Generation and packing algorithms are developed to create models of mesoscale heterogeneous concrete with randomly distributed elliptical/polygonal aggregates and circular/elliptical voids in two dimensions (2D) or ellipsoidal/polyhedral aggregates and spherical/ellipsoidal voids in three dimensions (3D). The generation process is based on the Monte Carlo simulation method wherein the aggregates and voids are generated from prescribed distributions of their size, shape, and volume fraction. A combined numericalstatistical method is proposed to investigate damage and failure of mesoscale heterogeneous concrete: the geometrical models are first generated and meshed automatically, simulated by using cohesive zone model, and then results are statistically analysed. Zerothickness cohesive elements with different tractionseparation laws within the mortar, within the aggregates, and at the interfaces between these phases are preinserted inside solid element meshes to represent potential cracks. The proposed methodology provides an effective and efficient tool for damage and failure analysis of mesoscale heterogeneous concrete, and a comprehensive study was conducted for both 2D and 3D concrete in this paper.
Concrete is the most widely used construction material in the world due to its good strength and durability relative to its cost. Numerical simulations coupled with theory and experiment are considered to be an important tool for examining the mechanical behaviour through computational materials science. Wittmann [
An extensive literature review recently carried out by Wang et al. [
Identification and generation of unit cell geometry are a vital step in the mesoscale analysis of concrete. Both shape and size of aggregates have significant influences on the stress distribution, crack initiation, and damage accumulation up to the macroscopic failure within the concrete material [
Several numerical models for crack propagation at mesoscopic level have been used to study the heterogeneity of concrete. Continuum based finite element models are the main approaches employed in the literature [
With the preprocessing functionality in ANSYS [
The size distribution of aggregates in concrete is often described by the Fuller curve [
Aggregate size distribution [
Sieve size (mm)  Total percentage retained (%)  Total percentage passing (%) 

12.70  0  100 
9.50  39  61 
4.75  90  10 
2.36  98.6  1.4 
From Xray CT images, it was found that gravel aggregates have circular/elliptical shapes in 2D and spherical/ellipsoidal shapes in 3D, while crushed aggregates have polygonal shapes in 2D and polyhedral shapes in 3D [
Tomographic crosssectional view of three concretes varying by coarse aggregate shape (after [
The basic idea is to create the aggregates and voids in the concrete in a repeated manner, until the target area/volume is filled with aggregates and voids. The generation starts with the information input process and is followed by taking and placing voids within size range; the aggregates within the grading segments are produced last. The “input” process reads the controlling parameters for generating a heterogeneous concrete; the “taking” process generates an individual void or aggregate in accordance with the random size and shape descriptions. The “placing” process positions the aggregates and voids into the predefined area/volume in a random manner, subject to the prescribed physical constraints. There are three physical conditions to be satisfied simultaneously: (1) the whole inclusion (void or aggregate) must be within the concrete, (2) there is no overlapping of inclusions, and (3) there is no intersection between any two inclusions. A threelevel hierarchical method is proposed to reduce the computational cost and is outlined in Figure
Flowchart to hierarchical intersection and overlapping checking.
Elliptical aggregates and circular/elliptical voids in 2D or ellipsoidal aggregates and spherical/ellipsoidal voids in 3D were used for concrete with gravel aggregates. A series of concrete samples with dimensions of 50 mm × 50 mm in 2D and 50 mm × 50 mm × 50 mm in 3D are generated. Figure
Numerically generated 2D and 3D models for concrete with gravel aggregates (
Elliptical aggregates and circular voids
Elliptical aggregates and elliptical voids
Ellipsoidal aggregates and spherical voids
Ellipsoidal aggregates and ellipsoidal voids
Polygonal aggregates and circular/elliptical voids in 2D or polyhedral aggregates and spherical/ellipsoidal voids in 3D are considered for concrete with crushed aggregates. A series of concrete samples with the same dimensions as the previous gravel aggregate samples are generated. Figure
Numerically generated 2D and 3D models for concrete with crushed aggregates (
Polygonal aggregates and circular voids
Polygonal aggregates and elliptical voids
Polyhedral aggregates and spherical voids
Polyhedral aggregates and ellipsoidal voids
A combined numericalstatistical method is proposed in this paper to study the material behaviour of concrete in a statistical sense. The detailed procedure is as follows:
Generate a model with prescribed variables, for example, sample size, aggregate volume fraction, void content, and aggregate shape.
Perform a finite element simulation of the sample for given boundary conditions.
Compute the mean value, standard deviation, and coefficient of variation (CoV) of effective property for the considered sample size.
Repeat steps (1) to (3) for sufficient number of random samples to meet the required precision, and conduct statistical analysis.
This procedure is automated by running a batch file in this study.
Results from all realisations are evaluated statistically. The standard deviation
To compare results obtained with different sample sizes quantitatively, we use the coefficient of variation given by
The cohesive zone model developed by Barenblatt [
Characterization of cohesive zone model (after [
The separation displacement is difficult to derive from experiments, so cohesive fracture energy and cohesive strength are usually used. Among them, only two parameters are independent, and the fracture energy can be obtained as
In the 3D cohesive zone model, it is assumed that there exist a normal traction
Bilinear softening laws for cohesive elements.
It is worth noting that the initial tensile stiffness
The areas under the curves in Figures
Cohesive elements in ABAQUS based on the cohesive zone model are used here. The damage is characterised by a scalar index
The damage evolution law is given by
The damage initiation and evolution will degrade the unloading and reloading stiffness coefficients
In this study, damage in the cohesive zone model is assumed to initiate when a quadratic interaction function involving the nominal stress ratios reaches a value of one
The material properties, such as density, Young’s modulus, Poisson’s ratio, tensile strength, and fracture energy, are set for continuum elements of aggregates and mortar, three different interface cohesive elements. The material heterogeneity is investigated by considering different phases in the concrete specimen with corresponding material properties.
In the fracture process zone for a 2D case, tractions exist in the normal direction
In this study, all finite element meshing is performed with the preprocessing functionality in commercial FE package ANSYS [
Inserting different cohesive elements to the initial mesh.
2D initial mesh
2D mesh with zerothickness CIEs
3D initial mesh
3D mesh with zerothickness CIEs
2D concrete specimens with elliptical aggregates and circular voids and 3D specimens with ellipsoidal aggregates and spherical voids (
Specimen dimensions, loading, and boundary conditions.
2D
3D
Generally, aggregates have much higher strength than mortar and interfaces in normal concrete. In this section, no cracks are allowed to initiate inside the aggregates by assuming elastic behaviour without damage in CIEAGG. The linear tension/shear softening laws described above were used to model CIEs with quadratic nominal stress initiation criterion and linear damage evolution criterion. Similar material properties extracted from [
Material properties.
Young’s modulus 
Poisson’s ratio 
Density 
Elastic stiffness 
Cohesive strength 
Fracture energy 


Aggregate  70000  0.2  2.8  —  —  — 
Mortar  25000  0.2  2.0  —  —  — 
CIEAGG  —  —  2.8  10^{6}  —  — 
CIEMOR  —  —  2.5  10^{6}  4  0.06 
CIEINT  —  —  2.0  10^{6}  2  0.03 
ABAQUS/explicit with small time increments (typically
Due to the statistical nature of mesostructure models, an extensive series of numerical simulations would be necessary to capture the range of behaviours. Fifty random samples were modelled in this study to ensure that the results are statistically converged.
In Figure
Comparison of stressdisplacement and toughnessdisplacement curves.
Stressdisplacement curves
Toughnessdisplacement curves
Comparison of standard deviationdisplacement curves.
The mean curve, mean value, and standard deviation of the stress and toughness shown in Figures
Influence of sample number on the CoV of stress.
Figure
Statistical distribution of peak stress from Monte Carlo simulation.
2D
3D
The complex mesoscale crack propagation is realistically simulated using the proposed method, and typical crack patterns for both 2D and 3D at failure are shown in Figure
Failure of concrete under tension.
2D model with damaged CIEs
2D models without CIEs
2D models with failed interfaces
3D model with damaged CIEs
3D model without CIEs
3D morphology of failed surface
Models of numerical concrete with random mesostructures comprising elliptical/polygonal aggregates and circular/elliptical voids in 2D and ellipsoidal/polyhedral aggregates and spherical/ellipsoidal voids in 3D have been generated in this study. Numericalstatistical analysis was carried out using a cohesive zone model to simulate damage and failure of concrete. Crack patterns are realistically simulated using the technique of preembedding cohesive interface elements. The main conclusions, based on the results obtained from the statistical analysis under a uniaxial tension loading, are as follows: (1) statistical analysis is necessary in both 2D and 3D due to the high dependence of material behaviour on different aggregate and void spatial distribution; (2) the third dimension is demonstrated to have a pronounced influence on both macroscopic mechanical properties and crack patterns in tension; (3) compared to 2D modelling, 3D modelling demonstrates a larger mean peak stress and a smaller standard deviation in the prepeak response, attributed to more uniformly distributed microcracks within the specimen; a larger standard deviation/CoV of stress in the postpeak response is attributed to the larger number of possibilities for microcrack coalescence under the constraint of randomly distributed features. It has to be pointed out that the conclusions obtained are based on mesoscale modelling with specific specimen and aggregate size. The variation on resulting mechanical behaviour is also associated with the size of specimen and aggregates, and this phenomenon may not exist when specimen is large enough, for example, at the length scale of engineering structures.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the EPS Faculty Ph.D. Studentship from the University of Manchester for Xiaofeng Wang. The authors would like to acknowledge the assistance given by IT Services and the use of the Computational Shared Facility (CSF) at The University of Manchester.