced from 3D characterization described above by takingone of the shear tractions and displacements out.

3.3. Cohesive Interface Elements Insertion. In this study, allfinite element meshing is performed with the preprocessingfunctionality in commercial FE package ANSYS [29]. Tomesh the mesoscopic structure of concrete, different materialparts (mortar and aggregates) should maintain continuity ontheir surfaces. Hence, the finite element boundaries are coin-cident with material surfaces and there are no material dis-continuities within the elements. The mortar and aggregatesare meshed with triangular elements (plain stress) in 2D andhexahedral elements in 3D so that more realistic crack pathscan be obtained. The specially developed ANSYS parametricdesign language (APDL) programs in combination with theANSYS batch processing provide a powerful tool of automatic

mesh generation for a large number of samples requiredfor statistical analysis. Following the meshing procedure, thegenerated mesh will automatically have shared nodes at theinterfaces between two elements. If the interfaces surround-ing the elements are to be modelled explicitly as potentialmicrocrack sources, a duplicate set of nodes will be requiredat the interface locations. Here 4-node or 6-node zero in-plane thickness CIEs are preinserted into the existing elementinterfaces in 2D or 3D. This is conducted by a purposewritten and flexible in-house computer program cohesiveinterface elements insertion (CIEIN).Three sets of CIEs withdifferent traction-separation softening laws are inserted (seeFigure 7), namely, CIE-AGG inside the aggregates (grey inFigures 7(b) and 7(d)), CIE-MOR inside themortar (green inFigures 7(b) and 7(d)), andCIE-INT on the aggregate-mortarinterfaces (yellow in Figures 7(b) and 7(d)). The elementand node numbers are denoted by 𝐸 and 𝑁, respectively.The detailed numbering of elements and nodes in the initialmesh and the mesh with inserted cohesive elements showsthe insertion procedure with the new nodes generated at thesame positions and CIEs between the continuum elements.

4. Numerical Simulation and Results

4.1. Description of the Numerical Model. 2D concrete spec-imens with elliptical aggregates and circular voids and 3Dspecimens with ellipsoidal aggregates and spherical voids(𝑃agg = 30%, 𝑃void = 2%, 𝑅

1

= 𝑅2

= [1, 2], 𝛾1

= 𝛾2

= 0.2mm)were modelled, loaded under uniaxial tension. 25mm ×

25mm concrete square in 2D and 25mm × 25mm × 25mmconcrete cubic in 3D are used here. All the other mesoscaleparameters were fixed to be the same for 2D and 3Dmodels. Figure 8 shows the geometry, boundary, and loadingconditions of the numericalmodel. Horizontal displacementswere prescribed to all nodes on the left and right surfacesof the specimen, with a value equal to zero for the leftsurface, and a uniformly distributed displacement on theright surface. Vertical displacements for the same nodes areleft free, except for the nodes at the left lower corner of thespecimen, which are fixed to prevent rigid body translation.A displacement controlled loading scheme was used and allthe analyses were ended at a displacement 𝑑 = 0.1mm andloading time 𝑡 = 0.005 s [3]. Considering the balance betweenaccuracy and efficiency, the element length 1mmwas used forall the meshes in this study [3].

Generally, aggregates have much higher strength thanmortar and interfaces in normal concrete. In this section,no cracks are allowed to initiate inside the aggregates byassuming elastic behaviour without damage in CIE-AGG.The linear tension/shear softening laws described abovewere used to model CIEs with quadratic nominal stressinitiation criterion and linear damage evolution criterion.Similar material properties extracted from [39] were used inthis study, which are listed in Table 2. The fracture propertiesrelated to the shear behaviour were assumed to be the sameas the normal ones due to the lack of experimental data. Bothinitial tensile and shear stiffness for all the cohesive elementswere set to 106MPa/mm after trial and error.


E14

E13

E16

E4

E2E3

E5

E1

N12

N7 N6

N1

N13 N11

N5N3

E8

E6 E9

E10

E11 E7

E12

E15

N8

N4

N9

N10

N2

MortarAggregate

(a) 2D initial mesh

E5

N44

E9

E1 E3

E15E16

E8

N25E6

E12

E10

N1

E11 E7

N15

E4

E2

E14

N2

N21

N20

N19N18N11N9 N10

N22

N42N43N48N46 N47N45

N24

N23

N13N12N5N3 N4N8N7N6 N16

N14

N17

N37

N34N32

N35N33

N36N29

N28

N31N30

N27N26

N39N40N38N41

CIE-AGG

E25E26 E27 E28

E32

E29

E33E36E35E34

E31

E30

E13E24

E22

E17

E23

E21

E18

E19 E20

MortarAggregate

CIE-MOR

CIE-INT

(b) 2D mesh with zero-thickness CIEs

Mortar

Aggregate

N7

N6N5

N1

N2

N3

N4 E4

E3E2

E1

(c) 3D initial mesh

N1

N7

N6

N9

N14

N5

N12 N13N15

N16

N4N3N2

N8 N11N10

E4E6

E7

E3E2

E1

E5

CIE-MOR

CIE-INT

CIE-AGG

Mortar

Aggregate

(d) 3D mesh with zero-thickness CIEs

Figure 7: Inserting different cohesive elements to the initial mesh.

Table 2: Material properties.

Young’smodulus 𝐸(MPa)

Poisson’sratio ]

Density 𝜌(10−9 tonne/mm3)

Elastic stiffness𝑘𝑛

(MPa/mm)

Cohesivestrength 𝑡

𝑛

(MPa)

Fractureenergy 𝐺

𝐹

(N/mm)Aggregate 70000 0.2 2.8 — — —Mortar 25000 0.2 2.0 — — —CIE-AGG — — 2.8 106 — —CIE-MOR — — 2.5 106 4 0.06CIE-INT — — 2.0 106 2 0.03


25m

m25mm

(a) 2D

25mm

25m

m

25mm

(b) 3D

Figure 8: Specimen dimensions, loading, and boundary conditions.

0.00 0.02 0.04 0.06 0.08

Mea

n str

ess (

MPa

)

Displacement (mm)

2D3D

0

1

2

3

4

A-3D

A-2D-

B-2D/3D

(a) Stress-displacement curves

0.00 0.02 0.04 0.06 0.08Displacement (mm)

0

1

2

3

4

Toug

hnes

s (kJ

/m3)

2D3D

(b) Toughness-displacement curves

Figure 9: Comparison of stress-displacement and toughness-displacement curves.

ABAQUS/explicit with small time increments (typicallyΔ𝑡 = 3 × 10

−8 s) was utilised to solve the highly nonlinearequation systems by the high performance computing (HPC)cluster at the University of Manchester computational sharedfacility (CSF). A typical run of simulations with 50 samples inthis study takes about 6 hours by parallel computation with32CPUs.

4.2. Tensile Behaviour. Due to the statistical nature ofmesostructuremodels, an extensive series of numerical simu-lationswould be necessary to capture the range of behaviours.

Fifty random samples were modelled in this study to ensurethat the results are statistically converged.

In Figure 9, resulting mean stress-displacement andtoughness-displacement curves, together with the variationrange for both 2D and 3D, are depicted. It can be seenthat the predicted stress-displacement curves are qualitativelysimilar with a clear peak and sharp initial postpeak dropfollowed by a long shallow tail. Numerical mean stress forevery curve is obtained by summing the nodal reactionforces in the constraints and dividing by the specimencross section area. The mean peak stress predicted by 2D


0.0

0.1

0.2

0.3

0.4

0.5

0.00 0.02 0.04 0.06 0.08

Stan

dard

dev

iatio

n (M

Pa)

Displacement (mm)

2D3D

B-3D

B-2D

A-2D

B-3D

Figure 10: Comparison of standard deviation-displacement curves.

and 3D modelling is 2.65MPa and 3.49MPa, respectively,representing an increase of 24.1%. The mean toughness(energy absorbed per unit volume up to the break point,i.e., the area under the stress-strain curve up to the breakpoint, measured in J/m3) predicted in 2D and 3D modellingis 2.47 KJ/m3 and 3.15 KJ/m3, respectively, representing anincrease of 21.6%; Figure 10 shows the standard deviation-displacement curves for both 2D and 3D modelling. Thestandard deviation of peak stress is 0.11MPa and 0.05MPa,respectively, representing a decrease of 54.5%.These increasesin peak stress or decreases in standard deviation are due tothe constraint effects in the thickness direction in 3D and theless smooth cracking surfaces from 3D modelling. However,the standard deviation of 3D modelling at softening rangeis larger than that in 2D modelling which indicates that 3Dmodelling is more complicated and dependent on differentsamples. This is due to the 3D heterogonous distributionof aggregates and voids. It is clear that the third dimensioncannot be neglected due to the nature of fracture process.Theimportance of the thickness effect has also been pointed outexperimentally by Van Mier and Schlangen [40].

The mean curve, mean value, and standard deviationof the stress and toughness shown in Figures 9 and 10 areextracted from the stress-displacement curves for 50 sampleswith different aggregate and void distribution. In order toevaluate the effect of sample number on simulation resultsas required for the proposed combined numerical-statisticalmethod, two special loading points, namely, point A at peakstress and point B at displacement 0.03mm where largeststandard deviation is observed (denoted in Figures 9(a) and10), are investigated. The influence of sample number onthe coefficient of variation (CoV) of stress at points A andB is illustrated in Figure 11. It can be seen that 50 samplesare enough to get convergent values of CoV of stress witha stable fluctuation. This is an indication that the combinednumerical-statistical method with 50 random realisationsoffers a good balance between general applicability and

0

20

40

60

80

0

1

2

3

4

5

6

0 10 20 30 40 50

CoV

-B (%

)

CoV

-A (%

)

Sample number

2D-A3D-A

2D-B3D-B

Figure 11: Influence of sample number on the CoV of stress.

testing of different samples. It can be observed that COV ofpeak stress for different void and aggregate distribution is low(4.2% for 2D modelling and 1.5% for 3D modelling), whichdemonstrates that the peak stress is relatively insensitive tothe void and aggregate distribution. The high CoV of stressat point B further demonstrates the necessity to conducta statistical analysis as the results vary greatly for differentaggregate and void distribution in softening range.

Figure 12 shows the statistical distribution of peak stress,together with the probability density and the best fit GaussianProbability Density Function (PDF). The probability densitycurve can be used to calculate structural reliability or failurereliability against given external loadings and material prop-erties for structural design.

4.3. Crack Patterns. The complex mesoscale crack propaga-tion is realistically simulated using the proposed method,and typical crack patterns for both 2D and 3D at failure areshown in Figure 13. To clearly visualise the fracture surfaces,models with damaged cohesive elements, models withoutcohesive elements, and failed interfaces only are used for 2Dmodels (see Figures 13(a)–13(c)), and models with damagedcohesive elements, models without cohesive elements, andmorphology of failed surface are used for 3D models (seeFigures 13(d)–13(f)).The cracks shown in Figures 13(a), 13(c),and 13(d) are represented by red CIEs with damage index𝐷 ≥ 0.95 (𝐷 = 1 means complete failure). A magnificationfactor of 20 and 200 was used for the deformed models withdamagedCIEs and thosewithout damagedCIEs, respectively.The noticeable difference of 2D and 3D modelling discussedabove could also be summarized in that the 3D modelingpredicts more realistic fracture surfaces in the thicknessdirection whereas 2D modeling only assumes plane fractureproblems. This result confirms the importance of includingthe third dimension into the analysis.


Mean: 2.65MPa, standard deviation: 0.11MPa

Prob

abili

ty d

ensit

y

0.25

0.20

0.15

0.10

0.05

0.00

Peak stress (MPa)2.3 2.4 2.5 2.6 2.7 2.8 2.9

(a) 2D

0.20

0.25

0.15

0.10

0.05

0.00

Peak stress (MPa)

Mean: 3.49MPa, standard deviation: 0.05MPa

Prob

abili

ty d

ensit

y

0.30

3.35 3.40 3.45 3.50 3.55 3.60

(b) 3D

Figure 12: Statistical distribution of peak stress fromMonte Carlo simulation.

(a) 2D model with damaged CIEs (b) 2D models without CIEs (c) 2D models with failed interfaces

(d) 3D model with damaged CIEs (e) 3D model without CIEs (f) 3D morphology of failed surface

Figure 13: Failure of concrete under tension.


5. Conclusions

Models of numerical concrete with random mesostruc-tures comprising elliptical/polygonal aggregates and circu-lar/elliptical voids in 2D and ellipsoidal/polyhedral aggre-gates and spherical/ellipsoidal voids in 3D have been gener-ated in this study. Numerical-statistical analysis was carriedout using a cohesive zone model to simulate damage andfailure of concrete. Crack patterns are realistically simulatedusing the technique of preembedding cohesive interface ele-ments. The main conclusions, based on the results obtainedfrom the statistical analysis under a uniaxial tension loading,are as follows: (1) statistical analysis is necessary in both 2Dand 3D due to the high dependence of material behaviouron different aggregate and void spatial distribution; (2) thethird dimension is demonstrated to have a pronouncedinfluence on both macroscopic mechanical properties andcrack patterns in tension; (3) compared to 2D modelling,3D modelling demonstrates a larger mean peak stressand a smaller standard deviation in the prepeak response,attributed to more uniformly distributed microcracks withinthe specimen; a larger standard deviation/CoV of stress inthe postpeak response is attributed to the larger number ofpossibilities for microcrack coalescence under the constraintof randomly distributed features. It has to be pointed out thatthe conclusions obtained are based on mesoscale modellingwith specific specimen and aggregate size. The variation onresulting mechanical behaviour is also associated with thesize of specimen and aggregates, and this phenomenon maynot exist when specimen is large enough, for example, at thelength scale of engineering structures.

Conflict of Interests

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

Acknowledgments

This work is supported by the EPS Faculty Ph.D. Studentshipfrom the University of Manchester for Xiaofeng Wang. Theauthors would like to acknowledge the assistance given byIT Services and the use of the Computational Shared Facility(CSF) at The University of Manchester.

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