Computational homogenization of effective permeability in three-phase mesoscale concrete

Construction and Building Materials 121 (2016) 100–111

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Computational homogenization of effective permeability in three-phase mesoscale concrete Xinxin Li, Yi Xu, Shenghong Chen ⇑ State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, Hubei 430072, PR China

h i g h l i g h t s
3D concrete mesostructure composed of three phases was modeled.
Extensive Monte Carlo simulations for permeability test were realized and conducted.
Effects of various mesostructural parameters on concrete permeability were investigated.
RVE size regarding concrete permeability was evaluated by computational homogenization.

a r t i c l e

i n f o

Article history: Received 28 March 2016 Received in revised form 9 May 2016 Accepted 24 May 2016

Keywords: Concrete Mesoscale model Effective permeability Computational homogenization Representative volume element (RVE)

a b s t r a c t Concrete is modeled on the mesoscale as a heterogeneous three-phase composite consisting of mortar, aggregates and the interfacial transition zone (ITZ). By exerting a steady state flow in the concrete sample, the effective permeability is estimated using finite element method (FEM). Extensive Monte Carlo (MC) simulations for more than 1000 concrete samples are carried out. The effects of the mesostructural parameters (i.e., the shape, gradation and volume fraction of aggregates and the thickness and permeability of ITZ) on the permeability of concrete are comprehensively investigated. For a specific set of mesostructural parameters, the size of the representative volume element (RVE) for concrete permeability is suggested in terms of the expected errors by numerical and statistical analysis. It shows that computational homogenization for estimating the effective permeability of concrete in three dimensions (3D) is absolutely necessary since the two dimensional (2D) results are less representative. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In the design of dams and other large hydraulic structures, the rate at which water passes through concrete that is subjected to relatively high hydraulic pressure, i.e., water permeability, has long been recognized as a significant parameter for seepage analysis [1– 3]. Permeability is also one of the most important transport properties influencing the durability and serviceability of concrete structures, since many durability problems (e.g., steel corrosion, sulfate or chloride attack, freezing-thawing cycles, abrasion and cavitations) are water-related [4–6]. The growing awareness of the role that permeability plays in the development of service life prediction models and optimum material design has led to the need for practical ways to quickly assess the permeability of concrete.

⇑ Corresponding author. E-mail address: [email protected] (S. Chen). http://dx.doi.org/10.1016/j.conbuildmat.2016.05.141 0950-0618/Ó 2016 Elsevier Ltd. All rights reserved.

Laboratory-based tests have been conducted for many years to measure concrete permeability. It is widely recognized that direct experimental measurement of water permeability usually encounters practical difficulties for concrete due to its low permeability and the complicated conditions of testing methods [7,8]. Since concrete possesses a highly complex mesostructure composed of mortar (or cement paste), aggregates and ITZ between them, its transport properties are influenced by many interacting parameters related to the mesoscopic compositions which are difficultly identified by physical experiments [9,10]. Therefore, it would be extremely helpful to predict the transport properties of concrete based on its mesostructure, either with analytical or numerical models. Traditional close-form models available focusing on the mixture proportion like the Parallel model, the Series model, the Maxwell-Eucken (ME) model and the Effective Medium Theory (EMT) model reviewed in literature [11], are restricted to simple geometrical arrangements and fail to reveal the complicated transport behavior of concrete [12]. Over the past decade, with the advantages of repeatability and efficiency, numerical modeling

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emerges as a promising way for the study of the composite behavior of concrete with mesostructure. Many efforts have been made to explore the macroscopic transport properties of heterogeneous concrete, such as capillary absorption, diffusivity and permeability [13–17]. The influence of the mesoscopic compositions on the properties of concrete is strongly sensitive when the sampling volume size is comparable to that of aggregates. To estimate the material property of such a random heterogeneous material like concrete convincingly, sample size need to be large enough to guarantee statistical representativeness [18]. For this purpose, the concept of representative volume element (RVE) is generally introduced for computational homogenization of the effective properties [9,19,20]. As the volume size gets larger than RVE, the influence of the mesoscopic compositions on the properties of concrete weakens considerably and it could be assumed that the composite material is homogeneous in a statistical sense. Although the RVE on the mechanical properties of concrete has been investigated extensively, yet only a few of the studies on the size effect in modeling transport process are reported insofar. Recently, some useful contributions have been made to determine the RVE from the perspective of transport properties by numerical simulation. For example, Keskin et al. [21] use a probabilistic model for 2D aggregate particles to evaluate the diffusivity of mortar and identify the smallest size needed for reliable estimations. Zhou and Li [22] conduct a numerical and statistical analysis of concrete permeability with a three-phase model in 2D. Nilenius et al. [23,24] study the effective moisture and chloride diffusivity coefficients in three-phase mesoscale concrete by computational homogenization. However, these simulations are carried out either in the case of 2D or using a composite sphere model in 3D, which might be less representative and insufficient for resembling the real concrete. Attributable to the extreme complexity of concrete material, these computer models still lack the required level of detail for taking into account the influential factors derived from its internal constituents, especially ITZ which is widely recognized as the zone of weakness in terms of fluid permeation. In order to gain a better knowledge of the RVE with regard to effective permeability of concrete, a realistic and sophisticated simulation of its mesostructure is urgently required. In this paper, an investigation into the RVE of concrete for a rational estimation of its effective permeability via numerical simulation is presented. The exquisite model of concrete with a 3D mesostructure is generated, which is realized as three-phase composite material comprising of coarse aggregates, mortar matrix and ITZ. A steady state flow in the mesostructure of concrete can be implemented for investigating the water transport behavior and assessing its permeability using FEM. With Monte Carlo (MC) simulations, numerical studies on various factors influencing the permeability of concrete, such as the shape, gradation and volume fraction of aggregates and the transport property of ITZ, have been carried out to understand their impacts comprehensively. Subsequently, computational homogenization using the three-phase composite model is undertaken to determine the RVE size of permeability.

a RAS resembling real concrete are the coarse aggregate configuration parameters, i.e., the size and spatial distribution, volume fraction and shape characteristic [13,25,26]. The size distribution of aggregates plays an important role in concrete mixture design, which affects the main properties of concrete such as workability of concrete mix, mechanical strength, permeability and durability [27,28]. In practice, concrete is most designed following Fuller curve [18,29] which is defined by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðDÞ ¼ 100 D=Dmax

ð1Þ

where PðDÞ is the cumulative percentage passing a sieve with aperture diameter D (L), and Dmax (L) is the maximum size of the aggregate particles. As one of the main constituents, coarse aggregates usually occupy around 0.4 of the concrete volume, which have proven to be an important influential factor on the overall transport properties of concrete [30–32]. Particles in various shapes are employed for modeling the aggregates, i.e., spherical or elliptical particles for gravel aggregates and convex polyhedral particles for crushed ones. These different shaped aggregates according to prescribed volume fractions and a specific size distribution (e.g., Fuller curve) are then randomly packed into a target cubic or cylindrical container for concrete samples, with the efficient approach of ‘‘occupation and removal method” [33,34]. 3D cubic geometrical models are generated in this study, with the detailed procedure described in our previous study [25]. Fig. 1 shows five typical models with particles of spherical shape in different sample sizes, Ls (L). Fig. 2 shows three models sized by Ls = 150 mm, containing spherical, elliptical and polyhedral shaped aggregates, respectively. In all these samples, the sizes of aggregates are scaling from Dmin = 5 mm to Dmax = 40 mm obeying Fuller curve and the aggregate volume fraction, f agg , is fixed as 0.4. 2.2. Development of FE model 2.2.1. Governing equation Suppose that the permeation of water in perfectly saturated porous medium obeys Darcy’s law, which allows for the flow process driven by pressure gradient. Combining the continuity equation for incompressible and steady flow, water transport in concrete under the assumption that concrete skeleton is rigid, is governed by the following partial differential equation [1]:

rðk  r/Þ ¼ 0

ð2Þ

where k (L T1) represents the permeability coefficient, / (L) is the hydraulic potential and r (L1) is the spatial gradient operator. Eq. (2) is subject to appropriate boundary conditions:

/jC1 ¼ /0 ðfirst typeÞ

ð3Þ

k  @/=@fngjC2 ¼ q ðsecond typeÞ

ð4Þ

T

in which fng ¼ flx ; ly ; lz g are direction cosines of the external normal to the boundary, /0 (L) and q (L T1) are specified hydraulic potential at the first type boundary C1 , and the flow rate through the second type boundary C2 , respectively.

2. The three-phase mesoscale model 2.1. Generation of geometrical models In the mesoscopic study, the 3D random aggregate structure (RAS) is first established for FE modeling. The geometrical model is the same as aforementioned: a three-phase composite consisting of homogeneous mortar matrix, randomly distributed coarse aggregates and ITZ of uniform thickness. The key factors for such

2.2.2. Discretization of the FE model For a refined and elegant simulation using FEM, the efficient discretization scheme of the generated concrete sample is demanded. Since the 3D RAS is a heterogeneous random model, the geometry of the mesostructure is usually complicated. In order to improve the mesh quality and solution efficiency, the two main phases of the model, aggregates and mortar, are meshed together with tetrahedral solid elements. In FE analysis, both the aggregates

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Fig. 1. Cubic samples with different sizes.

Fig. 2. Cubic samples containing different shaped aggregates.

and mortar can be treated as porous medium, therefore the water flow in these two phases is governed by Eq. (2). As known, the ITZ is of great importance to investigate the composite properties of concrete. However, as an extremely thin layer structure, the discretization and modeling of ITZ is still a challenging task. To address this issue, the zero-thickness interface element is used herein to simulate the ITZ phase (see Fig. 3) which is assumed as a conductive ‘‘layer” between aggregates and mortar. The transport behavior in ITZ domains is described by the following equation,

rs ðki  di rs /Þ ¼ f m þ f a

ð5Þ

1

where ki (L T ) is the permeability coefficient of ITZ and di (L) represents its hydraulic thickness. rs (L1) denotes the gradient operator restricted to the ITZ’s tangential plane. f m (L T1) and f a (L T1) are the flow rates penetrating into ITZ in the normal directions from mortar and aggregates respectively, which can be written as

8 < f m ¼ km @/up @nup

: f ¼ ka @/down a @n down

Fig. 3. Schematic diagram of the zero-thickness interface element.

ð6Þ

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103

in which n is the unit vector normal to the fracture plane. km (L T1) is the permeability coefficient for mortar and ka (L T1) for aggregates. The subscripts ‘‘up” and ‘‘down” respectively mean the up and down surface of the ITZ plane (see Fig. 3). Since that the thickness of ITZ is extremely small, it can be assumed that the hydraulic potentials at two sides of the ITZ plane are identical, i.e.,

/up ¼ /down ¼ /i

ð7Þ

where /i (L) is the hydraulic potential in ITZ. The detailed implementation of the aforementioned formulations in the commercial FE software, COMSOL Multiphysics, can be referred to the previous work [17]. 3. Numerical approach and analysis for the effective permeability 3.1. Constituent properties of the three-phase model Prior to applying the FE simulation, the water transport property of each component in the mesostructure must be determined (i.e., the permeability coefficients km , ka and ki for mortar, aggregates and ITZ respectively, and the thin thickness di for ITZ). As an important parameter in evaluating the transport property of concrete, the permeability of mortar matrix depends strongly on its pore structure and hydration degree, and thus varies in a large range. The inclusion of coarse aggregates and the formation of ITZ can reduce the water/cement ratio of the matrix, consequently affecting its permeability. Mostly, the effects of coarse aggregates and ITZ on the properties of mortar matrix are not considered explicitly in numerical simulations for the sake of simplicity [9,12–15]. Given this assumption, the permeability coefficient of mortar is fixed constantly in the current study. Since most coarse aggregates used in concrete are much denser than mortar matrix, it is assumed that they are basically impermeable with permeability coefficient 107 times lower than mortar’s for this study, namely, ka ¼ km /107. Another important phase in concrete is ITZ which has long been regarded as a zone of weakness, both in terms of strength and the permeation of fluid [35,36]. Because of its lower density and higher porosity, water flow in ITZ is thought to be much faster than that in the mortar matrix. The permeability coefficient of ITZ is in general substantially larger than mortar’s as ki ¼ kkm , where k > 1.0 represents the permeability ratio. Additionally, as an important modeling parameter, the thickness of ITZ should also be determined, which is widely reported in the range of 10–100 lm [36–38]. Different values for k and di are adopted to study the effects of ITZ on the macroscopic permeability, cf. Section 4.1 below. 3.2. Simulation of transport process With the presented FE model, numerical simulation for a steady flow can be employed to investigate water transport behavior and estimate the permeability of concrete. Two types of boundary conditions are conventionally applied to describe the water flow driven separately by a pressure gradient or a fixed flow rate profile at the inlet accompanied with a pressure imposing on the outlet, corresponding to constant pressure boundary condition (CPBC) and constant flow boundary condition (CFBC). Fig. 4 exhibits a schematic diagram of steady water flow in FE analysis to illustrate the determination of permeability in Z direction. For CPBC, the hydraulic potential /1 (L) and /2 (L) are respectively exerted to the bottom and top boundaries of the cubic sample, while the CFBC could be achieved by substituting /1 with a constant flow Q inlet (L3 T1) applying at the entrance of water

Fig. 4. Illustration of steady water flow in FE analysis.

flow. The other surfaces of the cube are taken as impervious condition. It is noted that the calculated permeability will not change with the hydraulic potential difference D/ (L) (or flow rate Q (L3 T1)), since a steady transport process is simulated. Once the seepage field of the steady flow is mathematically obtained, the effective permeability keff (L T1) of the concrete sample can be predicted from

keff ¼

QL D/  S

ð8Þ

outlet in which flow rate Q ¼ Q inlet þQ , S (L2) is the cross-sectional area 2 perpendicular to the flow direction, and L (L) is the length of the flow path. It is worth mentioning that numerical analysis in similar ways can be performed to obtain the permeability in X and Y directions of this sample. The detailed validation process of the numerical simulation against experimental data can be found in our previous work [17]. It has been proved that the presented model is capable of simulating the transport process in concrete on the mesoscale and evaluating its effective permeability.

3.3. Mesh dependences The precision level of numerical methods depends largely on the mesh density. Therefore, the calculated permeability by mesoscopic modeling may be mesh-dependent. To obtain a highly accurate solution, the second-order Lagrange element [39] is chosen as the element shape function to reduce the effects of mesh dependence in FE analysis. A mesh convergence study is further carried out in order to determine the maximum element size. For a particular concrete sample (see Fig. 2(a)) which comprises only two phases: aggregates and mortar, three FE meshes with element size hmin;max = 1.5– 12.0 mm, hmin;max = 0.6–8.25 mm and hmin;max = 0.225–5.25 mm, are modeled separately (see Fig. 5). The number of solid elements and nodes of Mesh 1, 2 and 3 are (659,164, 114,416), (1,098,264, 189,474) and (2,416,696, 412,796), respectively. The CPU time consumptions on a single 64-bit PC (4.0 GHz, 16 GB RAM) for numerical simulations with these three meshes are 130 s, 383 s and 1184 s, respectively. Fig. 6 shows the distribution of the hydraulic potential along the central axis for the sample with different mesh densities. It can be observed that the hydraulic

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Fig. 5. One sample containing 444 particles with different mesh densities.

Fig. 6. Hydraulic potential along the central axis for different meshes.

potential is not linearly proportional to the distance in flow direction due to the inclusion of aggregates, where a steeper gradient of the curve is revealed in comparison with the slope in mortar. Meanwhile, the results are quite similar with little mesh dependence exhibited for the three meshes. Considering the accuracy requirement and computing cost simultaneously, FE meshes with the mesh size of hmin;max = 1.5–12.0 mm are subjectively used in the subsequent numerical simulations. 3.4. Effects of boundary condition The effective properties of heterogeneous material like concrete may be very sensitive to different external boundary conditions [19,40]. And it is commonly suggested that the response of the RVE should be independent on the types of boundary conditions [20,41]. Therefore, the effects of boundary condition on effective permeability of concrete are investigated in this section. 40 cubic concrete samples (with f agg = 0.4 and Ls = 150 mm) are established and further analyzed numerically to obtain the permeability coefficients under both CPBC and CFBC. For CPBC (or CFBC), /1 = 1000 mm (or Q inlet ) and /2 = 0 mm are applied to the inlet and outlet of the mesoscale model, respectively. Under CFBC, Q inlet is derived from the inlet flow rate of the simulation with CPBC to make them comparable. Fig. 7 shows the distribution of hydraulic potential at the inlet surface of one typical sample under CPBC and CFBC. It is

Fig. 7. Distribution of hydraulic potential at the inlet surface under CPBC and CFBC.

demonstrated that the simulated hydraulic potential of CFBC model is non-uniformly distributed and not identical to the result of CPBC (1000 mm). The inhomogeneous mesostructure of concrete sample is apparently visualized, which induces the heterogeneity of transport property. And the intermediate parts of the inlet surface exhibit a larger hydraulic potential due to the higher density of aggregates situated in the core of the sample. Besides, the difference between maximum and minimum values of hydraulic potential is approximately 10% under CFBC, which may be appropriately used for explaining the deviation or error existed in experimental measurements of concrete permeability using a controlled constant flow method [8]. Fig. 8 shows the calculated permeability coefficients of the 40 concrete samples under both CFBC and CPBC. When calculating the permeability in the case of CFBC, the hydraulic potential at the inlet is obtained by averaging the values on the whole surface. It can be found that the computed permeability coefficient for each sample under CPBC is always larger than the value under CFBC. The result is in good accordance with the similar published results for elastic modulus [19] and plane Poiseuille flow [42]. It should also be noted that boundary conditions make limited difference in these simulations. For consistency, CPBC is used for the following study.

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Fig. 8. The calculated permeability coefficients for the 40 samples.

4. Mesostructural parameters affecting effective permeability of concrete Before the investigation of the size effect, the influences of several key factors are statistically studied with extensive Monte Carlo (MC) simulations, which will contribute to the identification of RVE with respect to the effective permeability of concrete. The MC simulations are automatically performed by running a set of Matlab codes so that the permeation process in a large number of samples can be quickly modeled for statistical analyses. The samples used in this section are generated with various mesostructural parameters (i.e., the shape, gradation and volume fraction of aggregates and the thickness and permeability of ITZ) in order to evaluate the different individual parts they act in the permeability of concrete. 4.1. Effects of ITZ The ITZ regions always potentially exist in concrete due to the presence of aggregates and the inefficient packing of cement grains at the aggregate surface [10,37]. Adding large amounts of impermeable or less permeable aggregates could reduce concrete permeability by decreasing the volume fraction of porous mortar (dilution effect) and further lengthen the flow path (tortuosity effect) [9,22,43]. On the other hand, a higher volume fraction of aggregates will lead to the increase of ITZ structures in concrete. With a lower density and higher porosity, ITZ is believed to possess a larger hydraulic conductivity and thus plays an important role in the transport property of concrete [10,14,35]. However, the effects of ITZ are interrelated tightly with aggregates, since ITZ forms, increases and varies in volume fraction in proportion to aggregate surface area. It is often difficult to purely extract the effects of ITZ phase from the composite properties of concrete for other influencing parameters inevitably vary [9,10,12,14]. Fortunately, the threephase mesoscale model can be readily used to examine the impacts of ITZ. In this section, one cubic concrete sample (Ls = 150 mm) with spherical aggregates in constant volume fraction of f agg = 0.4 is randomly generated and studied. The steady state flow process in the sample is simulated to predict its effective permeability. The ITZ phase is modeled with different transport properties, i.e., the thickness di ranging from 10 to 100 lm and four different values of the permeability ratio k as 5, 10, 20 and 50. A two-phase concrete sample with the same geometrical configuration but without ITZ is also tested in parallel. Fig. 9 shows the impacts of ITZ on the effective permeability coefficient for different thicknesses and permeability ratios.

Fig. 9. Impacts of ITZ on effective permeability coefficient.

It is shown that the effective permeability coefficient of concrete increases with the thickness and permeability of ITZ increasing. Also, it shows similar trend to the 2D numerical results in the literature [22]. Since the aggregate surface area is identical in this simulation, the thickness of ITZ directly represents its volume fraction. The ITZ’s volume fraction and permeability are the main factors affecting the permeability of concrete. In addition, the ratio of keff =km tends to be closely equivalent to 1.0 when both the thickness and permeability of ITZ are large enough. This means that the presence of ITZ almost overcomes the dilution and tortuosity effects from mixing impermeable aggregates, because of the much higher water conductivity ITZ possesses. Nevertheless, in most cases the contrast in transport properties between ITZ and the matrix is insufficiently large and ITZ may not significantly enhance the effective permeability of concrete [30,31]. Therefore, its thickness and permeability are respectively determined as relative low values as 30 lm and 10 times larger than mortar’s, which are also in consistency with some published studies [22,38]. The characteristics of ITZ are kept constant in the following study, in order to isolate its effects as much as possible when identifying the influences of other variables. Water permeation process through one typical sample is also illustrated in two scenarios (ITZ excluded and ITZ included with di = 30 lm and k = 10), with the different flux fields on the central profile presented in Fig. 10. Comparing Fig. 10(a) with (b), it is clearly visible that the flux vectors are significantly shifted toward the external surface of coarse aggregates when ITZ is included. The existence of ITZ has led to the formation of fast-conductive pathways around aggregates, allowing accelerated ingress and movement of water. Once the ITZ is percolated due to the extremely large volume fraction of aggregates, the ITZ regions would interconnect into a leaking path for the successful transport of fluids, which can dramatically increase the permeability [44–46]. Many results show that ITZ percolation phenomena may occur when the volume fraction of aggregates is relative high, and becomes deterministic for fractions larger than 0.5 [46,47]. The aggregate volume fractions used in this study will be controlled less than 0.5, since the ITZ percolation effect is not the main concern. 4.2. Effects of aggregate shape Neglecting the shape characteristics of coarse aggregates is a gross simplification since they are generally non-spherical or irregular, which might have considerable bearing on the composite

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Fig. 10. Flux field on the central profile of the sample.

behaviors of concrete. Numerous works based on numerical modeling have been devoted in the attempt to study the effects of aggregate shape on the effective mechanical properties of concrete [18,25,29]. In terms of the transport properties, it is reported that the shape of aggregate particles has a significant effect on the sorptivity [13] and diffusivity [9,48] of concrete, meanwhile the conflicting conclusions that aggregate shape acts a small influence on the diffusion coefficient of chlorides are also drawn [14,43,49]. These literatures regarding the effects of aggregate shape on the transport properties of concrete are controversial, which have not been fully understood as yet. Moreover, the aggregate shape may become an important influential factor when determining the RVE of concrete permeability. Therefore, numerical analysis for mesoscale models with different particle shapes, i.e., spherical, ellipsoidal and crushed aggregates (see Fig. 2), is performed to definitely examine the effects of aggregate shape on concrete permeability. For each type of particle shape, 40 realizations (with f agg = 0.4) are then generated and tested in order to minimize the numerical variance. Fig. 11 shows the plots of streamlines on the central profile of three samples with different shaped particles. It provides a snapshot in that how the streamline in the mortar matrix curves around the different shaped aggregates. It is further noticed that the larger flow velocity can be observed between two aggregates in closer distance. These results provide a deep understanding of the water transport behavior in concrete, which may be helpful for investigating the durability problems. Water as well as other fluid in heterogeneous concrete does not follow a straight line as it flows through the homogeneous media. Instead, it follows a tortuous path as shown in Fig. 11 and the tortuosity s in porous media can be defined by [50]:

s¼

 2 L Le

ð9Þ

where Le (L) represents the effective length of the flow path. To further explore the effects of aggregate shape, the tortuosities of all these samples are measured. From each sample, 2000 streamlines are uniformly spaced and utilized to calculate the tortuosity according to Eq. (9). Consequently, the average tortuosities of 40 samples with spherical, ellipsoidal and crushed aggregates are 0.856, 0.863 and 0.902, respectively. The flow path is shorter when the tortuosity is higher and it turns out to be a straight line when s = 1.0. It is shown that the samples with crushed aggregates can provide the shortest flow paths for water permeation, comparing with the other two shaped aggregates. It should be noted that the ITZ phase may be another factor when understanding the effects of different aggregate shapes, since

the total surface areas of aggregates with identical volume are different in these samples due to the various shapes, which is directly related with ITZ. So these concrete samples are also tested in the scenario without ITZ. Fig. 12 summaries the mean value and standard deviation of the estimated permeability for these concrete samples both with and without ITZ, as well as the volume fraction of ITZ (denoted as f itz ) included in these samples. Regardless of the presence or absence of ITZ, concrete samples with polyhedral aggregates exhibit the largest permeability while the samples using ellipsoidal aggregates show relatively lower value and the samples of spherical aggregates possess the lowest permeability. As omitting ITZ, the different permeability coefficients estimated by three types of aggregates may be mainly attributed to their corresponding tortuosities as mentioned before. When ITZ is included, the permeability coefficients of the samples with polyhedral aggregates increase dramatically, which might be primarily explained by the larger proportion of ITZ as depicted in Fig. 12. The results show more ITZ regions present in concrete when using irregular aggregates in the same volume fraction, which have enhanced the effects of aggregate shape on the transport property. Though the shape of aggregates may have influence to some extent on water transport process and permeability of concrete, it is not the main concern for the study on defining RVE. Moreover, a huge amount of calculation is demanded when introducing irregular aggregates in the numerical modeling. Therefore, the spherical particles are used in our study to simplify the calculations.

4.3. Effects of aggregate gradation The permeability coefficients of concrete material are strongly related to its density [1], which depends much on the grain composition of aggregate, i.e., the particle size distribution of aggregates. It is recognized that better aggregate packing or compaction improves the performance of concrete [28]. Recently, Warda and Munaz [51] have examined the effects of aggregate gradation on water permeability of concrete using laboratory tests and concluded that aggregate gradation is a significant factor for determining the concrete permeability. Therefore, the effects of aggregate gradation should be taken into account in the numerical modeling of concrete for studying its permeability RVE. 200 concrete samples (f agg = 0.4 and Ls = 150 mm) are generated and simulated in this section. Aggregates used in these samples are divided into two size segments of D1min;max = 5–20 mm and D2min;max = 20–40 mm, of which the volume fractions are represented by G1 and G2 , respectively. For each size segment, the particle size distribution obeys the law of Fuller curve. Five combinations of

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107

Fig. 12. Effects of aggregate shape on effective permeability coefficient.

Fig. 13. Effects of aggregate gradation on concrete permeability.

gradation on the estimated permeability coefficients of the concrete samples, and Fig. 14 presents the volume fraction of ITZ in these samples. It seems that when ignoring the ITZ, the effects of aggregate gradation are insignificantly small on the permeability, which nearly remains constant. However, in the presence of ITZ, the permeability increases basically in linearity with G1 (i.e., the content of smaller particles) (see Fig. 13). This is strongly related to the volume fraction of ITZ which is also scaling linearly with G1 (see Fig. 14). The results show that aggregate gradation has direct influences on permeability of concrete, which may be tightly controlled by the variance in the volume fraction of ITZ. Thus, in order to eliminate the effects of gradation, a consistent aggregate size distribution (e.g., the Fuller curve) should be used in the determination of RVE. 4.4. Effects of aggregate volume fraction

Fig. 11. Comparison of streamlines on the central profile of three samples with different shaped aggregates.

different aggregate gradations, i.e., G1 = 0.08, 0.16, 0.24, 0.32 and 0.40 (noting that G1 þ G2 = 0.4), are investigated, with 40 samples for each combination. Fig. 13 shows the effects of aggregate

Aggregate volume fraction is one of the most important parameters governing the transport behavior of concrete, which has always been concerned [4,13,30,31]. It is known that incorporating coarse aggregates in the mortar matrix have two opposite effects on the transport properties. The dilution and tortuosity effects reduce the permeability of concrete by blocking and redirecting conductive flow while the ITZ effects increase it. The comprehensive effects of aggregate volume fraction are quite complicated and worth to be studied when investigating the effective permeability and the size of RVE.

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by samples without ITZ are in good consistency with that of ME model, reflecting the validity of the numerical modeling. All of these estimations show that the effective permeability decreases with the increase of aggregate volume fraction. Similar observations can also be found in some laboratory tests [31,52,53] with respect to the transport properties of concrete. Therefore, the dilution effect and tortuosity effect are likely to significantly outweigh the impacts of ITZ, which is also why the effective permeability shows a small difference between the two cases with and without ITZ. Fig. 16 shows coefficients of variation (CoV) for these numerical realizations. It is found that the sample number of 40 would be sufficient to obtain convergent permeability coefficients with stable fluctuation. For various volume fractions of aggregates, different RVE sizes may be suggested. Fig. 14. Relationship between aggregate gradation and ITZ volume fraction.

5. RVE determination of effective permeability coefficient 5.1. Statistical and numerical algorithm for RVE determination In order to reasonably estimate the effective permeability coefficient of the concrete sample, numerical homogenization might be achieved through the RVE determination. In the statistical sense, the RVE must guarantee a given accuracy of the estimated effective properties obtained by spatial averaging. The definitions of the RVE vary widely as reviewed in literature [20]. A classical RVE definition [40] is introduced in this work as a sample which is representative of the overall permeability of concrete material. This definition should satisfy the two convergence criteria below [54]:

Fig. 15. Effective permeability coefficients keff =km obtained from different models.

Concrete samples containing spherical aggregates are tested with variable aggregate volume fractions, i.e., f agg = 0.2, 0.3 and 0.4, in this section. For a certain aggregate volume fraction, 40 samples are employed to obtain the average estimated value. Fig. 15 summarizes the effective permeability coefficients of these samples both in the cases with ITZ and no ITZ. In addition, the results of the numerical analysis are compared with the classical estimates of some analytical approaches including ME, EMT, and Series models (Vide [11,12]) for two-phase composite, which give theoretical ranges of the effective permeability with respect to a normal volume fraction of aggregates. It can be found that the results obtained

Fig. 16. The CoV for permeability coefficient with different aggregate volume fractions.

(1) The estimated properties of RVE are independent on the realizations with random distribution of aggregates, that is to say, the CoV of permeability coefficients from these realizations achieves a given accuracy, e1 . (2) The RVE is sufficiently large to be statistically representative with respect to the overall concrete permeability, according to the convergence criterion below,

  kðLi Þ  k   s t  6 e2    kt

ð10Þ

where kðLis Þ is the permeability coefficient of the sample sized by Lis , kt is overall permeability for concrete and e2 is relative error. Based on these criteria, MC simulations could be performed for determining RVE. The numerical and statistical algorithm is summarized in the flowchart shown in Fig. 17. In Sections 4.1–4.4, the effects of various mesostructural parameters on the effective permeability are comprehensively investigated by numerical modeling with concrete samples in a constant size of Ls = 150 mm. The roles they play may remain rather than vanish with the enlarging of the sample size. Actually, the RVE of concrete permeability is parameter-dependent, which means that different sizes of RVE might be identified by various mesostructural parameters. However, it is impossible to take all these parameters into account when determining the RVE, since the computational cost is too tremendous and marvelous. Thus, in the perspective of methodology, an example of RVE determination for effective permeability coefficient is provided herein through concrete samples with particular mesostructural parameters (i.e., aggregates: spherical shape, f agg = 0.4 and Dmin;max = 5– 40 mm obeying Fuller curve; ITZ: di = 30 lm and k = 10). In most experiments, the size of RVE is commonly determined in relation to the maximum aggregate size, which is expressed as the contrast ratio (i.e., Ls =Dmax ) for effective properties study. On the basis of this concept, 13 series of cubic samples are generated with different sizes ranging from 80 mm to 320 mm (or contrast

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109

Fig. 17. Flowchart for determining the size of RVE.

ratio from 2.0 to 8.0, noting that Dmax =40 mm) in an increment of 20 mm to quantify the size of RVE. For each series, N = 40 realizations are used as discussed aforementioned. Furthermore, the simulations of permeability test are conducted in all the three directions (X, Y and Z) since the permeability of these samples may be anisotropic. For comparison, the same samples without ITZ are also tested in parallel. Totally, 3120 MC simulations are organized for numerical homogenization in this section.

Fig. 18. Effects of sample size on effective permeability coefficient in X direction.

5.2. Results and analysis Fig. 18 shows the numerically estimated permeability coefficients in X direction of all the tested samples. And the results in three directions, sorted by magnitude as maximum, median and minimum for each sample, are summarized in Fig. 19. In Fig. 18(a), the estimated effective permeability coefficients from the samples without ITZ exhibit considerable dispersion when the sample size is small. With the increase of sample size, the dispersion decreases progressively. Due to the random distribution of aggregates in different realizations, this size-dependent dispersion may not vanish. Therefore, a tolerance error (i.e., e1 ), is expected herein for the numerical determination of RVE. It is also shown that the mean values of effective permeability coefficients have a little fluctuation with such a large sampling number. These observations are consistent with the 2D numerical results by Zhou and Li [22]. However, compared with the 2D results using nearly the same parameters, the maximum dispersion in 3D has been greatly reduced by 76.40% and the mean values of keff =km are about 24.61% greater. In comparison with the realizations omitting ITZ, slightly larger dispersion is observed in the obtained effective permeability coefficients from the ones with ITZ in Fig. 18(b). The corresponding difference for the maximum dispersion and mean values of keff =km are 72.07% and around 19.91% respectively, compared with the results of Zhou and Li [22]. Similar observations of the significant

Fig. 19. Effects of size sample on permeability in three directions.

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difference between the results from 3D and 2D models can also be found in the literature [24]. These comparisons demonstrate the importance of investigating and evaluating the permeability of concrete in three spatial dimensions, especially when calibrating the numerical permeability test with experimental data. Comparing with the single direction, the fluctuation of mean values with relatively small sample sizes for three directions is more intense as shown in Fig. 19. This may lead to a larger size of RVE for three directions than that in one direction. Thus, for rational determination of the RVE, the investigations for the three directions must be indispensable for the reason that the RVE of concrete is usually regarded as isotropic. It is also shown that as the sample size increases, the maximum, median and minimum values are gradually becoming stable. To assess the required sample size for statistical representativeness, an expected relative error (i.e., e2 ) is necessary as discussed in Section 5.1. From the aforementioned numerical simulation results, a statistical analysis is applied to define the size of RVE, with the given controlled errors of e1 and e2 as 0.2% and 0.4%, respectively. The results for CoV and relative error of keff =km for the samples with ITZ are shown in Fig. 20. As the sample size increases, both the values of CoV (Fig. 20(a)) and relative error (Fig. 20(b)) of permeability coefficient are diminishing rapidly. It can be observed that both of them are reduced below the estimation errors (horizontal dotted lines) when the sample size attains approximately 180 mm (equivalent to 4.5 times of Dmax = 40 mm). Therefore, the RVE size for these samples is suggested to be 4.5 times greater than the maximum size of aggregates, which complies with the widely reported values of Ls =Dmax as 4.0–8.0 for the effective properties study of concrete

Fig. 20. CoV and relative error of keff =km for the samples with ITZ.

[19,55–57]. The presented numerical model is proved to be capable of reliably evaluating the RVE of concrete permeability, and might be suitable for further application to permeability assessment in real concrete structures. 6. Conclusions In this paper, an exquisite FE model of concrete with a 3D mesostructure is generated, which is realized as three-phase composite of coarse aggregates, mortar matrix and ITZ modeled by zero-thickness element. The presented model is employed for estimating the effective permeability of concrete. With extensive Monte Carlo (MC) simulations, the effects of various mesostructural parameters (i.e., the shape, gradation and volume fraction of aggregates and the thickness and permeability of ITZ) on the permeability of concrete are comprehensively investigated. A numerical and statistical analysis is subsequently conducted to determine the RVE in sense of permeability. From the study, some conclusions can be drawn as follows: (1) The mesoscale concrete model shows to be useful for rationally simulating the water permeation process in concrete and efficiently evaluating its permeability. Some results obtained from this model exhibit similar trends with the published data available in the literature, suggesting that the numerical approach is reliable, which might be suitable for further application to the permeability assessment in real concrete structures. Additionally, water transport characteristics in concrete can be well captured via the mesoscopic approach, which provides a promising tool for further investigating the durability problems. (2) The parameters of ITZ (e.g., the thickness, permeability and volume fraction) play a vital role on the transport properties of concrete, which may considerably increase its permeability. Conversely, the impermeable or less permeable aggregates mixing into mortar could reduce the permeability by dilution and tortuosity effects. Both of these effects enhance with the increase of aggregate volume fraction, meanwhile the tortuosity effect is obviously pronounced when different aggregate shapes are employed. Moreover, the variation in aggregate shape, gradation and volume fraction leads to the difference in the volume fraction of ITZ and thus that in the effective permeability. Therefore, these important variables which influence the permeation behavior of concrete are strongly interacting. The mesoscopic modeling has proven to be capable of better understanding their significance and interactions. (3) Regarding computational homogenization of effective permeability, it is shown that 40 realizations are sufficient to obtain convergent results with stable fluctuation for each sample size. As the sample size increases, the dispersion of the predicted effective permeability decreases progressively. Compared with some available 2D results, the mean values of effective permeability coefficients in this study are approximately 19.91% and 24.61% greater respectively in the cases with ITZ and without ITZ, indicating the necessity of studying the effective properties in three spatial dimensions. Due to the anisotropy of numerical samples, it is also suggested that the investigations for the three directions are of importance to rationally determine the RVE size. (4) The numerical approach can provide a support for determining RVE size with a given error level. For concrete samples with particular mesostructural parameters (i.e., aggregates: spherical shape, f agg = 0.4 and Dmin;max = 5–40 mm obeying Fuller curve; ITZ: di = 30 lm and k = 10) in this paper, it is

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found that the permeability coefficients in three directions become stable when the sample size attaining 4.5 times of Dmax . Thus, the RVE size with respect to the permeability of numerical concrete sample is suggested as 4.5 times greater than the maximum size of aggregate.

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