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A new method to simulate permeability degradation of stressed concrete
Construction and Building Materials 174 (2018) 284–292
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
A new method to simulate permeability degradation of stressed concrete Manuka Wimalasiri, D.J. Robert, Chun-Qing Li ⇑, Hassan Baji School of Engineering, RMIT University, Melbourne, Australia
h i g h l i g h t s A methodology to determine the permeability degradation of stressed concrete. Easy to determine permeability with different mechanical properties and geometries. Permeability starts to degrade when stress reaches a certain threshold value. Aggregate fraction is the most influential factor for permeability degradation. Aggregates with sharper edges tend to result higher permeability values for concrete.
a r t i c l e
i n f o
Article history: Received 6 February 2018 Received in revised form 12 April 2018 Accepted 13 April 2018
Keywords: Monte Carlo simulation Concrete meso-structure Permeability degradation Stressed concrete
a b s t r a c t Concrete permeability is usually determined by testing unstressed, non-damaged specimens in laboratories as a constant. In reality, however, concrete is subjected to various operational loads in the time span of its service life. Therefore, the permeability of stressed concrete can be time-variant and completely different from that of unstressed conditions. This paper intends to develop a new methodology to determine the permeability of stressed concrete over time, based on the damage mechanics of concrete mesostructure in conjunction with Monte Carlo simulation and finite element analysis. It is found in this study that the permeability of concrete starts to increase when the applied stress in concrete reaches a certain threshold value for the given aggregate fraction. It is also found that the aggregate fraction is the most influential factor for permeability degradation of concrete. The significance of the developed methodology is that it can determine the permeability degradation of concrete with different mechanical properties and geometries of constituent materials under various applied loads over time, which would be otherwise impractical experimentally. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Permeability of concrete is considered as the most influential property which controls its strength, serviceability and durability [1,2]. It also governs the ingress of aggressive agents with water, oxygen or chlorides that can cause infiltration, corrosion of reinforcing steel and degradation of concrete strength [1]. As the strength of concrete decreases with the increase of its permeability, an increase in permeability can be considered as a degradation in its durability assessment. The concrete permeability is usually determined by testing well-cured, unstressed, non-damaged specimens in laboratories as a constant. However, in the time span of its service life, concrete structures are subjected to stresses in various forms such as mechanical, thermal, chemical, environmental or a combination ⇑ Corresponding author. E-mail address: [email protected] (C.-Q. Li). https://doi.org/10.1016/j.conbuildmat.2018.04.124 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.
of these [3]. Therefore, the permeability of stressed concrete due to various operational loads can be time-variant and completely different from that of unstressed laboratory conditions. Hence, the applicability of the permeability data produced on undamaged concrete in laboratories has limitations in real life scenarios of concrete in service. Some researchers [1,2,4,5] have attempted to investigate the time variant permeability, referred to as permeability degradation in this study, of stressed concrete experimentally. There are however some practical difficulties. One such difficulty is to conduct the permeability tests under an increasing applied load as setting up a test for increasing load is quite challenging. To overcome this difficulty, some researchers stressed the specimens first and carried out the permeability tests afterwards [1,4,6]. Very few researchers have succeeded in carrying out the tests while the load is being applied simultaneously [7]. Another difficulty is the extremely excessive time in conducting permeability tests. To overcome this, the water (or gas) has been pressurised to obtain a quick
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steady state flow and to have a sufficient outflow for volume measurements [1]. Based on published research, it can be observed that the water/gas permeability of concrete is related to the applied stress (e.g., compression, tension, flexure), rate and amount of loading (as a percentage of ultimate) in its direct resemblance to crack initiation and propagation in concrete [3]. Furthermore, it may not be practical to test the permeability of concrete with a number of different material constituents and property combinations, because of the excessive time required for achieving a steady state flow for each test. With these difficulties in testing concrete for its permeability, an increasing interest has been developed among researchers to determine the permeability of stressed concrete numerically with computer simulations. The significance of determining permeability through computer simulations can be twofold. One is that numerical methods can determine the permeability of concretes with different combinations of material properties and geometries at will which would otherwise be impractical not just due to excessive testing time. More importantly, the numerical method can determine the permeability of concretes under various applied loads over time, which can be replicated as practically as possible to the operational loads. The permeability of intact concrete is influenced by two primary factors [2]. One is the porosity and the interconnectivity of pores in cement matrix. The other factor is the pre-existing micro-cracks in concrete, mainly in the cement-aggregate interface, which is also termed as Interfacial Transition Zone (ITZ). The permeability of the same concrete under loading also depends heavily on the presence of stress induced cracks. Loading affects the crack formation and propagation by interconnecting the cracks and voids, which changes the transport properties of concrete and the durability of concrete structures [8]. For normal concrete, its permeability decreases with the application of load up to a certain threshold value of load level and then it increases rapidly afterwards. According to Hoseini, Bindiganavile and Banthia [3], the initial reduction of permeability is due to the contraction of voids upon the application of load. The slight increase afterwards is caused by the micro-cracks development within ITZs. Subsequently, with further increase in load, the micro-cracks initiated at ITZs become localized through the cement mortar and result in a rapid increase in permeability. Recently, with the advances in computational techniques, researchers have been trying to use meso-scale computer simulations in determining the permeability of concrete [9,10]. At meso-scale concrete is considered as a 3-phase heterogeneous material consisting of cement mortar, aggregate and ITZ. It is acknowledged that the research on meso/micro structural simulation of concrete as a multi-phase material has been undertaken but the knowledge on the change of concrete permeability at mesostructural level and under various loads is very scarce and there is no numerical method to determine the time variant permeability of concrete, i.e., permeability degradation, caused by stress induced micro-cracks. Moreover, there are some limitations on capturing the initiation, propagation and interconnecting of stress-induced cracks using existing models of meso/micro structural simulation, resulting in unreliable permeability predictions. Up to-date, especially for meso-scale concrete, the best approach to address these issues is to replace the fracture mechanics with the damage mechanics as suggested by many researchers [8,10,11]. This will be explored in the current study. The intention of this paper is to develop a new methodology to determine the permeability of stressed concrete over time, based on damage mechanics of meso-structure of concrete. Firstly, a representative meso-structure of concrete as a 3-phase material is simulated using Monte Carlo techniques. Secondly, based on the simulated meso-structure of concrete, finite element models are
developed for the stress analysis, in which a user subroutine is created for the identification of individual damage to each element of the mesh and for the determination of load-induced overall damage. The permeability of concrete corresponding to the overall damage of bulk concrete is determined as the final step. A worked example is presented as verification and also demonstration of the application of the developed methodology. Further, a parametric study on the effects of concrete constituent factors, including aggregate fraction, particle size distribution of aggregates and the distribution of aggregates on the permeability degradation of concrete is undertaken to identify the most influential factors. The merit of the developed methodology is that it can determine the permeability degradation of concrete with different mechanical properties and geometries of constituent materials under various applied loads over time, which would be otherwise impractical experimentally. 2. Simulation of meso-structure of concrete In order to represent concrete as realistically as possible, it has to be modelled as a heterogeneous material consisting of all 3phases of cement mortar, aggregate and ITZ. The meso-structure of concrete depends greatly on the aggregate fraction, aggregate particle size distribution and aggregate distribution [12]. Fine aggregates are usually considered within the cement mortar phase. For computational modelling of concrete at this level, aggregate particles are usually assumed as spheres for 3-D problems and circles for 2-D problems [13]. Aggregates in real concrete are neither circular nor spherical in shape. Though different shapes can be accommodated in computer simulations, this is more of an assumption for the convenience of aggregate distribution/generation and also for reducing the computational effect without compromising the technical development. However, the current study accommodates two other aggregate shapes in the simulations to see the effect of the aggregate shape on the permeability of concrete. The concept of these simulations is the same as circular shape. It is also assumed that the particle size and the distribution of aggregates are random. The size distribution of aggregate particles in a given concrete mix can be obtained experimentally by conducting a sieve analysis. However, based on the theory of stereology, particle size distribution of aggregates in concrete can be considered following Fuller mix [12]. For a Fuller mix, the cumulative distribution function (CDF) for the aggregates in concrete of a two-dimensional problem can be expressed as follows.
d f ða0 ; b0 Þ d0 f ða; bÞ 1:5
P2d ðdÞ ¼
1:5
ð1Þ
d f ða0 ; b0 Þ 1:5
where, d = diameter of aggregate (d0 d dm), d0 is the smallest aggregate diameter and dm is the largest aggregate diameter in the concrete mix. Coefficients a, a0, b, b0 and f(a, b), all related to d, d0 and dm, are expressed as follows [13,14].
f ða; bÞ ¼
pffiffiffi 2 2 2 1 1 cos1:5 asina þ 2E b; pffiffiffi F b; pffiffiffi 5 5 2 2
a ¼ arccos
d ; dm
a0 ¼ arccos
pffiffiffi a b ¼ arcsin 2sin ; 2
d0 dm
ð2Þ
pffiffiffi a 0 b0 ¼ arcsin 2sin 2
ð3Þ ð4Þ
E() and F() in Eq. (2) are the Legendre’s standard elliptical integrals [12]. The Eq. (1) for 2-D problems is developed based on the size distribution of intersecting circles of a randomly located p-plane. The conversion of aggregate volume fraction (AVF) to
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aggregate area fraction (AAF) also has been considered when deriving the Eq. (1) [13]. However, it should be acknowledged that the concrete in practice is of three dimensions and the 2-D representation of concrete in this research or even in mesoscale modelling of concrete in general is for the purpose of minimizing the computational time and capacities. With Eq. (1), it is possible to generate random values for the diameter of circular aggregates. For a required AAF, above equation and a similar process used by Zheng, Li and Zhao [14] can be employed to determine various aggregate diameters and the total number of aggregates (NA). Once the total number of aggregates and the diameters of those are known, coordinates for their random distribution within the representative meso-structure of concrete (RMSC) boundaries can be determined. Since the current study includes the ITZ, a new algorithm is developed to determine the distribution of aggregate particles considering ITZ thicknesses as well. The key steps of the developed algorithm for the simulation of a RMSC can be summarized as follows: 1. Set the width and height for the RMSC. 2. Arrange the generated NA aggregate diameters in descending order [14]. 3. Take the ith aggregate diameter (i = 1 to NA). 4. Determine the center coordinates (xi, yi) of ith aggregate by sampling from random variables Xi and Yi. The intervals of these random variables respectively depend on the required width and height of RMSC. In addition, the selected aggregate diameter and the minimum cement mortar cover to an aggregate also influence these intervals. 5. Use ITZ thickness (tITZ) and diameter of ith aggregate to determine the ITZ diameter. 6. Check whether the circle for the ITZ diameter of ith aggregate overlaps with all previously distributed ITZ circles or not. Currently, it is assumed that the ITZs for aggregates won’t merge with each other. If there is any overlap, return to step 4. 7. Otherwise, return to step 3 until all the NA aggregate particles are distributed within the RMSC boundaries.
START
Pre-determine the AAF, ITZ thickness, dimensions of RMSC, Max and Min diameters of Aggregate Determine the Aggregate diameter using Fuller Mix
Yes
Calculate the cumulative Area (AT) of aggregates
Check AT < AAF No Sort the generated diameters (NT) in descending order
Generate random numbers for x, y coordinates until Ni = NT Yes Check whether the outer surface of ITZ intersect with all previously placed ITZs No Record the diameter and the coordinates
END Fig. 1. Key steps in random aggregate diameter generation and distribution.
Aggregate
Cement Mortar
The key steps for the random generation of aggregate diameters using Fuller mix and distribution of aggregate particles are provided in Fig. 1.
3. Finite element models for concrete damage Because of the randomness of the aggregate diameters and the distribution, development of the finite element (FE) models for RMSC has to be carried out by adopting Abaqus Python scripting [15]. This python script incorporates all the necessary steps, which automates the creation of Abaqus FE model as it is essential to conduct number of analyses for different conditions. The 2-D 3-phase meso-structure of concrete is generated in Abaqus as shown in Fig. 2. Aggregate phase is placed in accordance with the data generated, as described in previous section, for diameters and center coordinates. ITZ phase is placed around the aggregates with the same center coordinates and the ITZ thickness. The remaining area is considered as the cement mortar (Fig. 2). Because of the randomness in geometric distribution of aggregates/ITZs, number of aggregates/ITZs and the geometry of cement mortar as an individual phase, the meshing of individual phases needs to be done through the developed python script. The element type, element creation technique and all the required parameter to create the mesh are given within the python script. The Mesh discretization of the RMSC is also shown in Fig. 2 with some zoomed mesh discretization views of cement mortar around aggregate, of aggregates and ITZs. When concrete is considered as a 3-phase heterogeneous
ITZ
Fig. 2. Geometrical arrangement and the Mesh Discretization of RMSC.
material in 2-D at mesoscale finite element analyses, with the cross-section varying along the thickness direction, generally the elements are considered as plain-stress [16,17]. Therefore, all three phases of concrete in the model are represented by 2-D planestress elements. The load is applied at the top boundary as a displacement load and the bottom boundary is restrained to represent the selected experimental testing setup as closely as possible [1]. At this stage, all the geometrical and mechanical properties for each individual phase should be identified and available for the finite element analysis of RMSC. As the aggregate phase is high in stiffness compared to other phases, it can be assumed that only the cement
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mortar and ITZ phases experience the load-induced damage. Therefore, the damage criterion is applied only to cement mortar and ITZ. Behaviour of concrete is simplified as an elastic material in the present study. To elaborate this further, the behaviour of Aggregate phase was assumed as pure elastic. Whereas behaviour of cement mortar and ITZ were assumed as brittle-elastic as those phases are checked against the damage criterion. This consideration is really important as the stress behaviour of an individual element upon reaching the damage criterion would affect the stress and damage progression of surrounding elements. Individual elements for cement mortar and ITZ are considered as damaged when tensile stress of an element exceeds the tensile strength or compressive strain exceeds the ultimate compressive strain. This definition of damage is incorporated using ABAQUS subroutine (USDFLD) written in a Fortran code to check/identify the load-induced damage for each element. In this way, an element can only fail due to either tension or compression. This damage identification process can be summarized as follows,
characteristics has attracted substantial popularity in the research community recently. Therefore, this paper only focuses on the stress induced damage on permeability of concrete. Several researchers have proposed various models relating different damage characteristics of concrete to the permeability. The research work conducted by Picandet, Khelidj and Bastian [20] has been accepted by many other researchers in this field because of its simplicity in application. Several other researchers have also contributed to this field in substantial terms [8–11]. Picandet, Khelidj and Bastian [20] conducted gas permeability tests on three different types of concretes (normal concrete, high performance concrete and high-performance concrete with steel fibre) with Nitrogen as the percolating gas. Load has been applied as a uniaxial compression load. The permeability has been measured after unloading in the direction of applied stress. An exponential relationship between damage and permeability was proposed by them based on regression as follows.
1. For the ith increment of load (i = 1 to N, N is the total number of increments required by Abaqus to apply the full compression load), take an element of ITZ/cement mortar and check that against the defined damage criterion. 2. If the element is identified as damaged, rank the damage variable of that element as 1. 3. Record the area of that element to an output file with the increment and element number. 4. For this load increment, repeat the check for all elements of cement mortar and ITZ. 5. Continue this check for all the increments until the application of the whole load.
kD and k0 are current and initial permeability coefficients respectively in the units of m2. D is the damage variable. a and b are constant regression coefficients from the regression analysis of their test data with suggested values of 11.3 and 1.64 for a and b respectively for damage range from 0 to 0.18. It should be noted that the way of defining the damage variable influences the its range. As the original Picandet, Khelidj and Bastian [20] relation is only applicable to damage range 0–0.18, Choinska, Dufour and Pijaudier-Cabot [8] applied the Taylor’s expansion to Eq. (6) to cover the full range of damage from 0 to 1. That negates the possibility of permeability reaching infinity when the damage is closer to 1. The extended form of Eq. (6) is as follows.
Once the output file for the area of damaged elements at each increment of load is generated by employing above process, the overall damage for a RMSC is determined by adopting the definition of damage variable based on effective undamaged area [16]. In this consideration, overall damage variable (D) is defined by the ratio of total damaged element area (AD) to the crosssectional area of all elements (A) as follows.
kD ¼ k0 1 þ ðaDÞb þ
D¼
AD A
ð5Þ
h h ii kD ¼ k0 exp ðaDÞb
"
ð6Þ
ðaDÞ2b ðaDÞ3b þ 2 6
# ð7Þ
Both Eqs. (6) and (7) have been used by many other researchers to relate the stress-induced damage with permeability [8– 11,18,19]. As the original Picandet, Khelidj and Bastian [20] relation (Eq. (6)) was based on gas permeability tests, a concern may arise over the use of this relation for water permeability problems like in Kermani [1]. To compare the concrete permeability obtained from different liquids it is necessary to identify the intrinsic permeability (k), which depends only on the pore structure of concrete and independent of flowing fluid [27]. For water, the intrinsic permeability coefficient (kw, measured in m2) is 1.02 107 times the Darcy coefficient (Kw, measured in m/s) [27]. It can be written as kw = 1.02 107 Kw (m2). When Nitrogen gas is used as the percolating fluid, it always gives higher intrinsic permeability coefficient values compared to water. This is due to viscous/slip flow behaviour of gases [28]. Bamforth [27] found that the difference is 5 to 6 times for high permeable concrete (in the range 1015–1012 m2) and around 60 times for low permeable concretes. However, the following equation can be used to convert the coefficient of permeability from gas to water or vice versa [27,28].
Once the overall damage values are found by Eq. (5) for each increment of load, those values are normalized between 0 and 1 to have the full range of damage variable. Damage variable for concrete has been defined in various ways by different researchers. Some researchers have defined the damage variable as the relative change in modulus of elasticity of concrete [18–20]. Some others have defined it as the ratio between stress to ultimate strength in post-peak region [21,22]. Strain based techniques also have been employed by many other researchers to define damage variable [23–25]. It has also been defined by considering energy-based techniques as well [26]. However, it should be noted that, whichever the way damage variable is defined, for the finite element analyses of 3-phase heterogeneous meso-structural concrete models, this has to be considered at element level. Therefore, the overall damage for a specimen needs to be calculated either by referring to average damage variable considering all the elements or by employing a weighted average method based on area/volume as has been used in this research work (Eq. (5)).
where, kl and kg are the liquid and gas permeability coefficients respectively. b is the Klinkenberg [29] coefficient and Pm is the mean atmospheric pressure.
4. Determination of permeability
5. Verification
Even though the load induced cracks contribute to the permeability of concrete, relating the concrete damage to water flow
The verification of the developed method to determine the concrete permeability subjected to unconfined compression loading is
b kl ¼ kg = 1 þ Pm
ð8Þ
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Table 1 Material Properties for the FE Model.
Modulus of Elasticity – E, GPa Tensile Strength, MPa Compressive Strength, MPa Ultimate compressive strain Poisson’s ratio – m Density – q, kg/m3
Aggregate
Cement Mortar
ITZ
80 [9] – – – 0.17 [35] 3000 [9]
18.57 [5] 1.85 (10% of Compressive Strength) 18.5 Calculated based on [34] 0.001 Calculated (r = Ee) 0.3 [10] 2200 [10]
Ei = 0.6 Em = 11.142 [35] 0.66 (10% of Compressive Strength) 6.6 Calculated (r = Ee) 0.0006 Assumed as 0.6 times Mortar 0.35 [35] 1320 Assumed as 0.6 times Mortar
Fig. 3. Damage distribution with stress percentage (a) 25%, D = 0.04, (b) 50%, D = 0.23, (c) 75%, D = 0.76 and (d) 100%, D = 0.97.
to be illustrated in this section by a worked example and compared with the experimental data published by Kermani [1]. The meso-structural model in this example was formulated as closely as possible to replicate the experimental concrete specimen used by Kermani [1]. Three different types of concrete were considered by Kermani [1] and only the data and results related to normal strength concrete were used in this study for the comparison and verification. A characteristic compressive strength of 30 MPa was specified when preparing the concrete mixes and a value of 34 MPa was obtained at 28 days when tested by Kermani [1]. As the exact mix proportions of this test were not provided, reasonable assumptions had to be made for carrying out an effective comparison for the verification. A tested compressive strength of 36 MPa for normal concrete was reported by Aldea, Shah and Karr [5] with an AVF of around 0.48. Even though a same compressive strength can be obtained by different mix proportions, the deviation of the proportions cannot be significant for the same grade of concrete. In addition, AVF is always smaller compared to AAF and the values can be considered similar when the particle size distribution of aggregates is very large [13]. Therefore, it is reasonable to assume an AAF value between 0.4 and 0.5 for this example to compare the results with Kermani [1]. With these considerations, AAFs of 0.4 and 0.5, minimum aggregate diameter of 5 mm and a maximum of 20 mm, same as reported in the experiment by Kermani [1], were considered for the comparison. The size of the RMSC was taken as 100 mm x 100 mm [16]. The thickness of the ITZ is affected by the packing of cement grains and extend at least to the size of the largest cement particle, which may be up to 100 lm [30]. Therefore, the thickness of the ITZ can be in the range of 9–100 lm [30,31]. As ITZ is considered as the weakest link of concrete matrix where damage initiates [32,33], conservatively, the thickness of ITZ for the finite element models was assumed as 100 lm. Compressive strength of Cement mortar, corresponding to 30 MPa compressive strength of concrete, was calculated as 18.5 MPa following the Fig. 1.27 of the book by Neville [34]. Since cement mortar was assumed to behave elastically, the ultimate compressive strain of cement mortar was calculated by the ratio
of compressive strength to elastic modulus (e = r/E). Average Elastic modulus of ITZ to Cement mortar ratio is 0.6 [35]. Following that, because of the unavailability of individual material properties for ITZ for 30 MPa compressive strength concrete, the ultimate compressive strain and the density of ITZ also were assumed as 0.6 times the cement mortar values (as in Table 1). Further, the ultimate tensile strengths were considered as 10% of the ultimate compressive strengths [34]. Material properties for all the three phases of concrete used in this study are summarized in Table 1. The other required parameters were obtained from reported studies in literature in the same context. All three phases of concrete in the model were represented by 4-node bilinear, reduced integration plane-stress quadrilateral elements (CPS4R) available in Abaqus. Surface interactions between individual phases were considered as non-over closure/ clearance during the application of loading. To represent the uni-axial compression test conditions, a displacement of 0.2 mm was applied at the top boundary, both x and y directions were restrained at bottom and the sides were not restrained in any direction (free ends). The damage identification criteria were only applied to the cement mortar and ITZ phases. Behaviour of all 3 phases was assumed as elastic in the current study as an initial proof of the concept. After stress analysis in Abaqus, each element was checked against the defined damage criterion by the adopted subroutine at the start of each load increment to identify the individual damage. Fig. 3 shows the progressive damage distribution (for AAF of 0.4) with the stress percentage (applied stress to the characteristic strength of concrete, 30 MPa in this example). Characteristic strength of concrete percentages of 25, 50, 75 and 100 results in an overall damage variable values of 0.04, 0.23, 0.76 & 0.97 respectively. The overall damage for each increment was determined by Eq. (5). Subsequently, the coefficient of permeability corresponding to each damage value at each increment of load was calculated using Eq. (7) in the form of Eq. (9) as follows.
" K ¼ K0
ðaDÞ2b ðaDÞ3b 1 þ ðaDÞ þ þ 2 6 b
# ð9Þ
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1.8
1.6
AAF 0.40
1.4
AAF 0.50
1.2
Kermani 1991
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 4. Degradation of permeability for AAFs of 0.4 and 0.5.
where, K and K0 are current and initial permeability coefficients respectively in the units of m/s. In doing that, the initial water permeability was taken as 2 1014 m/s, same as the initial permeability value obtained by Kermani [1], which equals 2 1021 m2 for water or 1 1019 m2 for Nitrogen gas if the gas to water conversion factor is assumed as 50. Fig. 4 indicates the degradation of permeability with the stress level for AAFs of 0.4 and 0.5 in this example. Stress level is calculated by the ratio between applied stresses and ultimate stress. In addition to the analysis results, the experimental results by Kermani [1] also is provided for a comparison. It can be observed from Fig. 4 that, similar to the experimental results, value of permeability coefficient increases (is degraded) with the increase in applied loading. As can be seen from Fig. 4, results by Kermani [1] are between AAF of 0.4 and 0.5 and very close to 0.4 which effectively verifies the developed method. Therefore, the developed method can be used to determine the permeability degradation of stressed concrete.
Coefficient of Permeability, K, (m/s) x 10-10
Coefficient of Permeability, K, (m/s) x 10-10
1.8
AAF 0.30
1.6
AAF 0.40
1.4
AAF 0.50
1.2
Kermani 1991
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 6. Degradation of permeability for different AAFs.
all the models in this particular study and those are expected to be investigated in a future study. It is acknowledged that when these parameters are changed individually, the material properties of the final concrete would be affected. For example, when the AAF is changed, the strength of concrete might change and that will affect the grade of concrete. However, when concrete is considered as a 3-phase heterogeneous material, the material properties of individual phases will not change significantly and can be considered constants for all the models. It should be noted that the parametric study in this section was conducted to show the capability of the developed method to identify the effect of those parameters on permeability degradation. Results from the developed method could not be compared with experimental results directly without knowing the exact experimental conditions, mix proportions and material properties of the tested concrete samples. However, the experimental results by Kermani [1] is presented together with the following analyses results on different parameters, just to show the trend of permeability degradation.
6. Parametric study The meso-structure of concrete depends greatly on the AAF, the aggregate particle size distribution and aggregate distribution [12]. Hence, by employing the developed method, which has been verified before, the effect of above parameters on the permeability degradation of stressed concrete was studied. The dimensions of the RMSC were kept constant at 100 mm for each side for all the analyses. In addition, as highlighted in the worked example, the behaviour of aggregate, ITZ and cement mortar were considered as elastic. The material properties were assumed as constants for
6.1. Effect of aggregate fractions The amount of aggregates present in a concrete can affect the properties of a concrete largely. Three different AAFs of 30%, 40% and 50% were simulated for the analysis (Fig. 5). The particle size distribution of aggregates was considered from 5 to 20 mm for all three cases. By employing the developed method, the degradation of permeability with the stress level for these different AAFs is presented in Fig. 6.
Fig. 5. Meso-models for AAFs (a) 30%, (b) 40%, (c) 50%.
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1.8 5-10mm 5-15mm 5-20mm Kermani 1991
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength
Coefficient of Permeability, K, (m/s) x 10-10
Fig. 7. Meso-models for aggregate size distributions (a) 5–10 mm, (b) 5–15 mm & (c) 5–20 mm.
1.8 1.6
Distribution 1 Distribution 2 Distribution 3 Kermani 1991
1.4 1.2 1 0.8 0.6
7% 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 8. Degradation of permeability for different aggregate size distributions. Fig. 10. Degradation of permeability for different aggregate distributions.
As can be seen from Fig. 6, coefficient of permeability increases with the applied loading for all the AAFs of 0.3, 0.4 and 0.5. However, with the increase in stress level, it can be observed that the higher the AAF is the higher the chance of concrete matrix is damaged and the higher the increase in permeability coefficient. Since, compressive strength of concrete is inversely proportional to AAF [16], higher AAFs have lower compressive strengths and are more susceptible to damage which will lead to higher permeability coefficients. Clearly the results from this research confirm this phenomenon. Further, at a stress level of 0.7, AAF of 0.3 has a permeability coefficient of around 1.5 1011 m/s. When the AAF is increased up to 0.4, the permeability coefficient is increased by 4 times up to a value of 6 1011 m/s. AAF of 0.5 has a permeability coefficient of 9.5 1011 m/s, an increase of 6.3 times to AAF of
0.3. Considering all, the results indicate that the effect of AAF on the degradation of permeability is significant. 6.2. Effect of aggregate size distribution The size distribution of aggregates also can influence the transport properties of concrete [12]. Hence, as shown in Fig. 7, aggregate size distributions of 5–10 mm, 5–15 mm and 5–20 mm were generated (using fuller distribution) to be analysed by employing the developed method. Because of the experimental results had a reasonably good agreement with the permeability degradation for AAF of 0.4, all three size distributions were considered to have
Fig. 9. Meso-models for aggregate distributions (a) Distribution 1, (b) Distribution 2, (c) Distribution 3.
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291
Coefficient of Permeability, K, (m/s) x 10-10
Fig. 11. Meso-models for aggregate shapes (a) Circular, (b) Pentagonal & (c) Hexagonal.
7% considering the overall increase in permeability (5500 times the initial permeability). Even though the differences in the curves are not that significant, the distribution of aggregates also influences the permeability degradation of stressed concrete.
1.8 Circular Hexagonal Pentagonal Kermani 1991
1.6 1.4 1.2
6.4. Effect of aggregate shape
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 12. Degradation of permeability for different aggregate shapes.
the same AAF of 0.4. Analysis results for the degradation of permeability with the stress level for the above aggregate size distributions can be seen in Fig. 8. It can be observed from the results in Fig. 8 that the effect of aggregate size distribution is not that significant on the degradation of permeability. Even though all the three distributions show seemingly similar results overall, at 0.7 stress level, 5–10 mm distribution has an increment in permeability of 5283 times the initial permeability. At same stress level, increase in permeability for 5–15 mm and 5–20 mm distributions are respectively 5572 times (5.5% more permeable compared to 5–10 mm distribution) and 5690 times (7.7% more permeable compared to 5–10 mm distribution) the initial permeability. Until a stress level of around 0.73, the aggregate size distribution with smaller variation shows lesser permeability coefficient increment. 6.3. Effect of aggregate distribution Three different purposely selected aggregate distributions (Distribution 1–3) as shown in Fig. 9 were also studied on the permeability degradation of stressed concrete. Same AAF of 0.4 and size distribution from 5 to 20 mm were considered for all the cases. The obtained analysis results for the permeability degradation are presented in Fig. 10. As can be observed from results in Fig. 10, different distributions of aggregates provide slightly different variation of permeability with the increase in load. However, the differences are negligibly small compared to the overall increase in permeability due to loading. For example, at 0.7 stress level, the maximum difference of permeability coefficients of two distributions is roughly
As mentioned before, even though this particular research work considered circular shaped aggregates for all above analyses, the effect of the aggregate shape on the permeability of concrete also was investigated by simulating different shapes e.g. regular hexagonal and irregular pentagonal aggregates as shown in Fig. 11. The AAF and the particle size distribution of aggregates were kept constant at 0.4 and 5–20 mm respectively for all three models. It can be observed by the results shown in Fig. 12 that the influence of aggregate shape has some influence on the permeability with the applied load. Though the curves look similar in Fig. 12, results show that the pentagonal shaped aggregate has the highest permeability degradation, then the hexagonal shape and the lowest is the circular shape. For example, at the 0.7 stress level, the permeability with pentagonal and hexagonal shapes increases 14.78% and 12.34% respectively compared with the circular shape. With these results, it can be said that the aggregate shapes with lesser sides (consist of more sharper corners) experience higher permeability degradation as oppose to more rounded shapes. Higher applied stresses can result high stress concentrations near the sharper edges which causes more elements to get damaged in these locations and leads to a higher permeability degradation. This makes sense since the degradation of permeability is primarily caused by stress induced cracks. 7. Conclusions A methodology to determine the permeability degradation of stressed concrete has been developed in this paper and verified with experimental results. The developed methodology is based on the damage mechanics of concrete meso-structure. The significance of the developed methodology is that it can determine the permeability of concrete with different mechanical properties and geometries of constituent materials under various applied loads over time, which would be otherwise impractical experimentally. A parametric study has also been undertaken in the paper to identify the most important factors that affect permeability degradation of concrete. It has been found in this study that the permeability of concrete starts to degrade when the stress in concrete reaches a certain threshold value for a given aggregate fraction and that the aggregate fraction is the most influential factor for permeability degradation of concrete. It has also been
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found that the particle size and geometrical distribution of aggregates only affect the permeability degradation of concrete marginally. Further, it has been found that the aggregate shape influences the permeability degradation as well and the aggregates with more sharper edges tend to result higher permeabilities as oppose to rounded shapes. It can be concluded that the developed methodology can determine the permeability degradation of stressed concrete with reasonable accuracy and hence contribute to the body of knowledge of concrete technology. Conflict of interest None. Acknowledgements Financial support from Metro Trains Melbourne, Australia and Australian Research Council, Australia under DP140101547, LP150100413 and DP17010224 is gratefully acknowledged. References [1] A. Kermani, Permeability of stressed concrete, Build. Res. Inf. 19 (6) (1991) 360–366. [2] N. Banthia, A. Biparva, S. Mindess, Permeability of concrete under stress, Cem. Concr. Res. 35 (9) (2005) 1651–1655. [3] M. Hoseini, V. Bindiganavile, N. Banthia, The effect of mechanical stress on permeability of concrete: a review, Cem. Concr. Compos. 31 (4) (2009) 213– 220. [4] N. Hearn, Effect of shrinkage and load-induced cracking on water permeability of concrete, Mater. J. 96 (2) (1999) 234–241. [5] C.M. Aldea, S.P. Shah, A. Karr, Permeability of cracked concrete, Mat. Struct. 32 (5) (1999) 370–376. [6] C.M. Aldea, S.P. Shah, A. Karr, Effect of cracking on water and chloride permeability of concrete, J. Mater. Civil Eng. 11 (3) (1999) 181–187. [7] N. Banthia, A. Bhargava, Permeability of stressed concrete and role of fiber reinforcement, ACI Mater. J. 104 (1) (2007) 70–76. [8] M. Choinska, F. Dufour, G. Pijaudier-Cabot, Matching permeability law from diffuse damage to discontinuous crack opening, Fract. Mech. Concr. Concr. Struct. 1 (2007) 541–547. [9] D. Niknezhad, B. Raghavan, F. Bernard, S. Kamali-Bernard, Towards a realistic morphological model for the meso-scale mechanical and transport behavior of cementitious composites, Compos. Part B: Eng. 81 (2015) 72–83. [10] B. Raghavan, D. Niknezhad, F. Bernard, S. Kamali-Bernard, Combined mesoscale modeling and experimental investigation of the effect of mechanical damage on the transport properties of cementitious composites, J. Phys. Chem. Solids 96–97 (2016) 22–37. [11] G. Pijaudier-Cabot, F. Dufour, M. Choinska, Permeability due to the Increase of damage in concrete: from diffuse to localized damage distributions, J. Eng. Mech. 135 (9) (2009) 1022–1028.
[12] C.Q. Li, J.J. Zheng, X. Zhou, M. McCarthy, A numerical method for the prediction of elastic modulus of concrete, Mag. Concr. Res. 55 (6) (2003) 497–505. [13] J.J. Zheng, Mesostructure of concrete-stereological analysis and some mechanical implications, Delft University, The Netherlands, 2000. [14] J.J. Zheng, C.Q. Li, L. Zhao, Simulation of two-dimensional aggregate distribution with wall effect, J. Mater. Civil Eng. 15 (5) (2003) 506–510. [15] ABAQUS, ABAQUS V.6.14 User’s Manual, Providence, Rhode Island, 2014. [16] S.-M. Kim, R.K. Abu Al-Rub, Meso-scale computational modeling of the plasticdamage response of cementitious composites, Cem. Concr. Res. 41 (3) (2011) 339–358. [17] M. Rupasinghe, P. Mendis, T. Ngo, T.N. Nguyen, M. Sofi, Compressive strength prediction of nano-silica incorporated cement systems based on a multiscale approach, Mater. Des. 115 (2017) 379–392. [18] A. Djerbi Tegguer, S. Bonnet, A. Khelidj, V. Baroghel-Bouny, Effect of uniaxial compressive loading on gas permeability and chloride diffusion coefficient of concrete and their relationship, Cem. Concr. Res. 52 (2013) 131–139. [19] G. Chatzigeorgiou, V. Picandet, A. Khelidj, G. Pijaudier-Cabot, Coupling between progressive damage and permeability of concrete: analysis with a discrete model, Int. J. Numer. Anal. Meth. Geomech. 29 (10) (2005) 1005–1018. [20] V. Picandet, A. Khelidj, G. Bastian, Effect of axial compressive damage on gas permeability of ordinary and high-performance concrete, Cem. Concr. Res. 31 (11) (2001) 1525–1532. [21] T. Yu, J.G. Teng, Y.L. Wong, S.L. Dong, Finite element modeling of confined concrete-II: plastic-damage model, Eng. Struct. 32 (3) (2010) 680–691. [22] T. Jankowiak, T. Lodygowski, Identification of parameters of concrete damage plasticity constitutive model, Found. Civil Environ. Eng. 6 (1) (2005) 53–69. [23] B. Alfarah, F. López-Almansa, S. Oller, New methodology for calculating damage variables evolution in Plastic Damage Model for RC structures, Eng. Struct. 132 (2017) 70–86. [24] J. Lubliner, J. Oliver, S. Oller, E. Oñate, A plastic-damage model for concrete, Int. J. Solids Struct. 25 (3) (1989) 299–326. [25] J. Lee, G.L. Fenves, Plastic-damage model for cyclic loading of concrete structures, J. Eng. Mech. 124 (8) (1998) 892–900. [26] W.B. Krätzig, R. Pölling, An elasto-plastic damage model for reinforced concrete with minimum number of material parameters, Comput. Struct. 82 (15) (2004) 1201–1215. [27] P. Bamforth, The relationship between permeability coefficients for concrete obtained using liquid and gas, Mag. Concr. Res. 39 (138) (1987) 3–11. [28] V. Picandet, A. Khelidj, H. Bellegou, Crack effects on gas and water permeability of concretes, Cem. Concr. Res. 39 (6) (2009) 537–547. [29] L. Klinkenberg, The permeability of porous media to liquids and gases, Drilling and production practice, American Petroleum Institute, 1941. [30] K.L. Scrivener, K.M. Nemati, The percolation of pore space in the cement paste/ aggregate interfacial zone of concrete, Cem. Concr. Res. 26 (1) (1996) 35–40. [31] J.J. Zheng, C.Q. Li, X.Z. Zhou, Thickness of interfacial transition zone and cement content profiles around aggregates, Mag. Concr. Res. (2005). [32] J. Hu, P. Stroeven, Properties of the interfacial transition zone in model concrete, Interface Sci. 12 (4) (2004) 389–397. [33] K.L. Scrivener, A.K. Crumbie, P. Laugesen, The interfacial transition zone (ITZ) between cement paste and aggregate in concrete, Interface Sci. 12 (4) (2004) 411–421. [34] A.M. Neville, Properties of concrete, 5th ed., Pearson, Harlow, England, 2011. [35] J.J. Zheng, C.Q. Li, X.Z. Zhou, An analytical method for prediction of the elastic modulus of concrete, (2006).
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
A new method to simulate permeability degradation of stressed concrete Manuka Wimalasiri, D.J. Robert, Chun-Qing Li ⇑, Hassan Baji School of Engineering, RMIT University, Melbourne, Australia
h i g h l i g h t s A methodology to determine the permeability degradation of stressed concrete. Easy to determine permeability with different mechanical properties and geometries. Permeability starts to degrade when stress reaches a certain threshold value. Aggregate fraction is the most influential factor for permeability degradation. Aggregates with sharper edges tend to result higher permeability values for concrete.
a r t i c l e
i n f o
Article history: Received 6 February 2018 Received in revised form 12 April 2018 Accepted 13 April 2018
Keywords: Monte Carlo simulation Concrete meso-structure Permeability degradation Stressed concrete
a b s t r a c t Concrete permeability is usually determined by testing unstressed, non-damaged specimens in laboratories as a constant. In reality, however, concrete is subjected to various operational loads in the time span of its service life. Therefore, the permeability of stressed concrete can be time-variant and completely different from that of unstressed conditions. This paper intends to develop a new methodology to determine the permeability of stressed concrete over time, based on the damage mechanics of concrete mesostructure in conjunction with Monte Carlo simulation and finite element analysis. It is found in this study that the permeability of concrete starts to increase when the applied stress in concrete reaches a certain threshold value for the given aggregate fraction. It is also found that the aggregate fraction is the most influential factor for permeability degradation of concrete. The significance of the developed methodology is that it can determine the permeability degradation of concrete with different mechanical properties and geometries of constituent materials under various applied loads over time, which would be otherwise impractical experimentally. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Permeability of concrete is considered as the most influential property which controls its strength, serviceability and durability [1,2]. It also governs the ingress of aggressive agents with water, oxygen or chlorides that can cause infiltration, corrosion of reinforcing steel and degradation of concrete strength [1]. As the strength of concrete decreases with the increase of its permeability, an increase in permeability can be considered as a degradation in its durability assessment. The concrete permeability is usually determined by testing well-cured, unstressed, non-damaged specimens in laboratories as a constant. However, in the time span of its service life, concrete structures are subjected to stresses in various forms such as mechanical, thermal, chemical, environmental or a combination ⇑ Corresponding author. E-mail address: [email protected] (C.-Q. Li). https://doi.org/10.1016/j.conbuildmat.2018.04.124 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.
of these [3]. Therefore, the permeability of stressed concrete due to various operational loads can be time-variant and completely different from that of unstressed laboratory conditions. Hence, the applicability of the permeability data produced on undamaged concrete in laboratories has limitations in real life scenarios of concrete in service. Some researchers [1,2,4,5] have attempted to investigate the time variant permeability, referred to as permeability degradation in this study, of stressed concrete experimentally. There are however some practical difficulties. One such difficulty is to conduct the permeability tests under an increasing applied load as setting up a test for increasing load is quite challenging. To overcome this difficulty, some researchers stressed the specimens first and carried out the permeability tests afterwards [1,4,6]. Very few researchers have succeeded in carrying out the tests while the load is being applied simultaneously [7]. Another difficulty is the extremely excessive time in conducting permeability tests. To overcome this, the water (or gas) has been pressurised to obtain a quick
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steady state flow and to have a sufficient outflow for volume measurements [1]. Based on published research, it can be observed that the water/gas permeability of concrete is related to the applied stress (e.g., compression, tension, flexure), rate and amount of loading (as a percentage of ultimate) in its direct resemblance to crack initiation and propagation in concrete [3]. Furthermore, it may not be practical to test the permeability of concrete with a number of different material constituents and property combinations, because of the excessive time required for achieving a steady state flow for each test. With these difficulties in testing concrete for its permeability, an increasing interest has been developed among researchers to determine the permeability of stressed concrete numerically with computer simulations. The significance of determining permeability through computer simulations can be twofold. One is that numerical methods can determine the permeability of concretes with different combinations of material properties and geometries at will which would otherwise be impractical not just due to excessive testing time. More importantly, the numerical method can determine the permeability of concretes under various applied loads over time, which can be replicated as practically as possible to the operational loads. The permeability of intact concrete is influenced by two primary factors [2]. One is the porosity and the interconnectivity of pores in cement matrix. The other factor is the pre-existing micro-cracks in concrete, mainly in the cement-aggregate interface, which is also termed as Interfacial Transition Zone (ITZ). The permeability of the same concrete under loading also depends heavily on the presence of stress induced cracks. Loading affects the crack formation and propagation by interconnecting the cracks and voids, which changes the transport properties of concrete and the durability of concrete structures [8]. For normal concrete, its permeability decreases with the application of load up to a certain threshold value of load level and then it increases rapidly afterwards. According to Hoseini, Bindiganavile and Banthia [3], the initial reduction of permeability is due to the contraction of voids upon the application of load. The slight increase afterwards is caused by the micro-cracks development within ITZs. Subsequently, with further increase in load, the micro-cracks initiated at ITZs become localized through the cement mortar and result in a rapid increase in permeability. Recently, with the advances in computational techniques, researchers have been trying to use meso-scale computer simulations in determining the permeability of concrete [9,10]. At meso-scale concrete is considered as a 3-phase heterogeneous material consisting of cement mortar, aggregate and ITZ. It is acknowledged that the research on meso/micro structural simulation of concrete as a multi-phase material has been undertaken but the knowledge on the change of concrete permeability at mesostructural level and under various loads is very scarce and there is no numerical method to determine the time variant permeability of concrete, i.e., permeability degradation, caused by stress induced micro-cracks. Moreover, there are some limitations on capturing the initiation, propagation and interconnecting of stress-induced cracks using existing models of meso/micro structural simulation, resulting in unreliable permeability predictions. Up to-date, especially for meso-scale concrete, the best approach to address these issues is to replace the fracture mechanics with the damage mechanics as suggested by many researchers [8,10,11]. This will be explored in the current study. The intention of this paper is to develop a new methodology to determine the permeability of stressed concrete over time, based on damage mechanics of meso-structure of concrete. Firstly, a representative meso-structure of concrete as a 3-phase material is simulated using Monte Carlo techniques. Secondly, based on the simulated meso-structure of concrete, finite element models are
developed for the stress analysis, in which a user subroutine is created for the identification of individual damage to each element of the mesh and for the determination of load-induced overall damage. The permeability of concrete corresponding to the overall damage of bulk concrete is determined as the final step. A worked example is presented as verification and also demonstration of the application of the developed methodology. Further, a parametric study on the effects of concrete constituent factors, including aggregate fraction, particle size distribution of aggregates and the distribution of aggregates on the permeability degradation of concrete is undertaken to identify the most influential factors. The merit of the developed methodology is that it can determine the permeability degradation of concrete with different mechanical properties and geometries of constituent materials under various applied loads over time, which would be otherwise impractical experimentally. 2. Simulation of meso-structure of concrete In order to represent concrete as realistically as possible, it has to be modelled as a heterogeneous material consisting of all 3phases of cement mortar, aggregate and ITZ. The meso-structure of concrete depends greatly on the aggregate fraction, aggregate particle size distribution and aggregate distribution [12]. Fine aggregates are usually considered within the cement mortar phase. For computational modelling of concrete at this level, aggregate particles are usually assumed as spheres for 3-D problems and circles for 2-D problems [13]. Aggregates in real concrete are neither circular nor spherical in shape. Though different shapes can be accommodated in computer simulations, this is more of an assumption for the convenience of aggregate distribution/generation and also for reducing the computational effect without compromising the technical development. However, the current study accommodates two other aggregate shapes in the simulations to see the effect of the aggregate shape on the permeability of concrete. The concept of these simulations is the same as circular shape. It is also assumed that the particle size and the distribution of aggregates are random. The size distribution of aggregate particles in a given concrete mix can be obtained experimentally by conducting a sieve analysis. However, based on the theory of stereology, particle size distribution of aggregates in concrete can be considered following Fuller mix [12]. For a Fuller mix, the cumulative distribution function (CDF) for the aggregates in concrete of a two-dimensional problem can be expressed as follows.
d f ða0 ; b0 Þ d0 f ða; bÞ 1:5
P2d ðdÞ ¼
1:5
ð1Þ
d f ða0 ; b0 Þ 1:5
where, d = diameter of aggregate (d0 d dm), d0 is the smallest aggregate diameter and dm is the largest aggregate diameter in the concrete mix. Coefficients a, a0, b, b0 and f(a, b), all related to d, d0 and dm, are expressed as follows [13,14].
f ða; bÞ ¼
pffiffiffi 2 2 2 1 1 cos1:5 asina þ 2E b; pffiffiffi F b; pffiffiffi 5 5 2 2
a ¼ arccos
d ; dm
a0 ¼ arccos
pffiffiffi a b ¼ arcsin 2sin ; 2
d0 dm
ð2Þ
pffiffiffi a 0 b0 ¼ arcsin 2sin 2
ð3Þ ð4Þ
E() and F() in Eq. (2) are the Legendre’s standard elliptical integrals [12]. The Eq. (1) for 2-D problems is developed based on the size distribution of intersecting circles of a randomly located p-plane. The conversion of aggregate volume fraction (AVF) to
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aggregate area fraction (AAF) also has been considered when deriving the Eq. (1) [13]. However, it should be acknowledged that the concrete in practice is of three dimensions and the 2-D representation of concrete in this research or even in mesoscale modelling of concrete in general is for the purpose of minimizing the computational time and capacities. With Eq. (1), it is possible to generate random values for the diameter of circular aggregates. For a required AAF, above equation and a similar process used by Zheng, Li and Zhao [14] can be employed to determine various aggregate diameters and the total number of aggregates (NA). Once the total number of aggregates and the diameters of those are known, coordinates for their random distribution within the representative meso-structure of concrete (RMSC) boundaries can be determined. Since the current study includes the ITZ, a new algorithm is developed to determine the distribution of aggregate particles considering ITZ thicknesses as well. The key steps of the developed algorithm for the simulation of a RMSC can be summarized as follows: 1. Set the width and height for the RMSC. 2. Arrange the generated NA aggregate diameters in descending order [14]. 3. Take the ith aggregate diameter (i = 1 to NA). 4. Determine the center coordinates (xi, yi) of ith aggregate by sampling from random variables Xi and Yi. The intervals of these random variables respectively depend on the required width and height of RMSC. In addition, the selected aggregate diameter and the minimum cement mortar cover to an aggregate also influence these intervals. 5. Use ITZ thickness (tITZ) and diameter of ith aggregate to determine the ITZ diameter. 6. Check whether the circle for the ITZ diameter of ith aggregate overlaps with all previously distributed ITZ circles or not. Currently, it is assumed that the ITZs for aggregates won’t merge with each other. If there is any overlap, return to step 4. 7. Otherwise, return to step 3 until all the NA aggregate particles are distributed within the RMSC boundaries.
START
Pre-determine the AAF, ITZ thickness, dimensions of RMSC, Max and Min diameters of Aggregate Determine the Aggregate diameter using Fuller Mix
Yes
Calculate the cumulative Area (AT) of aggregates
Check AT < AAF No Sort the generated diameters (NT) in descending order
Generate random numbers for x, y coordinates until Ni = NT Yes Check whether the outer surface of ITZ intersect with all previously placed ITZs No Record the diameter and the coordinates
END Fig. 1. Key steps in random aggregate diameter generation and distribution.
Aggregate
Cement Mortar
The key steps for the random generation of aggregate diameters using Fuller mix and distribution of aggregate particles are provided in Fig. 1.
3. Finite element models for concrete damage Because of the randomness of the aggregate diameters and the distribution, development of the finite element (FE) models for RMSC has to be carried out by adopting Abaqus Python scripting [15]. This python script incorporates all the necessary steps, which automates the creation of Abaqus FE model as it is essential to conduct number of analyses for different conditions. The 2-D 3-phase meso-structure of concrete is generated in Abaqus as shown in Fig. 2. Aggregate phase is placed in accordance with the data generated, as described in previous section, for diameters and center coordinates. ITZ phase is placed around the aggregates with the same center coordinates and the ITZ thickness. The remaining area is considered as the cement mortar (Fig. 2). Because of the randomness in geometric distribution of aggregates/ITZs, number of aggregates/ITZs and the geometry of cement mortar as an individual phase, the meshing of individual phases needs to be done through the developed python script. The element type, element creation technique and all the required parameter to create the mesh are given within the python script. The Mesh discretization of the RMSC is also shown in Fig. 2 with some zoomed mesh discretization views of cement mortar around aggregate, of aggregates and ITZs. When concrete is considered as a 3-phase heterogeneous
ITZ
Fig. 2. Geometrical arrangement and the Mesh Discretization of RMSC.
material in 2-D at mesoscale finite element analyses, with the cross-section varying along the thickness direction, generally the elements are considered as plain-stress [16,17]. Therefore, all three phases of concrete in the model are represented by 2-D planestress elements. The load is applied at the top boundary as a displacement load and the bottom boundary is restrained to represent the selected experimental testing setup as closely as possible [1]. At this stage, all the geometrical and mechanical properties for each individual phase should be identified and available for the finite element analysis of RMSC. As the aggregate phase is high in stiffness compared to other phases, it can be assumed that only the cement
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mortar and ITZ phases experience the load-induced damage. Therefore, the damage criterion is applied only to cement mortar and ITZ. Behaviour of concrete is simplified as an elastic material in the present study. To elaborate this further, the behaviour of Aggregate phase was assumed as pure elastic. Whereas behaviour of cement mortar and ITZ were assumed as brittle-elastic as those phases are checked against the damage criterion. This consideration is really important as the stress behaviour of an individual element upon reaching the damage criterion would affect the stress and damage progression of surrounding elements. Individual elements for cement mortar and ITZ are considered as damaged when tensile stress of an element exceeds the tensile strength or compressive strain exceeds the ultimate compressive strain. This definition of damage is incorporated using ABAQUS subroutine (USDFLD) written in a Fortran code to check/identify the load-induced damage for each element. In this way, an element can only fail due to either tension or compression. This damage identification process can be summarized as follows,
characteristics has attracted substantial popularity in the research community recently. Therefore, this paper only focuses on the stress induced damage on permeability of concrete. Several researchers have proposed various models relating different damage characteristics of concrete to the permeability. The research work conducted by Picandet, Khelidj and Bastian [20] has been accepted by many other researchers in this field because of its simplicity in application. Several other researchers have also contributed to this field in substantial terms [8–11]. Picandet, Khelidj and Bastian [20] conducted gas permeability tests on three different types of concretes (normal concrete, high performance concrete and high-performance concrete with steel fibre) with Nitrogen as the percolating gas. Load has been applied as a uniaxial compression load. The permeability has been measured after unloading in the direction of applied stress. An exponential relationship between damage and permeability was proposed by them based on regression as follows.
1. For the ith increment of load (i = 1 to N, N is the total number of increments required by Abaqus to apply the full compression load), take an element of ITZ/cement mortar and check that against the defined damage criterion. 2. If the element is identified as damaged, rank the damage variable of that element as 1. 3. Record the area of that element to an output file with the increment and element number. 4. For this load increment, repeat the check for all elements of cement mortar and ITZ. 5. Continue this check for all the increments until the application of the whole load.
kD and k0 are current and initial permeability coefficients respectively in the units of m2. D is the damage variable. a and b are constant regression coefficients from the regression analysis of their test data with suggested values of 11.3 and 1.64 for a and b respectively for damage range from 0 to 0.18. It should be noted that the way of defining the damage variable influences the its range. As the original Picandet, Khelidj and Bastian [20] relation is only applicable to damage range 0–0.18, Choinska, Dufour and Pijaudier-Cabot [8] applied the Taylor’s expansion to Eq. (6) to cover the full range of damage from 0 to 1. That negates the possibility of permeability reaching infinity when the damage is closer to 1. The extended form of Eq. (6) is as follows.
Once the output file for the area of damaged elements at each increment of load is generated by employing above process, the overall damage for a RMSC is determined by adopting the definition of damage variable based on effective undamaged area [16]. In this consideration, overall damage variable (D) is defined by the ratio of total damaged element area (AD) to the crosssectional area of all elements (A) as follows.
kD ¼ k0 1 þ ðaDÞb þ
D¼
AD A
ð5Þ
h h ii kD ¼ k0 exp ðaDÞb
"
ð6Þ
ðaDÞ2b ðaDÞ3b þ 2 6
# ð7Þ
Both Eqs. (6) and (7) have been used by many other researchers to relate the stress-induced damage with permeability [8– 11,18,19]. As the original Picandet, Khelidj and Bastian [20] relation (Eq. (6)) was based on gas permeability tests, a concern may arise over the use of this relation for water permeability problems like in Kermani [1]. To compare the concrete permeability obtained from different liquids it is necessary to identify the intrinsic permeability (k), which depends only on the pore structure of concrete and independent of flowing fluid [27]. For water, the intrinsic permeability coefficient (kw, measured in m2) is 1.02 107 times the Darcy coefficient (Kw, measured in m/s) [27]. It can be written as kw = 1.02 107 Kw (m2). When Nitrogen gas is used as the percolating fluid, it always gives higher intrinsic permeability coefficient values compared to water. This is due to viscous/slip flow behaviour of gases [28]. Bamforth [27] found that the difference is 5 to 6 times for high permeable concrete (in the range 1015–1012 m2) and around 60 times for low permeable concretes. However, the following equation can be used to convert the coefficient of permeability from gas to water or vice versa [27,28].
Once the overall damage values are found by Eq. (5) for each increment of load, those values are normalized between 0 and 1 to have the full range of damage variable. Damage variable for concrete has been defined in various ways by different researchers. Some researchers have defined the damage variable as the relative change in modulus of elasticity of concrete [18–20]. Some others have defined it as the ratio between stress to ultimate strength in post-peak region [21,22]. Strain based techniques also have been employed by many other researchers to define damage variable [23–25]. It has also been defined by considering energy-based techniques as well [26]. However, it should be noted that, whichever the way damage variable is defined, for the finite element analyses of 3-phase heterogeneous meso-structural concrete models, this has to be considered at element level. Therefore, the overall damage for a specimen needs to be calculated either by referring to average damage variable considering all the elements or by employing a weighted average method based on area/volume as has been used in this research work (Eq. (5)).
where, kl and kg are the liquid and gas permeability coefficients respectively. b is the Klinkenberg [29] coefficient and Pm is the mean atmospheric pressure.
4. Determination of permeability
5. Verification
Even though the load induced cracks contribute to the permeability of concrete, relating the concrete damage to water flow
The verification of the developed method to determine the concrete permeability subjected to unconfined compression loading is
b kl ¼ kg = 1 þ Pm
ð8Þ
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Table 1 Material Properties for the FE Model.
Modulus of Elasticity – E, GPa Tensile Strength, MPa Compressive Strength, MPa Ultimate compressive strain Poisson’s ratio – m Density – q, kg/m3
Aggregate
Cement Mortar
ITZ
80 [9] – – – 0.17 [35] 3000 [9]
18.57 [5] 1.85 (10% of Compressive Strength) 18.5 Calculated based on [34] 0.001 Calculated (r = Ee) 0.3 [10] 2200 [10]
Ei = 0.6 Em = 11.142 [35] 0.66 (10% of Compressive Strength) 6.6 Calculated (r = Ee) 0.0006 Assumed as 0.6 times Mortar 0.35 [35] 1320 Assumed as 0.6 times Mortar
Fig. 3. Damage distribution with stress percentage (a) 25%, D = 0.04, (b) 50%, D = 0.23, (c) 75%, D = 0.76 and (d) 100%, D = 0.97.
to be illustrated in this section by a worked example and compared with the experimental data published by Kermani [1]. The meso-structural model in this example was formulated as closely as possible to replicate the experimental concrete specimen used by Kermani [1]. Three different types of concrete were considered by Kermani [1] and only the data and results related to normal strength concrete were used in this study for the comparison and verification. A characteristic compressive strength of 30 MPa was specified when preparing the concrete mixes and a value of 34 MPa was obtained at 28 days when tested by Kermani [1]. As the exact mix proportions of this test were not provided, reasonable assumptions had to be made for carrying out an effective comparison for the verification. A tested compressive strength of 36 MPa for normal concrete was reported by Aldea, Shah and Karr [5] with an AVF of around 0.48. Even though a same compressive strength can be obtained by different mix proportions, the deviation of the proportions cannot be significant for the same grade of concrete. In addition, AVF is always smaller compared to AAF and the values can be considered similar when the particle size distribution of aggregates is very large [13]. Therefore, it is reasonable to assume an AAF value between 0.4 and 0.5 for this example to compare the results with Kermani [1]. With these considerations, AAFs of 0.4 and 0.5, minimum aggregate diameter of 5 mm and a maximum of 20 mm, same as reported in the experiment by Kermani [1], were considered for the comparison. The size of the RMSC was taken as 100 mm x 100 mm [16]. The thickness of the ITZ is affected by the packing of cement grains and extend at least to the size of the largest cement particle, which may be up to 100 lm [30]. Therefore, the thickness of the ITZ can be in the range of 9–100 lm [30,31]. As ITZ is considered as the weakest link of concrete matrix where damage initiates [32,33], conservatively, the thickness of ITZ for the finite element models was assumed as 100 lm. Compressive strength of Cement mortar, corresponding to 30 MPa compressive strength of concrete, was calculated as 18.5 MPa following the Fig. 1.27 of the book by Neville [34]. Since cement mortar was assumed to behave elastically, the ultimate compressive strain of cement mortar was calculated by the ratio
of compressive strength to elastic modulus (e = r/E). Average Elastic modulus of ITZ to Cement mortar ratio is 0.6 [35]. Following that, because of the unavailability of individual material properties for ITZ for 30 MPa compressive strength concrete, the ultimate compressive strain and the density of ITZ also were assumed as 0.6 times the cement mortar values (as in Table 1). Further, the ultimate tensile strengths were considered as 10% of the ultimate compressive strengths [34]. Material properties for all the three phases of concrete used in this study are summarized in Table 1. The other required parameters were obtained from reported studies in literature in the same context. All three phases of concrete in the model were represented by 4-node bilinear, reduced integration plane-stress quadrilateral elements (CPS4R) available in Abaqus. Surface interactions between individual phases were considered as non-over closure/ clearance during the application of loading. To represent the uni-axial compression test conditions, a displacement of 0.2 mm was applied at the top boundary, both x and y directions were restrained at bottom and the sides were not restrained in any direction (free ends). The damage identification criteria were only applied to the cement mortar and ITZ phases. Behaviour of all 3 phases was assumed as elastic in the current study as an initial proof of the concept. After stress analysis in Abaqus, each element was checked against the defined damage criterion by the adopted subroutine at the start of each load increment to identify the individual damage. Fig. 3 shows the progressive damage distribution (for AAF of 0.4) with the stress percentage (applied stress to the characteristic strength of concrete, 30 MPa in this example). Characteristic strength of concrete percentages of 25, 50, 75 and 100 results in an overall damage variable values of 0.04, 0.23, 0.76 & 0.97 respectively. The overall damage for each increment was determined by Eq. (5). Subsequently, the coefficient of permeability corresponding to each damage value at each increment of load was calculated using Eq. (7) in the form of Eq. (9) as follows.
" K ¼ K0
ðaDÞ2b ðaDÞ3b 1 þ ðaDÞ þ þ 2 6 b
# ð9Þ
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1.8
1.6
AAF 0.40
1.4
AAF 0.50
1.2
Kermani 1991
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 4. Degradation of permeability for AAFs of 0.4 and 0.5.
where, K and K0 are current and initial permeability coefficients respectively in the units of m/s. In doing that, the initial water permeability was taken as 2 1014 m/s, same as the initial permeability value obtained by Kermani [1], which equals 2 1021 m2 for water or 1 1019 m2 for Nitrogen gas if the gas to water conversion factor is assumed as 50. Fig. 4 indicates the degradation of permeability with the stress level for AAFs of 0.4 and 0.5 in this example. Stress level is calculated by the ratio between applied stresses and ultimate stress. In addition to the analysis results, the experimental results by Kermani [1] also is provided for a comparison. It can be observed from Fig. 4 that, similar to the experimental results, value of permeability coefficient increases (is degraded) with the increase in applied loading. As can be seen from Fig. 4, results by Kermani [1] are between AAF of 0.4 and 0.5 and very close to 0.4 which effectively verifies the developed method. Therefore, the developed method can be used to determine the permeability degradation of stressed concrete.
Coefficient of Permeability, K, (m/s) x 10-10
Coefficient of Permeability, K, (m/s) x 10-10
1.8
AAF 0.30
1.6
AAF 0.40
1.4
AAF 0.50
1.2
Kermani 1991
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 6. Degradation of permeability for different AAFs.
all the models in this particular study and those are expected to be investigated in a future study. It is acknowledged that when these parameters are changed individually, the material properties of the final concrete would be affected. For example, when the AAF is changed, the strength of concrete might change and that will affect the grade of concrete. However, when concrete is considered as a 3-phase heterogeneous material, the material properties of individual phases will not change significantly and can be considered constants for all the models. It should be noted that the parametric study in this section was conducted to show the capability of the developed method to identify the effect of those parameters on permeability degradation. Results from the developed method could not be compared with experimental results directly without knowing the exact experimental conditions, mix proportions and material properties of the tested concrete samples. However, the experimental results by Kermani [1] is presented together with the following analyses results on different parameters, just to show the trend of permeability degradation.
6. Parametric study The meso-structure of concrete depends greatly on the AAF, the aggregate particle size distribution and aggregate distribution [12]. Hence, by employing the developed method, which has been verified before, the effect of above parameters on the permeability degradation of stressed concrete was studied. The dimensions of the RMSC were kept constant at 100 mm for each side for all the analyses. In addition, as highlighted in the worked example, the behaviour of aggregate, ITZ and cement mortar were considered as elastic. The material properties were assumed as constants for
6.1. Effect of aggregate fractions The amount of aggregates present in a concrete can affect the properties of a concrete largely. Three different AAFs of 30%, 40% and 50% were simulated for the analysis (Fig. 5). The particle size distribution of aggregates was considered from 5 to 20 mm for all three cases. By employing the developed method, the degradation of permeability with the stress level for these different AAFs is presented in Fig. 6.
Fig. 5. Meso-models for AAFs (a) 30%, (b) 40%, (c) 50%.
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1.8 5-10mm 5-15mm 5-20mm Kermani 1991
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength
Coefficient of Permeability, K, (m/s) x 10-10
Fig. 7. Meso-models for aggregate size distributions (a) 5–10 mm, (b) 5–15 mm & (c) 5–20 mm.
1.8 1.6
Distribution 1 Distribution 2 Distribution 3 Kermani 1991
1.4 1.2 1 0.8 0.6
7% 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 8. Degradation of permeability for different aggregate size distributions. Fig. 10. Degradation of permeability for different aggregate distributions.
As can be seen from Fig. 6, coefficient of permeability increases with the applied loading for all the AAFs of 0.3, 0.4 and 0.5. However, with the increase in stress level, it can be observed that the higher the AAF is the higher the chance of concrete matrix is damaged and the higher the increase in permeability coefficient. Since, compressive strength of concrete is inversely proportional to AAF [16], higher AAFs have lower compressive strengths and are more susceptible to damage which will lead to higher permeability coefficients. Clearly the results from this research confirm this phenomenon. Further, at a stress level of 0.7, AAF of 0.3 has a permeability coefficient of around 1.5 1011 m/s. When the AAF is increased up to 0.4, the permeability coefficient is increased by 4 times up to a value of 6 1011 m/s. AAF of 0.5 has a permeability coefficient of 9.5 1011 m/s, an increase of 6.3 times to AAF of
0.3. Considering all, the results indicate that the effect of AAF on the degradation of permeability is significant. 6.2. Effect of aggregate size distribution The size distribution of aggregates also can influence the transport properties of concrete [12]. Hence, as shown in Fig. 7, aggregate size distributions of 5–10 mm, 5–15 mm and 5–20 mm were generated (using fuller distribution) to be analysed by employing the developed method. Because of the experimental results had a reasonably good agreement with the permeability degradation for AAF of 0.4, all three size distributions were considered to have
Fig. 9. Meso-models for aggregate distributions (a) Distribution 1, (b) Distribution 2, (c) Distribution 3.
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Coefficient of Permeability, K, (m/s) x 10-10
Fig. 11. Meso-models for aggregate shapes (a) Circular, (b) Pentagonal & (c) Hexagonal.
7% considering the overall increase in permeability (5500 times the initial permeability). Even though the differences in the curves are not that significant, the distribution of aggregates also influences the permeability degradation of stressed concrete.
1.8 Circular Hexagonal Pentagonal Kermani 1991
1.6 1.4 1.2
6.4. Effect of aggregate shape
1 0.8 0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stress Level, ratio of applied stress to ultimate strength Fig. 12. Degradation of permeability for different aggregate shapes.
the same AAF of 0.4. Analysis results for the degradation of permeability with the stress level for the above aggregate size distributions can be seen in Fig. 8. It can be observed from the results in Fig. 8 that the effect of aggregate size distribution is not that significant on the degradation of permeability. Even though all the three distributions show seemingly similar results overall, at 0.7 stress level, 5–10 mm distribution has an increment in permeability of 5283 times the initial permeability. At same stress level, increase in permeability for 5–15 mm and 5–20 mm distributions are respectively 5572 times (5.5% more permeable compared to 5–10 mm distribution) and 5690 times (7.7% more permeable compared to 5–10 mm distribution) the initial permeability. Until a stress level of around 0.73, the aggregate size distribution with smaller variation shows lesser permeability coefficient increment. 6.3. Effect of aggregate distribution Three different purposely selected aggregate distributions (Distribution 1–3) as shown in Fig. 9 were also studied on the permeability degradation of stressed concrete. Same AAF of 0.4 and size distribution from 5 to 20 mm were considered for all the cases. The obtained analysis results for the permeability degradation are presented in Fig. 10. As can be observed from results in Fig. 10, different distributions of aggregates provide slightly different variation of permeability with the increase in load. However, the differences are negligibly small compared to the overall increase in permeability due to loading. For example, at 0.7 stress level, the maximum difference of permeability coefficients of two distributions is roughly
As mentioned before, even though this particular research work considered circular shaped aggregates for all above analyses, the effect of the aggregate shape on the permeability of concrete also was investigated by simulating different shapes e.g. regular hexagonal and irregular pentagonal aggregates as shown in Fig. 11. The AAF and the particle size distribution of aggregates were kept constant at 0.4 and 5–20 mm respectively for all three models. It can be observed by the results shown in Fig. 12 that the influence of aggregate shape has some influence on the permeability with the applied load. Though the curves look similar in Fig. 12, results show that the pentagonal shaped aggregate has the highest permeability degradation, then the hexagonal shape and the lowest is the circular shape. For example, at the 0.7 stress level, the permeability with pentagonal and hexagonal shapes increases 14.78% and 12.34% respectively compared with the circular shape. With these results, it can be said that the aggregate shapes with lesser sides (consist of more sharper corners) experience higher permeability degradation as oppose to more rounded shapes. Higher applied stresses can result high stress concentrations near the sharper edges which causes more elements to get damaged in these locations and leads to a higher permeability degradation. This makes sense since the degradation of permeability is primarily caused by stress induced cracks. 7. Conclusions A methodology to determine the permeability degradation of stressed concrete has been developed in this paper and verified with experimental results. The developed methodology is based on the damage mechanics of concrete meso-structure. The significance of the developed methodology is that it can determine the permeability of concrete with different mechanical properties and geometries of constituent materials under various applied loads over time, which would be otherwise impractical experimentally. A parametric study has also been undertaken in the paper to identify the most important factors that affect permeability degradation of concrete. It has been found in this study that the permeability of concrete starts to degrade when the stress in concrete reaches a certain threshold value for a given aggregate fraction and that the aggregate fraction is the most influential factor for permeability degradation of concrete. It has also been
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found that the particle size and geometrical distribution of aggregates only affect the permeability degradation of concrete marginally. Further, it has been found that the aggregate shape influences the permeability degradation as well and the aggregates with more sharper edges tend to result higher permeabilities as oppose to rounded shapes. It can be concluded that the developed methodology can determine the permeability degradation of stressed concrete with reasonable accuracy and hence contribute to the body of knowledge of concrete technology. Conflict of interest None. Acknowledgements Financial support from Metro Trains Melbourne, Australia and Australian Research Council, Australia under DP140101547, LP150100413 and DP17010224 is gratefully acknowledged. References [1] A. Kermani, Permeability of stressed concrete, Build. Res. Inf. 19 (6) (1991) 360–366. [2] N. Banthia, A. Biparva, S. Mindess, Permeability of concrete under stress, Cem. Concr. Res. 35 (9) (2005) 1651–1655. [3] M. Hoseini, V. Bindiganavile, N. Banthia, The effect of mechanical stress on permeability of concrete: a review, Cem. Concr. Compos. 31 (4) (2009) 213– 220. [4] N. Hearn, Effect of shrinkage and load-induced cracking on water permeability of concrete, Mater. J. 96 (2) (1999) 234–241. [5] C.M. Aldea, S.P. Shah, A. Karr, Permeability of cracked concrete, Mat. Struct. 32 (5) (1999) 370–376. [6] C.M. Aldea, S.P. Shah, A. Karr, Effect of cracking on water and chloride permeability of concrete, J. Mater. Civil Eng. 11 (3) (1999) 181–187. [7] N. Banthia, A. Bhargava, Permeability of stressed concrete and role of fiber reinforcement, ACI Mater. J. 104 (1) (2007) 70–76. [8] M. Choinska, F. Dufour, G. Pijaudier-Cabot, Matching permeability law from diffuse damage to discontinuous crack opening, Fract. Mech. Concr. Concr. Struct. 1 (2007) 541–547. [9] D. Niknezhad, B. Raghavan, F. Bernard, S. Kamali-Bernard, Towards a realistic morphological model for the meso-scale mechanical and transport behavior of cementitious composites, Compos. Part B: Eng. 81 (2015) 72–83. [10] B. Raghavan, D. Niknezhad, F. Bernard, S. Kamali-Bernard, Combined mesoscale modeling and experimental investigation of the effect of mechanical damage on the transport properties of cementitious composites, J. Phys. Chem. Solids 96–97 (2016) 22–37. [11] G. Pijaudier-Cabot, F. Dufour, M. Choinska, Permeability due to the Increase of damage in concrete: from diffuse to localized damage distributions, J. Eng. Mech. 135 (9) (2009) 1022–1028.
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