Uncertainty quantification of the mechanical properties of lightweight concrete using micromechanical modelling

Uncertainty quantification of the mechanical properties of lightweight concrete using micromechanical modelling

Journal Pre-proof

Uncertainty quantification of the mechanical properties of lightweight concrete using micromechanical modelling Tuan Nguyen, Abdallah Ghazlan, Thang Nguyen, Huu-Tai Thai, Tuan Ngo PII: DOI: Reference:

S0020-7403(19)33147-9 https://doi.org/10.1016/j.ijmecsci.2020.105468 MS 105468

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date:

22 August 2019 3 January 2020

Please cite this article as: Tuan Nguyen, Abdallah Ghazlan, Thang Nguyen, Huu-Tai Thai, Tuan Ngo, Uncertainty quantification of the mechanical properties of lightweight concrete using micromechanical modelling, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105468

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Highlights • A micromechanical modelling framework is developed to analyse the effects of microstructural uncertainties of lightweight EPS concrete.

• The random size, spatial distribution and volume fraction of EPS inclusions, as well as the variation of cement matrix properties are considered as microstructural uncertainties.

• Extensive Monte Carlo simulations are performed for uncertainty quantification using finite element modelling.

• The numerical results are compared with compressive test results of EPS concrete. • The influence of each microstructural uncertainty on the mechanical behaviour of EPS concrete is investigated.

1

Graphical Abstract

2

Uncertainty quantification of the mechanical properties of lightweight concrete using micromechanical modelling Tuan Nguyena , Abdallah Ghazlana , Thang Nguyenb , Huu-Tai Thaia , Tuan Ngoa,∗ a Department

b Faculty

of Infrastructure Engineering, The University of Melbourne, Melbourne, Australia of Building and Industrial Engineering, National University of Civil Engineering, Hanoi, Vietnam

Abstract This paper presents a micromechanical modelling framework for analysing the effects of microstructural uncertainties on the mechanical properties of heterogeneous materials. Specialised to lightweight concrete, the proposed framework addresses numerous microstructural uncertainties including the random size, spatial distribution and volume fraction of lightweight expanded polystyrene (EPS) inclusions, as well as the variation of cement matrix properties. A ”takingplacing” algorithm was used to realistically generate three-dimensional, heterogeneous microstructures of EPS concrete with EPS particles embedded into the cement matrix. The nonlinear behaviour of the cement matrix and EPS particles were taken into account by damage and plasticity material laws. By modelling the size, volume fraction and spatial distribution of EPS particles, as well as the properties of the cement matrix as stochastic parameters, the developed model can quantify the inherent uncertainty of EPS concrete. Extensive Monte Carlo simulations for more than 900 EPS concrete samples were performed for uncertainty quantification using finite element modelling. Enabled by a developed automated workflow, the proposed method can simultaneously perform a large number of computational steps on a single platform. A good agreement between the numerical and experimental results was obtained. The significance of each microstructural uncertainty on the mechanical behaviour of EPS concrete was examined in detail. In addition, the effect of particle size on EPS concrete was investigated. While this study focuses on the uncertainty quantification of EPS concrete, the proposed method can be applicable to a wide range of heterogeneous materials. ∗ Corresponding

author: Tuan Ngo. Email: [email protected]

Preprint submitted to International Journal of Mechanical Sciences

January 21, 2020

Keywords: Lightweight EPS concrete, Microstructural uncertainty, Micromechanical modelling, Monte Carlo simulations, Size effect 1

1. Introduction

2

Lightweight concrete, a special type of concrete, has recently gained popularity as an alter-

3

native material to traditional concrete in the construction industry due to its low density, multi-

4

functionality and sound/thermal insulation [1, 2, 3, 4, 5]. Additionally, lightweight concrete is

5

an excellent material for prefabricated components due to extensive transportation and lifting re-

6

quirements in modular, off-site construction [6]. Lightweight concrete is usually considered as a

7

heterogeneous, composite material, which is composed of a cement binder matrix and lightweight

8

inclusions such as expanded clay, expanded slag, pumice and expanded polystyrene (EPS). Similar

9

to traditional concrete, the mechanical properties of lightweight concrete are strongly affected by

10

its microstructural characteristics [7, 8, 9]. In recent years, researchers have expended significant

11

efforts to develop models for simulating the microstructural behaviour of concrete and lightweight

12

concrete [10, 11, 12, 13, 14, 15].

13

Among the different types of lightweight concrete, EPS concrete has attracted significant interest

14

due to its high strength-to-weight ratio, energy absorption, thermal and acoustic insulation and

15

low environmental impact [16, 17]. EPS concrete sandwich panels have been successfully used

16

as load-bearing structures and partition walls in the construction industry [18, 19]. The mechan-

17

ical behaviours of EPS concrete panels have been extensively investigated in the literature. The

18

feasibility of using EPS concrete as structural elements was examined by Fernando et al. [20]. It

19

is concluded that the EPS concrete panels can have sufficient mechanical properties to be used

20

as structural components for buildings [20]. Lee et al. [21, 22] investigated axial and flexural be-

21

haviour of sandwich panels with high performance EPS concrete core. It is found that an optimised

22

EPS concrete core material and suitable face sheets will provide an eco-friendly, lightweight panel

23

systems with high mechanical properties and good insulation [22]. Mousavi et al. [23] presented an

24

investigation on the behaviour of EPS concrete shear walls under cyclic tests. The results showed

25

that the EPS concrete shear wall system has a satisfactory behaviour for seismic-prone structures

26

[23]. 4

1

However, it is well-established that the mechanical behaviour of EPS concrete differs from tradi-

2

tional concrete due to its low density, and smooth, rounded and very light EPS inclusions [16, 24]

3

and more research is needed to fully understand and correctly predict the properties of EPS con-

4

crete [18, 23]. Bouvard et al. [25] showed that the compressive strength of EPS concrete decreases

5

when its density decreases. The size effect of the EPS particles on the compressive strength of EPS

6

concrete has been the focus of many researchers. Experimental studies have shown that, for the

7

same mixture density, the compressive strength of EPS concrete increases with a decrease in the

8

size of EPS inclusions [26]. Despite many research efforts, few computational studies have been

9

conducted to accurately understand and predict the mechanical behaviour of EPS concrete. Song

10

et al. [27] reported an investigation into the mechanical properties of EPS concrete. To under-

11

stand the underlying mechanisms of materials, an inclusion model based on the boundary element

12

method was developed. The mismatch between the elastic moduli of the EPS particles and ce-

13

ment matrix was modelled by an eigenstrain, which is a fictitious non-mechanical strain. A limited

14

number of inclusions (1, 8 and 27 particles) was artificially generated and simulated to account for

15

understanding the elastic behaviour of the EPS samples [27]. Nguyen et al. [28] studied the failure

16

initiation and propagation in EPS concrete using the phase field method, which was coupled with

17

a Computed Tomography (micro-CT) compression test. The crack initiation and propagation cap-

18

tured by the numerical model was validated by the micro-CT test. Miled et al. [29] developed an

19

idealised 2D numerical model of EPS concrete, using uniform particle size and spatial distribution.

20

The results from Miled et al. [29] surprisingly showed that there was no particle size effect for the

21

idealised EPS concrete samples. They concluded that this contradictory finding can be attributed

22

to a lack of randomness in the inclusion network of the idealised EPS concrete model.

23

Random microstructural characteristics can significantly affect the mechanical properties of any

24

type of heterogeneous materials [30, 31, 32], including porous, lightweight materials like EPS

25

concrete in this study [33, 34, 35]. Although the uncertainty in the mechanical properties of het-

26

erogeneous materials can be quantified using a large number of experiments, this approach is

27

impractical due to time and resource constraints. To overcome these limitations, many researchers

28

have coupled micromechanical modelling with the Monte Carlo simulation (MCS) method to study

5

1

the effect of microstructural uncertainties on the mechanical properties of heterogeneous materi-

2

als. Among several statistical techniques, MCS is the most prevalent method due to its robustness

3

and ability to tackle high-dimensional uncertainties [36, 37]. For example, Wang and colleagues

4

[38, 39, 40] used the MCS method to quantify the uncertainties of aggregates (size, shape and

5

spatial distribution) on the mechanical properties of concrete. Their results agreed well with ex-

6

perimental data obtained from the literature [40]. Li et al. [41] modelled the three dimensional (3D)

7

microstructure of concrete as a three-phase composite consisting of mortar matrix, aggregates and

8

the interfacial transition zone (ITZ) to investigated the effect of microstructural characteristics on

9

the effective permeability of concrete. Li et al. [42] studied the fracture behaviour of concrete by

10

means of micromechanical modelling, MCS method and experiments. They observed that the het-

11

erogeneous microstructure resulted in the discrepancies between the actual and predicted cracks

12

by deterministic model, which can be covered under the spectrum of MCS results. The use of

13

lightweight inclusions, particularly EPS particles in EPS concrete, significantly increases the vari-

14

ability of its properties due to high heterogeneity and uncertainties at the microscale [43]. However,

15

to the authors’ knowledge, no study has been developed to investigate the microstructural uncer-

16

tainties of EPS concrete has not been investigated in the literature. It should be mentioned that

17

existing computational models [28, 27, 29] for EPS concrete are based on deterministic frame-

18

works.

19

The goal of this paper is to develop a stochastic, micromechanical framework to capture the

20

microstructural uncertainties of heterogeneous materials using both experimental and numerical

21

techniques. The developed model, which is specialised to EPS concrete, captures numerous mi-

22

crostructural uncertainties including the random size, spatial distribution and volume fraction of

23

EPS particles as well as, variations in the cement matrix properties. The micromechanical model

24

of EPS concrete was accurately generated by adopting a ”taking-placing” algorithm, which was in-

25

troduced by Wang et al. [44] to generate 2D random aggregates of concrete. The concrete damage

26

plasticity (CDP) and crushable foam material laws were used to describe the mechanical behaviour

27

of the cement matrix phase and EPS particles, respectively. The proposed micromechanical model

28

was validated using experimental results. By modelling the size, volume fraction and spatial dis-

6

1

tribution of EPS particles, and the properties of the cement matrix as stochastic parameters, the

2

developed micromechanical model was extended to a stochastic framework to analyse the inherent

3

microstructural uncertainties of EPS concrete. To this effect, more than 900 samples were simu-

4

lated by a automated MCS workflow for uncertainty quantification.

5

The rest of the paper is organised as follows. The next section presents the sample preparation

6

and experiments of EPS concrete. Section 3 describes the micromechanical modelling of EPS

7

concrete, which includes the generation of numerical EPS samples, and the CDP and crushable

8

foam material laws. In Section 4, the stochastic, computational framework for quantifying the mi-

9

crostructural uncertainties of EPS concrete is presented. The results are presented and discussed in

10

Section 5, which includes the analysis of the size effect of EPS particles. Finally, conclusions and

11

an outlook for future works are given in Section 6.

12

2. Materials and Experiments

13

Table 1 shows the mixture proportion of EPS concrete used in this paper. The EPS particles

14

used in the experimental program were provided by a local supplier. The EPS particles had a

15

diameter between 7 − 9 mm and a density of 36 kg/m3 . The volume fraction of EPS particles is

16

20% of the total volume of the sample.

Table 1: The mixture proportion of EPS concrete.

Mass (g)

Cement Silica fume

Accelerator admixture

Water

EPS

1000.0

5.0

400.0

9.1

50.0

7

(a)

(b)

Figure 1: Compressive test (a) and typical failure pattern (red lines) (b) of EPS concrete.

1

Six cylindrical samples of EPS concrete were cast with a diameter of 50 mm and height of 100

2

mm. A group of six dense cement paste samples without EPS particles were also made to obtain

3

the input parameters for the cement matrix phase in the micromechanical model. The samples

4

were cured in sealed plastic bags at ambient temperature. Compression tests were conducted after

5

28 days of curing using the MTS Universal Testing System at the University of Melbourne. The

6

displacement-controlled tests were performed at a velocity of 0.03 mm/s until the samples com-

7

pletely failed, to obtain the stress-strain responses of EPS concrete. The experimental setup and

8

the typical failure patterns observed from the experiments are shown in Fig. 1

8

1

3. Micromechanical modelling of EPS concrete

2

3.1. Generating the numerical model of EPS concrete

Figure 2: A typical frequency diagram of EPS size distribution in the numerical simulations.

3

This section presents the process of generating micromechanical models of EPS concrete using

4

a taking-placing process. As aforementioned, the shape of EPS particles in this study is perfectly

5

spherical with a range of diameters between [7 − 9] mm. It is assumed that the size (diameter)

6

of EPS particles follows a random uniform distribution. A typical frequency diagram of the size

7

distribution used in the numerical simulations of EPS concrete is presented in Fig. 2. EPS con-

8

crete is usually considered as a two-phase composite material composed of a cement binder matrix

9

and EPS particles at the microscale. It is commonly assumed that the inherent pores at the sub-

10

millimetre and sub-micrometer scales are ”smeared” and considered as components of the cement

11

matrix phase [28, 27, 29]. [45] also reported that EPS particles are not overlapped.

12

The uncertainty quantification of EPS concrete requires a large number of numerical simulations

13

to accurately capture its microstructure and obtain reliable statistical outputs. Therefore, it is rea-

14

sonable to use a 2D modelling methodology to reduce computational costs. A 2D axisymmetric

15

model is commonly used to simulate homogeneous cylindrical concrete samples at the macroscale

16

[46]. However, this simplification is not valid when simulating cylindrical concrete samples at

17

the microscale. In fact, in a 2D axisymmetric, micromechanical model, the EPS particles are

18

genuinely realised and formulated as continuous circular rings [47]. Moreover, it is shown that 9

1

2D micromechanical models inaccurately capture the behaviours of materials at the microscale

2

[42, 40]. Therefore, 3D micromechanical modelling is employed in this study.

3

In this research, 3D EPS concrete models were generated by adopting the taking-placing algorithm.

4

In this process, a particle with random geometrical parameters (e.g. spatial coordinates, diameter)

5

is generated and subsequently placed into the sample domain (i.e. a cylinder with a diameter of

6

50mm and height of 100mm). Each particle is checked to ensure no overlap occurs. For spherical

7

EPS particles, the non-overlapping condition between two particles can be easily checked by the

8

following equation: di j =

q

xi − x j


2

+ yi − y j


2

+ zi − z j


2

Di + D j ≥ 2




(1)

9

where di j is the centre-to-centre distance of two EPS particle i and j; x, y, z and D in Eq. (1) are the

10

spatial coordinates and diameter of the particle. The ”taking-placing” process is repeated until the

11

target volume fraction of particles is reached. The flowchart of the ”taking-placing” process for

12

the realistic representation of EPS concrete is depicted in Fig. 3. Start Read input information: Specimen size, target volume fraction, inclusion size distribution (e.g.  normal, log‐normal), inclusion size range (e.g. maximum and minimum  diameter) Generating an  EPS particle

NO Check constraints?

NO

YES Stopping criterion: Reaching the target volume fraction? YES Stop

Figure 3: Flowchart of the ”taking-placing” process for the realistic generation of EPS concrete.

13

When the required parameters of EPS particles are obtained from the ”taking-placing” pro-

14

cess, the geometry of EPS concrete can be generated in the finite element modelling (FEM) code

10

1

ABAQUS [48]. The cement matrix part is obtained by subtracting the particles from a solid cylin-

2

der using the ”cutting” Boolean operation. A EPS concrete specimen in the numerical simulation

3

is obtained by applying the ”merging” Boolean operation to combine the EPS particles and cement

4

matrix part. As a result, perfect bonding is assumed between EPS particles and cement matrix.

5

Fig. 4 shows a few examples of generated EPS concrete models. The volume fraction of EPS

6

particles is 20% with a diameter range of [7 − 9] mm.

Figure 4: Numerical models of EPS concrete with different realisations of particles randomness.

Particle information

ABAQUS  CAE

Numerical  EPS  concrete  samples

Cut a  cylinder  by the  particles

Create a  cylinder

Merge  cut  cylinder  and the  particles

Figure 5: The procedure of generating numerical models of EPS concrete.

11

1

The process of generating an EPS concrete model is illustrated in Fig. 5. In this study, the

2

displacement-controlled compression experiments were simulated by imposing clamped boundary

3

conditions at the bottom of the specimens and prescribing a displacement of a reference point at

4

the top of the sample. The micromechanical model of EPS concrete is meshed using tetrahedral

5

elements and solved using the FEM code ABAQUS/EXPLICIT [48]. The effects of mesh size and

6

loading time, which is associated with loading (strain) rate in ABAQUS/EXPLICIT, are investi-

7

gated in Section 5.

8

3.2. Material laws

9

The concrete damage plasticity (CDP) and crushable foam material laws are used to model the

10

mechanical behaviour of the cement matrix and EPS particles, respectively. The detailed formula-

11

tion of the material laws is presented below.

12

3.2.1. Cement matrix phase

̃

1

, ̃ 1

, 

)

Figure 6: The typical stress-strain response of the CDP model in tension and compression [48].

13

The CDP material law has been widely used to model the behaviour of the cement matrix phase

14

in the micromechanical modelling of concrete due to its robustness and effectiveness [13, 48, 49, 12

1

50]. Fig. 6 illustrates the stress-strain curve representing the CDP material law. Under uni-axial

2

compressive loading, the material response is linear until the initial yield stress, σc0 , followed by

3

stress hardening in the plastic regime and strain softening beyond the ultimate stress, σcu . The

4

response under uni-axial compressive loading is defined by a linear elastic relationship until the

5

tensile failure stress, σt0 , is reached [13, 48]. At this point, it is assumed that micro-cracks in the

6

material are formed. The CDP model assumes the non-associated potential plastic flow rule as

7

follows:

∂G ε˜˙ pl = λ˙ ∂ σ¯

(2)

8

where σ¯ and ε˙˜ pl are the effective stress and plastic strain rate tensors, respectively; σ¯ is given by

9

stress-strain constitutive relation as follows: σ¯ = D el : ε − ε˜ pl

10 11




(3)

where D el is the degraded stiffness and ε is the total strain. λ˙ in Eq. (2) is a plastic multiplier, and G is the Drucker-Prager hyperbolic function, which is represented as follows: q G = (σt0 tan ψ)2 + q¯2 − p¯ tan ψ

(4)

12

where  is the eccentricity that defines the rate at which the function approaches the asymptote; ψ

13

is the dilation angle measured in the q¯ p¯ plane at a high confining pressure. q¯ and p¯ are the Von

14

Mises equivalent effective stress and hydrostatic pressure and calculated by Eq. (5): r 3 1 kdev (σ¯ )k p¯ = − tr (σ¯ ) q¯ = 2 3

(5)

15

In Eq. (4), σt0 is the uni-axial tensile strength as depicted in Fig. 6. It is common that the ten-

16

sile strength is assumed as 7-10% of the compressive strength, if the uni-axial tensile test is not

17

18

available [13, 50]. The yield function used in the CDP model is as follows:  

  

i 1 h F= q¯ − 3α p¯ + β ε˜tpl , ε˜cpl σˆ¯ max − γ −σˆ¯ max − σˆ c εcpl 1−α

where

σb0 −1 σc0 ; 0 ≤ α ≤ 0.5 α= σ b0 2 −1 σc0 13

(6)

1

2 3

σb0 and σˆ¯ max is the maximum principal effective stress; is the ratio of the initial equi-bi-axial σc0 σb0 is usually set to the compressive yield stress to the initial uni-axial compressive yield stress. σc0 default value of 1.16 [13, 48]. In Eq. (6), β and γ are given as follows:   ¯ σc ε˜cpl 3 (1 − Kc ) β =   (1 − α) − (1 + α) ; γ = (7) pl 2Kc − 1 σ¯ ε˜ t

4 5

t

    where σ¯ c ε˜cpl and σ¯ t ε˜tpl are the effective cohesion stress in tension and compression, respec-

tively; and Kc in Eq. (7) is the ratio of the second stress invariant on the tensile meridian, q( T M),

6

to that on the compressive meridian, q(CM), at any hydrostatic stress p¯ [48].

7

In the CDP material law, two scalar damage indices, namely dc and dt , are used to model the loss

8

of stiffness and failure of the material under compression and tension [48], respectively. The com-

9

pressive damage variable is given as a function of the crushing (inelastic) strain, which is defined as

10

the total strain minus the elastic strain corresponding to an undamaged material as given in Eq. (8): ε˜cin = εc −

11

σc E0

(8)

The compressive damage variable is calculated using the following linear expression: dc = 1 −

σc σcu

(9)

12

The plastic strain under compression ε˜cpl can be calculated from the crushing strain ε˜cin and the

13

compressive damage variable as follows [48, 50]: ε˜cpl = εc − εcel = ε˜cin +

14 15

σc σc dc σc − = ε˜cin − E0 (1 − dc ) E0 1 − dc E0

(10)

Similarly, the tensile damage can be defined as a function of the crack-opening displacement wt , which can be calculated using the following relationship: "   #  
σt wt 3 wt wt = 1 + c1 exp −c2 − 1 + c31 exp (−c2 ) σt0 wcr wcr wcr

(11)

16

where σt is the tensile stress that is normal to a crack direction; wcr , c1 and c2 are given in Nguyen

17

et al. [13], Huang et al. [50]. The damage variable in tension is calculated as follows: dt = 1 − 14

σt σt0

(12)

Young’s modulus (GPa)

6.25-7.45

Compressive strength (MPa)

37.9-44.1

Eccentricity

0.1

Dilation angle ψ σb0 ratio σc0 Kc

10

0.6

35

0.5

30 25

0.4

20

0.3

15

0.2

10

Compressive stress

5

0.1

3.0

1.2

2.5

1

2.0

0.8 Tensile stress

1.5

0.6

Tensile damage

1.0

0.4

0.5

0.2

Tensile damage

0.7

40

0.667

Tensile stress (MPa)

45

1.16

Compressive damage

Compressive stress (MPa)

Table 2: The input parameters of the CDP material law in the numerical model.

Compressive damage

0 0

0.002

0.004 0.006 Crushing strain

0 0.008

0.0

0 0

(a)

0.02 0.04 0.06 Crack-opening displacement (mm)

0.08

(b)

Figure 7: The behaviours of the cement matrix phase in the CDP material law: (a) Compressive behaviour; (b) Tensile behaviour.

1

The use of the CDP material law requires specifying several input parameters listed in Table 2,

4

and the compressive and tensile behaviour as illustrated in Fig. 7. In this study, the input parameters σb0 , ψ, , Kc of the CDP material law are referred to the literature [13] and calibrated based on σc0 the experimental results of the dense cement paste samples as discussed in Section 2. It should

5

be noted that the compressive strength and Young’s modulus are two physical parameters obtained

6

from the experiments of the dense cement paste samplesto account for the variation of the cement

7

matrix properties. Therefore, they are considered as stochastic parameters in the next section to

8

quantify the uncertainty of EPS concrete. It should be mentioned that Fig. 7 is plotted with the

9

mean compressive strength and Young’s modulus.

2 3

15

1

3.2.2. EPS particles

Stress

Densification Plateau

 pl Linear elasticity 0

0.2

0.4

0.6

0.8

1

Strain

Figure 8: A typical stress-strain response of EPS material under compression [51].

2

EPS is a type of foam material and its mechanical behaviour is extensively studied in the lit-

3

erature [52, 53, 54]. A typical stress-strain curve of EPS is depicted in Fig. 8, which consists

4

of three distinct regions of linear elastic behaviour, plateau plastic deformation and densification.

5

The material law, which is capable of capturing the mechanical behaviour of foam material, as de-

6

scribed in Fig. 8, has been studied by many researchers, including Deshpande and Fleck [55], who

7

proposed an isotropic hardening model for metallic foams, and Zhang et al. [56], who proposed a

8

volumetric hardening model for polymer foams. In this study, the behaviour of EPS is modelled

9

as a crushable foam with volumetric hardening. The model uses a yield surface with an elliptical

10

dependence of the deviatoric stress on the pressure [48]. It assumes that the evolution of the yield

11

surface is controlled by the volumetric compacting plastic strain experienced by the material. For

12

the plateau plastic deformation behaviour, the model requires two parameters [48, 51]: k=

σc0 pt ; k = t p0c p0c

(13)

13

where σc0 is the initial yield stress in uni-axial compression, p0c is the initial yield stress in hydro-

14

static compression, and pt is the yield strength in hydrostatic tension. In this study, the compression

15

yield stress ratio k and hydrostatic yield stress ratio kt are assumed to be 1.1 and 0.1, respectively,

16

as used in previous studies for EPS with a density of 30-40 kg/m3 [57, 58, 59]. The initial yield 16

1

stress in uni-axial compression and the Young’s modulus of EPS are set to 0.22 MPa and 3 MPa,

2

respectively, as experimentally measured by Cui et al. [51]. The input parameters of EPS particles

3

in the numerical model are summarised in Table 3. Table 3: The input parameters of EPS particles in the numerical model.

Density (kg/m3 )

36

Young’s modulus (MPa)

3

Initial uni-axial compression stress (MPa) 0.22

4

Compression yield stress ratio k

1.1

Hydrostatic yield stress ratio kt

0.1

4. Stochastic framework for describing the behaviours of EPS concrete Table 4: Uncertainties in the experimental parameters of EPS concrete.

Input

Output

Minimum

Maximum

Mean

SDV

Compressive strength of matrix (MPa)

37.9

44.1

40.3

2.3

Young’s modulus of matrix (GPa)

6.25

7.45

6.84

0.4

EPS volume fraction (%)

17.8

22.4

20.9

1.6

Compressive strength of EPS concrete

10.0

12.2

11.1

0.7

(MPa)

5

The behaviour of EPS concrete can be evaluated based on the deterministic, micromechanical

6

model presented in the previous section. Nonetheless, the deterministic model only provides a

7

rough estimation and cannot capture the significant randomness and variations in the material. In

8

fact, Table 4 shows the variation of the volume fraction and mechanical properties of the cement

9

matrix obtained from the experiments. Although the target volume fraction of EPS particles in the

10

mixture formulation (Table 1) is 20%, the actual EPS volume fraction, evaluated from the density,

11

varies from 17.8% to 22.4 % for the EPS concrete specimens with a mean of 20.9% and standard 17

1

deviation (SDV) of 1.6%. The variation of the compressive strength and Young’s modulus of the

2

cement matrix is also observed in Table 4. As the result of these uncertain inputs, the compressive

3

strength of EPS concrete (output) varies from 10.0-12.2 MPa. It is observed that the variation

4

in the compressive strength of EPS is significant with a 20.2% difference between the upper and

5

lower experimental data. Fig. 9 presents a spectrum of the compressive stress-strain response of

6

EPS concrete obtained from the experiments. 12 Experiment

Compressive stress (MPa)

10

8

6

4

2

0 0

0.0005

0.001

0.0015 Strain

0.002

0.0025

0.003

Figure 9: The stress-strain response of EPS concrete obtained from the experiments.

7

In this section, the deterministic, micromechanical model is combined with the MCS method-

8

ology in order to quantify the effects of microstructural uncertainties on EPS concrete. For this

9

purpose, the size, volume fraction and spatial distribution of EPS particles, and the properties of

10

cement matrix are comprehensively modelled as stochastic parameters. The material properties of

11

EPS particles are assumed as deterministic parameters. It should be mentioned that previous stud-

12

ies only consider the size and spatial distribution of inclusions as stochastic parameters whilst the

13

volume fraction and the properties of the cement matrix are assumed as deterministic parameters

14

[39, 41, 60]. In this study, it is assumed that the major sources of uncertainty in EPS concrete are

15

(as shown in Table 4):

16

• Random EPS particles (i.e. the random size distribution ([7 − 9] mm) and the random spatial 18

1

distribution);

2

• The variation of EPS volume fraction ([17.8 − 22.4]%);

3

• The variation of the mechanical properties of the cement matrix ([37.9 − 44.1]MPa for the

4

compressive strength and [6.25 − 7.45]GPa for the Young’s modulus).

5

Latin Hypercube Sampling (LHS) is used to achieve the efficiency of random sampling in this

6

study. LHS is an improved sampling strategy that aims to spread the sample points more evenly

7

across all possible values [61]. It is reported that more than 50% of the computational effort can

8

be saved by using LHS instead of simple random sampling methods [62]. In this paper, MCS

9

are conducted for a large number of numerical simulations for uncertainty quantification. For

10

each numerical simulation, a realistic 3D microstructural model of EPS concrete is generated with

11

the random input variables. The behaviour of the generated EPS concrete specimen is evaluated

12

through FEM simulations. After N samples of EPS concrete have been generated and analysed,

13

statistical analysis can be performed to estimate the statistical characteristics of EPS concrete, such

14

as the mean and SDV of the compressive strength [38], as follows: 1 N Y¯ = ∑ yi N i=1 v u N u u ∑ (yi − Y¯ ) t SDV = i=1 N

(14)

15

where yi is the compressive strength of sample i obtained from the simulation, Y¯ is the mean com-

16

pressive strength.

17

The uncertainty quantification of EPS concrete requires a large number of simulations with many

18

time-consuming and labour-intensive steps of generating stochastic parameters, generating random

19

EPS particles, creating micromechanical models, conducting FEM, extracting results and evaluat-

20

ing statistical characteristics. Therefore, it is crucial to develop an automated framework, which

21

links different steps, for the uncertainty analysis of EPS concrete in this study, as well as various

22

engineering applications. The automated workflow is developed using ABAQUS Scripting Inter-

23

face (ASI) [48] and illustrated in Fig. 10. For numerous numerical simulations in MCS method, the 19

1

automated workflow effectively performs the following steps: (i) generating EPS particle (Fig. 3);

2

(ii) creating the micromechanical model of EPS concrete (Fig. 5); (iii) solving the micromechan-

3

ical model by FEM; and (iv) extracting results and evaluating the statistical characteristics. The

4

automated workflow proposed in this paper is essential to facilitate a large number of numerical

5

simulations required in the MCS method, which enables the uncertainty analysis of EPS concrete

6

in the next section.

Start

Read input  information: ‐ Specimen  size, target  volume  fraction,  particle size  distribution,  particle size  range ‐ Compressive  strength,  Young’s  modulus of  cement matrix ‐ The number  of MCS: N

i = i +1 or increase N

Generating  micro‐ structures  of EPS concrete For 0
NO

Micromechanical  Solving model with  random inputs

Outputs 

Extracting results

i = N?  YES

Evaluate  statistical  characteristics 

Generating the  input parameters of  the CDP

END

Figure 10: Flowchart of the automated workflow for the uncertainty analysis of EPS concrete.

7

5. Results and discussion

8

5.1. Analysis of mesh size and loading time

9

In this section, the performance of the deterministic, micromechanical model is examined in

10

terms of mesh size and loading time. The mean value of EPS volume fraction and the cement

11

matrix properties in Table 9, and one random size and distribution of EPS particles are used. It is

12

known that the mesh size of FEM simulations is associated with the formulation of the material

13

laws and stable time increment, thereby affecting accuracy and efficiency (computational time)

14

[48]. Moreover, it is well-known that the loading time significantly affects accuracy and efficiency

15

due to dynamic effects in the FEM code ABAQUS/EXPLICIT [48]. In theory, the loading time

16

must be sufficiently long to minimise any dynamic effects. Nonetheless, a long loading time

17

leads to considerably higher computational demand. This issue is exacerbated in the proposed 20

1

framework which requires a large number of numerical simulations for the MCS method. Hence,

2

a balance, appropriate mesh size and loading time must be investigated prior to the uncertainty

3

quantification.

(a)

(b)

(c)

Figure 11: Three different mesh sizes with an average element size of: (a) 2 mm (Mesh 1); (b) 1.75 mm (Mesh 2); and (c) 1.5 mm (Mesh 3).

12 Mesh 1 (2mm) Mesh 2 (1.75mm) Mesh 3 (1.5mm)

Compressive stress (MPa)

10

8

6

4

2

0 0

0.0005

0.001

0.0015 Strain

0.002

0.0025

0.003

Figure 12: The stress-strain results obtained from the numerical simulation of mesh sensitivity analysis.

4

Fig. 11 shows three meshes with different average element sizes of 2 mm, 1.75 mm and 1.5 mm.

5

Mesh 1, 2 and 3 have 323,887 C3D4 elements (56,859 nodes), 347,756 C3D4 elements (61,030

6

nodes) and 369,976 C3D4 elements (64,836 nodes), respectively. The compressive stress-strain 21

1

responses from the simulations of the three mesh configurations are presented in Fig. 12, which

2

show that mesh dependency is negligible for the three element sizes. Moreover, Fig. 13 shows the

3

failure pattern (DAMAGEC) at the middle of the cylindrical sample for the three meshes. It can

4

be observed that Mesh 1 (2 mm) produces a distinctive failure pattern compared to Mesh 2 (1.75

5

mm) and Mesh 3 (1.5 mm).

(a)

(b)

(c)

Figure 13: The effect of different mesh sizes: (a) Mesh 1; (b) Mesh 2; and (c) Mesh 3; on the failure pattern (DAMAGEC) in the simulations.

6

The stress-strain response from the simulations with three different loading times of 1 mil-

7

lisecond (ms), 5 ms and 10 ms is presented in Fig. 14. The failure pattern for the three cases is

8

depicted in Fig. 15, which shows that the loading time of 1 ms results in a distinctive failure pat-

9

tern compared to the cases of 5 and 10 ms. The loading times of 5 and 10 ms show two diagonal

10

failure patterns (Fig. 15 (b) and (c)), which are also observed from the experiment (Fig. 1 (b)). In

11

addition, it should be noted that the random distribution of EPS particles significantly affects the

12

failure patterns as discussed in the next section. Fig. 14 and Fig. 15 show that the loading time of

13

5 ms is appropriate to minimise any dynamic effects. From the mesh dependence and loading time

14

analysis, the average element size of 1.75 mm and loading time of 5 ms are selected to balance

15

accuracy and efficiency in the following numerical examples. The computational time of a typical

16

numerical simulation is 0.5 hour using a workstation with 8 CPUs and 48 GB RAM. 22

12 Loading time 1ms Loading time 5ms Loading time 10ms

Compressive stress (MPa)

10

8

6

4

2

0 0

0.0005

0.001

0.0015 Strain

0.002

0.0025

0.003

Figure 14: The stress-strain results obtained from the numerical simulation of three different loading durations.

(a)

(b)

(c)

Figure 15: The effect of different loading durations: (a) 1 ms; (b) 5 ms; and (c) 10 ms; on the failure pattern (DAMAGEC) in the simulations.

1

5.2. The effects of microstructural uncertainties on EPS concrete

2

5.2.1. Random EPS particles

3

This section analyses the effect of microstructural uncertainties on EPS concrete. To quantify

4

the effect of random EPS particles, 100 simulations are conducted with random EPS particle size 23

1

and distribution whilst the EPS volume fraction and the properties of cement matrix are set using

2

the mean value in Table 4. Fig. 16 shows the failure partern (DAMAGEC) of three realisations

3

of random EPS particles. Fig. 16 demonstrates the effect of random EPS particles on the failure

4

pattern of EPS concrete.

(a) Front view

(b) At the middle section Figure 16: The effect of particle randomness on failure pattern (DAMAGEC) of three EPS particle realisations.

5

Fig. 17a presents the stress-strain curves from 100 simulations in this example, together with

6

the experimental results (Fig. 9). It can be observed that the numerical results fit well within the 24

1

experimental spectrum. Moreover, the random size and distribution of EPS particles significantly

2

affect the peak stress, and the strain at the peak stress, whilst the elastic modulus is slightly affected

3

as depicted in Fig. 17a. Fig. 17b clearly shows that the majority of compressive strengths predicted

4

by MCS lie between the upper and lower bound of the experimental results, whilst a few samples

5

have a compressive strength below the lower bound. 13

12 Experiment

Experiment‐Upper Compressive strength (MPa)

12.5

Compressive stress (MPa)

10

8

Experiment‐Lower

MC Samples

12

11.5

6

11

10.5

4

2

10 9.5 0

0 0

0.0005

0.001

0.0015 Strain

0.002

0.0025

0.003

20

40 60 Sample number

80

100

(b) Compressive strengths

(a) Stress-strain curves

Figure 17: The results obtained from the MCS of random EPS particles.

6

In order to investigate the effect of the number of samples on the convergence of MCS, the mean

7

and SDV of the compressive strength of EPS concrete are calculated and presented in Fig. 18a.

8

Apparently, the mean and SDV become stable when the number of MCS samples is about 55. The

9

SDV, which is an indicator for the scattering of outputs, gradually decreases when the number of

10

MCS samples increases. The results indicate the convergence of MCS analysis for the random

11

EPS particles. Fig. 18b shows the histogram frequency of the compressive strength from 100

12

simulations. The mean of 10.9 MPa and SDV of 0.35 MPa obtained from 100 EPS samples,

13

and the best-fit normal distribution are shown. It is observed that the histogram frequency of

14

results appears to follow the normal distribution. From Fig. 18b, the confidence interval (CI) of

15

the compressive strength can be calculated using the following equation: SDV Y¯ ± z × N

16

(15)

where Y¯ is the mean compressive strength, N is the number of samples in the MCS method and z is 25

1

the value for different confidence levels (z = 1.96 for 95% confidence). Using Eq. (15), it is found

2

that within a CI of 95%, the effective compressive strength of EPS concrete is 10.9±0.07 MPa in

3

this study. 40 0.50 35

11.05

0.40

Mean (MPa)

Mean (MPa)

11.05

11.00 10.95 10.90

11.00 10.95

Mean

0.30

Mean Standard Deviation Standard Deviation 0.20

10.90

0.10

10.85

Standard deviation (MPa) Histogram frequency

11.10

10.85

10.80

0.00

5

10.80 15 25 35 45 55 65 75 85 95 5 15 25 35 45 55 65 75 85 95 Sample number Sample number

30 0.40 25 0.30 20 15 0.20 10 0.10 5 0 0.00

(a) The effect of the MCS sample on the mean and SDV

40

Mean: 10.9 SD: 0.35Mean: 10.9 SD: 0.35

35 Standard deviation (MPa) Histogram frequency

0.50

11.10

30 25

95% CI 20

95% CI

15 10 5

9.9

0 10.2 10.5 10.8 11.1 11.4 11.7 9.9 10.2 10.5 10.8 11.1 11.4 11.7 Effective compressive strength(MPa) Effective compressive strength(MPa)

(b) Histogram frequency of 100 MCS samples

Figure 18: The statistical results of the compressive strength from the uncertainty analysis of random EPS particles.

4

5.2.2. The variation of EPS volume fraction and cement matrix properties

5

This study will separately examine the effects of the variation of EPS volume fraction and

6

cement matrix properties on EPS concrete. For the first analysis, the volume fraction is varied

7

between from 17.8% to 22.4 % whilst the compressive strength and Young’s modulus of the cement

8

matrix are set as the mean value. For each sample, only one random size and distribution of EPS

9

particles is considered in order to isolate the effect of EPS volume fraction. For the subsequent

10

analysis, the compressive strength and Young’s modulus of the cement matrix are varied from 37.9

11

MPa to 44.1 MPa and 6.25 GPa to 7.45 GPa, respectively, whilst the EPS volume fraction is set as

12

the mean values. Again,one random size and distribution of EPS particles is used.

26

12.5

12 Experiment

Experiment 10 Compressive stress (MPa)

Compressive stress (MPa)

10

7.5

5

2.5

8

6

4

2

0

0 0

0.0005

0.001

0.0015 Strain

0.002

0.0025

0.003

0

0.0005

0.001

0.0015 Strain

0.002

0.0025

0.003

(b)

(a)

Figure 19: The stress-strain results obtained from the MCS of EPS volume fraction (a) and cement matrix properties (b).

Figure 20: The negligible effect of cement matrix properties on theV02‐Matrix1 failure pattern (DAMAGEC) of two EPS concrete V02‐Matrix0 samples obtained from the MCS.

1

The stress-strain responses of the aforementioned analyses are presented in Fig. 19. Fig. 19

2

shows that the stress-strain curves for the two cases fit well within the experimental scatter. Unlike

3

the case of random EPS particles, the variation of EPS volume fraction and cement matrix prop-

4

erties has a profound effect on both the effective compressive strength (peak stress) and Young’s 27

1

modulus of EPS concrete as illustrated in Fig. 19. Remarkably, the results obtained from the uncer-

2

tainty quantification of the cement matrix almost overlay the entire spectrum of Young’s moduli of

3

EPS concrete obtained from the experiments in Fig. 19b. It is noted that the stress-strain responses

4

of 190 samples are very similar, which implies that the variation of cement matrix properties only

5

affects the magnitude but not the stress-strain responses. The failure patterns for the two samples

6

from the analysis presented in Fig. 20 are identical, which solidify the observation. The mean and

7

SDV of the compressive strength of EPS concrete are shown in Fig. 21. It can be observed from

8

Fig. 21 that the mean and SDV of the two analyses tend to be stable when the number of samples

9

is about 65 (for the variation of EPS volume fraction) and 125 (for the variation of cement matrix

10

properties). There are small oscillations in the mean and SDV the uncertainty analysis of EPS vol-

11

ume fraction (Fig. 21a) due to the assumption of a single particle distribution for each EPS volume

12

fraction input.

10.80 5

0.20

10.85

0.10

0.30 0.20 0.10

10.80 0.00 0.00 5 35 15 25 15 25 45 35 55 45 65 55 75 65 85 75 95 85 95 Sample number Sample number

11.35

11.30

11.30

11.25 11.20 11.15 11.10

11.25 11.20

11.05 11.00

5

(a)

Mean 0.30 Standard Deviation Standard Deviation

Mean

0.20

11.10

11.00 10.95

0.40

11.15

11.05

10.95 35 5

0.50

0.50

0.10

95 155 125 185 155 6535 9565 125 Sample number Sample number

0.00 185

0.40 0.30 0.20 0.10 0.00

(b)

Figure 21: The effect of the number of EPS concrete samples on the statistical results of the MCS of EPS volume fraction (a) and cement matrix properties (b).

13

5.2.3. The combined effect of microstructural uncertainties

14

In the previous numerical studies, the effects of microstructural uncertainties on EPS concrete

15

are examined separately. This numerical study presents the combined effects of three microstruc-

16

tural uncertainties. Fig. 22a shows the mean and SDV of the compressive strength resulting from

17

the three microstructural uncertainties, which were investigated simultaneously. Due to the inter-

18

action between the uncertain factors, more than 300 samples are required to achieve convergence 28

Standard deviation (MPa)

10.90

0.40

11.40

11.35

Standard deviation (MPa)

10.85

10.95

0.40 Mean Standard Deviation Standard Deviation 0.30 Mean

11.40

Mean CS (MPa)

10.90

11.00

0.50

Mean CS (MPa)

10.95

Mean CS (MPa)

Mean CS (MPa)

11.00

0.50

Standard deviation (MPa)

11.05

11.05

0.60

0.60

Standard deviation (MPa)

11.10

11.10

of the uncertainty analysis. The compressive strengths of 500 EPS concrete samples are shown in

2

Fig. 22b. Although the majority of outputs is within the experimental range, a significant number

3

of samples have a compressive strength that is out of the experimental upper and lower boundary.

4

This highlights the importance of simultaneously considering uncertain parameters in practical

5

engineering problems. 11.10 11.10

0.7

0.7

11.00 11.00

0.6

10.90 10.90 10.80 10.80

0.5

10.70 10.60 10.50

0.4 Mean Mean Standard Deviation 0.3 Standard Deviation

10.70

0.2

10.60

0.1

10.50

0

0 100

100 200 300 200 300 400 Sample number Sample number

400 500

0.0 500

0.6 0.5 0.4 0.3 0.2 0.1 0.0

14

14 13 12 11 10

Compressive strength (Mpa)

0.8

Compressive strength (Mpa) Standard deviation (MPa)

0.8

Standard deviation (MPa)

11.20 11.20

Mean CS (MPa)

Mean CS (MPa)

1

13

Experiment‐Upper Experiment‐Lower MC Samples Experiment‐Upper Experiment‐Lower MC Samples

12 11 10 9

9 0

0

100

(a) Mean and SDV

100

200 300 400 200 300 400 Sample number Sample number

500

500

(b) Compressive strengths

Figure 22: The statistical results obtained from the MCS of combined three microstructural uncertainties of EPS concrete.

Table 5: Summary of the effects of microstructural uncertainties on EPS concrete.

Minimum

Maximum

Mean

SDV

10.0

12.2

11.1

0.70

Random EPS particle

9.9

11.8

10.9

0.35

Volume fraction

9.9

12.3

10.9

0.51

Cement matrix

10.2

11.7

11.0

0.41

Combined effect

9.1

12.6

11.0

0.68

Experiment

Numerical

6

Table 5 and Fig. 17a show that the uncertainty of random EPS particles has the smallest effect

7

on the compressive strength and Young’s modulus of EPS concrete (with the lowest SDV of 0.35

8

MPa). The variation of EPS volume fraction has the most profound influence on EPS concrete

9

with the highest SDV among the three separate analyses. It can be deduced that the consistency of 29

1

inclusions in each specimen during the preparation of EPS concrete is the most important factor

2

to achieve consistent results. In addition, the combined effect of microstructural uncertainties

3

accurately captures the scatter of experimental results, with an SDV of 0.68 MPa, compared to

4

0.70 MPa from the experiment.

5

5.3. Size effect of EPS particles

6

The particle size effect on EPS concrete has been one of the most important research topics

7

in the literature. Through experimental investigations, many researchers have concluded that the

8

compressive strength of EPS concrete increases with a decrease in EPS particle size for the same

9

mixture density (i.e the same EPS volume fraction) [26, 45]. Miled et al. [29] conducted a numer-

10

ical investigation to analysis an idealised 2D model of EPS concrete with a uniform particle size

11

and spatial distribution. However, the numerical results surprisingly showed that there is no size

12

effect for the idealised EPS concrete sample, which is explained by a lack of randomness of EPS

13

particles. Nonetheless, the particle size effect on the mechanical properties of EPS concrete has

14

not been adequately studied in the literature. To this effect, this section investigates the effect of

15

particle size on EPS concrete using the developed stochastic, micromechanical framework. Two

16

groups are investigated: large particles with a diameter of [7 − 9] mm (from the previous analysis)

17

and small particles with a diameter of [4 − 6] mm. The investigation takes into account the ran-

18

domness of size and spatial distribution of EPS particles. The volume fraction and cement matrix

19

properties are set as the mean values in Table 4.

30

11.00 10.95 10.90 10.85

10 8

0.4 Mean Standard Deviation

6 4 2

10.80 5

15 0 25 35 45 55 65 75 85 95 0 0.0005 0.001 0.0015 0.002 0.0025 Sample number Strain

0.3 0.2 0.1

13.10

0.5

13.00

0.4

12.90

0.3 Mean Standard Deviation

12.80

0.0

12.70

0.1 5

0.003

0.2

Standard deviation (MPa)

Compressive stress (MPa)

12

Mean (MPa)

Large particle group Small particle group

11.05 Mean (MPa)

0.5

14

Standard deviation (MPa)

11.10

15 25 35 45 55 65 75 85 95 Sample number

(a) The stress-strain results from the MCS of two par- (b) The statistical results of the MCS of small particle ticle size groups.

group.

Figure 23: The results obtained from the analysis of EPS particle size effect on the compressive behaviour of EPS concrete.

1

Fig. 23a shows the stress-strain responses of 100 samples for the MCS analysis of each group.

2

It is apparent that the small particle group not only yields the higher compressive strength but also

3

the higher Young’s modulus with the same density. The mean and SDV obtained from the MCS

4

analysis are presented in Fig. 23b for the small particle group. Similar to the previous analysis, the

5

mean and SDV of the small particle group show a stable trend when the number of MCS samples

6

is about 65. The mean compressive strength of the small and large particle group is 13.1 MPa and

7

10.9 MPa, respectively. Hence, it is statistically concluded that using small particles can improve

8

the compressive strength of EPS concrete by 20%.

9

6. Conclusions

10

Uncertainty is the intrinsic feature of heterogeneous materials. Specialised to lightweight ex-

11

panded polystyrene (EPS) concrete, a stochastic, micromechanical modelling framework based

12

on the Monte Carlo simulation (MCS) method was proposed for quantifying the effect of mi-

13

crostructural uncertainties in this paper. The micromechanical model was developed based on the

14

”taking-placing” algorithm to generate the 3D, heterogeneous microstructure of EPS concrete. The

15

concrete damage plasticity (CDP) and crushable foam material laws were used to model the me31

1

chanical behaviour of the cement matrix and EPS particles. The size, volume fraction and spatial

2

distribution of EPS particles, and the properties of the cement matrix were modelled as stochastic

3

parameters to quantify the effect of microstructural uncertainties. An automated workflow was

4

developed using ABAQUS Scripting Interface to conduct extensive MCS for quantifying uncer-

5

tainties in more than 900 samples.

6

The results show that the EPS volume fraction has a profound effect on the compressive strength

7

of EPS concrete with the highest SDV of 0.51 MPa. The random size and distribution of EPS parti-

8

cles has a modest effect on the compressive strength and Young’s modulus but strongly influences

9

the failure pattern in the material material. Moreover, the variation of cement matrix properties

10

can remarkably capture the entire scatter observed from the experiments but slightly affects the

11

failure pattern in the material. In addition, the combined effect of three microstructural uncertain-

12

ties is significant and should be considered in the uncertainty analysis of concrete in future studies.

13

By analysing EPS particle size effect using the proposed stochastic, micromechanical model, it

14

was concluded that using small particles can improve the mechanical properties of EPS concrete

15

by 20%. The results show that the proposed stochastic, micromechanical framework is an effec-

16

tive tool to quantify the effect of microstructural uncertainties on the properties of heterogeneous

17

materials.

18

Acknowledgements

19

The first author would like to thank the University of Melbourne for offering the Melbourne Re-

20

search Scholarship. This work was mainly supported by the CRC-P for Advanced Manufacturing

21

of High Performance Building Envelope project, funded by the CRC-P program of the Depart-

22

ment of Industry, Innovation and Science, Australia, and the Asia Pacific Research Network for

23

Resilient and Affordable Housing (APRAH) grant, funded by the Australian Academy of Science,

24

Australia. This work was also supported by the ARC Training Centre for Advanced Manufacturing

25

of Prefabricated Housing (ARC IC150100023) and ARC Discovery Project (ARC DP170100851)

26

at the University of Melbourne.

32

1

References

2

[1] Nguyen, T., Kashani, A., Ngo, T., Bordas, S.. Deep neural network with high-order neuron for

3

the prediction of foamed concrete strength. Computer-Aided Civil and Infrastructure Engineering

4

2019;34(4):316–332.

5

[2] Hajimohammadi, A., Ngo, T., Mendis, P., Nguyen, T., Kashani, A., van Deventer, J.S.. Pore

6

characteristics in one-part mix geopolymers foamed by h2o2: the impact of mix design. Materials &

7

Design 2017;130:381–391.

8

[3] Bayat, A., Liaghat, G.H., Ghalami-Choobar, M., Ashkezari, G.D., Sabouri, H.. Analytical modeling

9

of the high-velocity impact of autoclaved aerated concrete (aac) blocks and some experimental results.

10

International Journal of Mechanical Sciences 2019;159:315 – 324.

11

[4] Hajimohammadi, A., Ngo, T., Mendis, P., Kashani, A., van Deventer, J.S.. Alkali activated

12

slag foams: the effect of the alkali reaction on foam characteristics. Journal of cleaner production

13

2017;147:330–339.

14

[5] Kashani, A., Ngo, T.D., Mendis, P., Black, J.R., Hajimohammadi, A.. A sustainable application

15

of recycled tyre crumbs as insulator in lightweight cellular concrete. Journal of cleaner production

16

2017;149:925–935.

17

[6] Ferdous, W., Bai, Y., Ngo, T.D., Manalo, A., Mendis, P.. New advancements, challenges and

18

opportunities of multi-storey modular buildings–a state-of-the-art review. Engineering Structures

19

2019;183:883–893.

20 21 22 23

[7] Hajimohammadi, A., Ngo, T., Mendis, P.. Enhancing the strength of pre-made foams for foam concrete applications. Cement and Concrete Composites 2018;87:164–171. [8] Kashani, A., Ngo, T.D., Nguyen, T.N., Hajimohammadi, A., Sinaie, S., Mendis, P.. The effects of surfactants on properties of lightweight concrete foam. Magazine of Concrete Research 2018;:1–10.

24

[9] Nguyen, T.T., Bui, H.H., Ngo, T.D., Nguyen, G.D.. Experimental and numerical investigation

25

of influence of air-voids on the compressive behaviour of foamed concrete. Materials & Design

26

2017;130:103–119. 96.

27

[10] Nguyen, T.T., Bui, H.H., Ngo, T.D., Nguyen, G.D., Kreher, M.U., Darve, F.. A micromechan-

28

ical investigation for the effects of pore size and its distribution on geopolymer foam concrete under

29

uniaxial compression. Engineering Fracture Mechanics 2019;209:228–244.

30

[11] Mauludin, L.M., Zhuang, X., Rabczuk, T.. Computational modeling of fracture in encapsulation-

33

1

based self-healing concrete using cohesive elements. Composite Structures 2018;196:63–75.

2

[12] Cook, A.C., Vel, S.S., Johnson, S.E.. Pervasive cracking of heterogeneous brittle materials us-

3

ing a multi-directional smeared crack band model. International Journal of Mechanical Sciences

4

2018;149:459 – 474.

5

[13] Nguyen, T., Ghazlan, A., Kashani, A., Bordas, S., Ngo, T.. 3d meso-scale modelling of foamed

6

concrete based on X-ray Computed Tomography. Construction and Building Materials 2018;188:583–

7

598.

8

[14] Rupasinghe, M., Mendis, P., Ngo, T., Nguyen, T.N., Sofi, M.. Compressive strength prediction

9

of nano-silica incorporated cement systems based on a multiscale approach. Materials & Design

10 11 12

2017;115:379–392. [15] Quayum, M.S., Zhuang, X., Rabczuk, T.. Computational model generation and rve design of selfhealing concrete. Frontiers of Structural and Civil Engineering 2015;9(4):383–396.

13

[16] Nikbin, I.M., Golshekan, M.. The effect of expanded polystyrene synthetic particles on the frac-

14

ture parameters, brittleness and mechanical properties of concrete. Theoretical and Applied Fracture

15

Mechanics 2018;94:160–172.

16 17 18

[17] Nogueira, C.L., Rens, K.L.. Ultrasonic wave propagation in EPS lightweight concrete and effective elastic properties. Construction and Building Materials 2018;184:634–642. [18] Ramli Sulong,

N.H., Mustapa,

S.A.S., Abdul Rashid,

M.K..

Application of expanded

19

polystyrene (eps) in buildings and constructions: A review. Journal of Applied Polymer Science

20

2019;136(20):47529.

21 22

[19] Rahim, J.A., Hamzah, S.H., Saman, H.M.. Expanded polystyrene fibred lightweight concrete (epsflwc) as a load bearing wall panel. Jurnal Teknologi 2015;76(9).

23

[20] Fernando, P., Jayasinghe, M., Jayasinghe, C.. Structural feasibility of expanded polystyrene (eps)

24

based lightweight concrete sandwich wall panels. Construction and building materials 2017;139:45–

25

51.

26 27

[21] Lee, J.H., Hong, S.G., Ha, Y.J.. Sandwich panels of ultra-high performance concrete composite with expanded polystyrene. Journal of Asian Concrete Federation 2017;3(2):90–97.

28

[22] Lee, J.H., Kang, S.H., Ha, Y.J., Hong, S.G.. Structural behavior of durable composite sandwich pan-

29

els with high performance expanded polystyrene concrete. International Journal of Concrete Structures

30

and Materials 2018;12(1):21.

34

1 2 3 4

[23] Mousavi, S.A., Zahrai, S.M., Bahrami-Rad, A.. Quasi-static cyclic tests on super-lightweight eps concrete shear walls. Engineering structures 2014;65:62–75. [24] Cui, C., Huang, Q., Li, D., Quan, C., Li, H.. Stressstrain relationship in axial compression for EPS concrete. Construction and Building Materials 2016;105:377–383.

5

[25] Bouvard, D., Chaix, J.M., Dendievel, R., Fazekas, A., Ltang, J.M., Peix, G., et al. Characterization

6

and simulation of microstructure and properties of EPS lightweight concrete. Cement and Concrete

7

Research 2007;37(12):1666–1673.

8 9

[26] Le Roy, R., Parant, E., Boulay, C.. Taking into account the inclusions’ size in lightweight concrete compressive strength prediction. Cement and Concrete Research 2005;35(4):770–775.

10

[27] Song, G., Wang, L., Deng, L., Yin, H.M.. Mechanical characterization and inclusion based bound-

11

ary element modeling of lightweight concrete containing foam particles. Mechanics of Materials

12

2015;91:208–225.

13

[28] Nguyen, T.T., Yvonnet, J., Bornert, M., Chateau, C.. Initiation and propagation of complex 3d

14

networks of cracks in heterogeneous quasi-brittle materials: Direct comparison between in situ testing-

15

microCT experiments and phase field simulations. Journal of the Mechanics and Physics of Solids

16

2016;95(Supplement C):320–350.

17 18 19 20 21 22 23 24

[29] Miled, K., Le Roy, R., Sab, K., Boulay, C.. Compressive behavior of an idealized EPS lightweight concrete: size effects and failure mode. Mechanics of Materials 2004;36(11):1031–1046. [30] Hauseux, P., Hale, J.S., Cotin, S., Bordas, S.P.. Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters. Applied Mathematical Modelling 2018;62:86–102. [31] Zhu, W., Chen, N., Liu, J., Xia, S.. The effective elastic properties analysis of periodic microstructure with hybrid uncertain parameters. International Journal of Mechanical Sciences 2018;148:39 – 49. [32] Akmar, A.I., Lahmer, T., Bordas, S.P.A., Beex, L., Rabczuk, T.. Uncertainty quantification of dry woven fabrics: A sensitivity analysis on material properties. Composite Structures 2014;116:1–17.

25

[33] Masmoudi, M., Kaddouri, W., Kanit, T., Madani, S., Ramtani, S., Imad, A.. Modeling of the effect

26

of the void shape on effective ultimate tensile strength of porous materials: Numerical homogenization

27

versus experimental results. International Journal of Mechanical Sciences 2017;130:497 – 507.

28

[34] Sakata, S., Ashida, F., Ohsumimoto, K.. Stochastic homogenization analysis of a porous material

29

with the perturbation method considering a microscopic geometrical random variation. International

30

Journal of Mechanical Sciences 2013;77:145 – 154.

35

1

[35] Takano, N., Takizawa, H., Wen, P., Odaka, K., Matsunaga, S., Abe, S.. Stochastic prediction of

2

apparent compressive stiffness of selective laser sintered lattice structure with geometrical imperfection

3

and uncertainty in material property. International Journal of Mechanical Sciences 2017;134:347 –

4

356.

5

[36] Ding, C., Deokar, R.R., Ding, Y., Li, G., Cui, X., Tamma, K.K., et al. Model order reduction acceler-

6

ated monte carlo stochastic isogeometric method for the analysis of structures with high-dimensional

7

and independent material uncertainties. Computer Methods in Applied Mechanics and Engineering

8

2019;349:266–284.

9

[37] Chandrashekhar, M., Ganguli, R.. Nonlinear vibration analysis of composite laminated and sandwich

10

plates with random material properties. International Journal of Mechanical Sciences 2010;52(7):874

11

– 891.

12

[38] Wang, X., Yang, Z., Yates, J., Jivkov, A., Zhang, C.. Monte Carlo simulations of mesoscale

13

fracture modelling of concrete with random aggregates and pores. Construction and Building Materials

14

2015;75:35–45.

15

[39] Wang, X., Yang, Z., Jivkov, A.P.. Monte Carlo simulations of mesoscale fracture of concrete with

16

random aggregates and pores: a size effect study. Construction and Building Materials 2015;80:262–

17

272.

18

[40] Wang, X., Zhang, M., Jivkov, A.P.. Computational technology for analysis of 3d meso-structure

19

effects on damage and failure of concrete. International Journal of Solids and Structures 2016;80:310–

20

333.

21 22

[41] Li, X., Xu, Y., Chen, S.. Computational homogenization of effective permeability in three-phase mesoscale concrete. Construction and Building Materials 2016;121:100–111.

23

[42] Li, G., Yu, J., Cao, P., Ren, Z.. Experimental and numerical investigation on III mixed-mode frac-

24

ture of concrete based on the Monte Carlo random aggregate distribution. Construction and Building

25

Materials 2018;191:523–534.

26

[43] Tenza-Abril, A.J., Villacampa, Y., Solak, A.M., Baeza-Brotons, F.. Prediction and sensitivity

27

analysis of compressive strength in segregated lightweight concrete based on artificial neural network

28

using ultrasonic pulse velocity. Construction and Building Materials 2018;189:1173–1183.

29 30

[44] Wang, Z., Kwan, A., Chan, H.. Mesoscopic study of concrete i: generation of random aggregate structure and finite element mesh. Computers & structures 1999;70(5):533–544.

36

1 2

[45] Miled, K., Sab, K., Le Roy, R.. Particle size effect on EPS lightweight concrete compressive strength: Experimental investigation and modelling. Mechanics of Materials 2007;39(3):222–240.

3

[46] Lepp¨anen, J.. Concrete subjected to projectile and fragment impacts: Modelling of crack softening

4

and strain rate dependency in tension. International Journal of Impact Engineering 2006;32(11):1828–

5

1841.

6

[47] Song, Z., Lu, Y.. Mesoscopic analysis of concrete under excessively high strain rate compression and

7

implications on interpretation of test data. International Journal of Impact Engineering 2012;46:41–55.

8

[48] ABAQUS, . Dassault Systmes. Abaqus Documentation. 2017.

9

[49] Sarikaya, A., Erkmen, R.. A plastic-damage model for concrete under compression. International

10

Journal of Mechanical Sciences 2019;150:584 – 593.

11

[50] Huang, Y., Yang, Z., Ren, W., Liu, G., Zhang, C.. 3d meso-scale fracture modelling and validation

12

of concrete based on in-situ X-ray Computed Tomography images using damage plasticity model.

13

International Journal of Solids and Structures 2015;6768:340–352.

14 15

[51] Cui, L., Kiernan, S., Gilchrist, M.D.. Designing the energy absorption capacity of functionally graded foam materials. Materials Science and Engineering: A 2009;507(1-2):215–225.

16

[52] Ling, C., Ivens, J., Cardiff, P., Gilchrist, M.D.. Deformation response of EPS foam under combined

17

compression-shear loading. Part I: Experimental design and quasi-static tests. International Journal of

18

Mechanical Sciences 2018;144:480–489.

19

[53] Ling, C., Ivens, J., Cardiff, P., Gilchrist, M.D.. Deformation response of EPS foam under combined

20

compression-shear loading. Part II: High strain rate dynamic tests. International Journal of Mechanical

21

Sciences 2018;145:9–23.

22 23 24 25

[54] Zhang, B., Zhang, X., Wu, S., Zhang, H.. Indentation of expanded polystyrene foams with a ball. International Journal of Mechanical Sciences 2019;161-162:105030. [55] Deshpande, V., Fleck, N.. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids 2000;48(6-7):1253–1283.

26

[56] Zhang, J., Kikuchi, N., Li, V., Yee, A., Nusholtz, G.. Constitutive modeling of polymeric foam ma-

27

terial subjected to dynamic crash loading. International journal of impact engineering 1998;21(5):369–

28

386.

29 30

[57] Ozturk, U.E., Anlas, G.. Finite element analysis of expanded polystyrene foam under multiple compressive loading and unloading. Materials & Design 2011;32(2):773–780.

37

1 2 3 4

[58] Ozturk, U.. Mechanical behavior of low density polymeric foams under multiple loading and unloading. MS, Bogazici University 2008;. [59] Ling,

C., Cardiff,

P., Gilchrist,

M.D..

Mechanical behaviour of EPS foam under combined

compression-shear loading. Materials Today Communications 2018;16:339–352.

5

[60] Wang, C., Wu, Y., Xiao, J.. Three-scale stochastic homogenization of elastic recycled aggregate

6

concrete based on nano-indentation digital images. Frontiers of Structural and Civil Engineering

7

2018;12(4):461–473.

8 9 10 11

[61] McKay, M.D., Beckman, R.J., Conover, W.J.. Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 1979;21(2):239–245. [62] Olsson, A., Sandberg, G., Dahlblom, O.. On latin hypercube sampling for structural reliability analysis. Structural safety 2003;25(1):47–68.

38

1

Credit Author Statement Tuan Nguyen: Methodology, Investigation, Software, Visualization, Writing - Original Draft Abdallah Ghazlan: Software, Writing - Review & Editing Thang Nguyen: Investigation, Visualization Huu-Tai Thai: Methodology, Software, Writing - Review & Editing Tuan Ngo: Conceptualization, Methodology, Supervision, Writing - Review & Editing

1

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Prof Tuan Ngo (Corresponding author) Director, Advanced Protective Technologies for Engineering Structures (APTES) Group The University of Melbourne