- Email: [email protected]
Modeling influence of hysteretic moisture behavior on distribution of chlorides in concrete
Cement and Concrete Composites 67 (2016) 73e84
Contents lists available at ScienceDirect
Cement and Concrete Composites journal homepage: www.elsevier.com/locate/cemconcomp
Modeling influence of hysteretic moisture behavior on distribution of chlorides in concrete Josko O zbolt a, b, *, Filip Orsani c a, Gojko Balabani cb a b
Institute of Construction Materials, University of Stuttgart, Pfaffenwaldring 4, 70560 Stuttgart, Germany Faculty of Civil Engineering, University of Rijeka, Radmile Matejcic 3, 51000 Rijeka, Croatia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 July 2015 Received in revised form 28 December 2015 Accepted 1 January 2016 Available online 9 January 2016
Aggressive environmental conditions, such as exposure to the sea climate or use of de-icing salts, can have a considerable influence on the durability of reinforced concrete structures due to corrosioninduced damage of reinforcement. Recently, the coupled 3D chemo-hygro-thermo-mechanical (CHTM) model for simulation of processes related to the chloride induced corrosion of steel reinforcement in concrete was developed. In the model, it is assumed that for wetting and drying of concrete, the transport of water is controlled by a single sorption curve. However, it is well known that concrete exhibits a hysteretic moisture behaviour, which significantly influences the distribution of moisture and chlorides. To account for the hysteretic moisture behaviour of concrete and for simulating a more realistic time and space distribution of moisture, the CHTM model was further improved. The proposed hysteretic model is implemented into a 3D finite element code and it is validated using a numerical example, which shows reasonably good agreement with the available test results. Similar to what is observed in the experimental tests, it is shown that due to the wetting and drying of the concrete surface, the peak concentration of chloride moves progressively deeper into the concrete specimen. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Reinforced concrete Corrosion Chemo-hygro-thermo-mechanical model Hysteretic moisture behaviour Microplane model Finite elements
1. Introduction Corrosion-induced damage of concrete is a major problem for durability of reinforced concrete (RC) elements exposed to severe climate conditions. The onset of corrosion can be caused by carbonation of the concrete cover or by reaching a critical concentration of free chloride ions in the vicinity of the reinforcement bar. Here presented work is focussing on the processes related to the chloride-induced corrosion. Generally, computation of corrosion current density depends upon the following physical, electrochemical and mechanical processes: (1) Transport of capillary water, oxygen and chloride through the concrete cover; (2) Immobilization of chloride in the concrete; (3) Drying and wetting of concrete and related hysteretic property of concrete; (4) Transport of OH ions through electrolyte in concrete pores; (5) Cathodic and anodic polarization; (6) Transport of corrosion products; (7) Creep of concrete around reinforcement bars and (8) Damage of
* Corresponding author. Institute of Construction Materials, University of Stuttgart, Pfaffenwaldring 4, 70560 Stuttgart, Germany. E-mail addresses: [email protected] (J. O zbolt), fi[email protected] (F. Orsani c), [email protected] (G. Balabani c). http://dx.doi.org/10.1016/j.cemconcomp.2016.01.004 0958-9465/© 2016 Elsevier Ltd. All rights reserved.
concrete due to mechanical and non-mechanical actions [1]. In partially saturated concrete, the main physical phenomenon governing transport of chlorides in concrete are diffusion in the capillary pore water and convection due to flow of the pore water as a consequence of capillary suction and diffusion. A moisture hysteretic behaviour typical for cementitious material affects the transport of water and consequently the ingress of chlorides. For more accurate prediction of chloride's distribution in the concrete cover, binding of chlorides on the pore walls also needs to be taken into account. Currently, there are a number of models in the literature that are able to simulate processes before and after depassivation of reinforcement in concrete [1e15]. A recently developed 3D chemohygro-thermo mechanical model [13e15] is able to simulate nonmechanical and mechanical processes and their interaction before and after depassivation of steel reinforcement. The model is implemented into a 3D finite element (FE) code and the results have shown that it is able to realistically replicate relevant processes before [13] and after depassivation of reinforcement [14,15]. However, in the original model, the hysteretic behaviour of concrete, which is relevant for the distribution of moisture and chlorides and consequently also for the processes related to the active
74
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
corrosion phase, was not included. Therefore, the model was recently extended to account for this hysteretic moisture response of concrete [16]. It is well known that the transport processes important for the onset and corrosion progression of steel reinforcement in concrete are highly dependent on the pore moisture content. With the change of the environmental relative humidity (RH), the relative pore pressure is correspondingly changing and hence, affecting the water content in the concrete pores. Unsaturated conditions in concrete specimens have a strong influence on the distribution of chlorides. Convective flux of chloride ions due to moisture transport becomes more dominant compared to a slower, diffusive flux. Hence, the ingress of chlorides is more pronounced compared to the chloride attack under saturated conditions [17]. Furthermore, exposure to wetting and drying cycles can result in characteristic “peak” chloride profiles along the concrete depth [18e20], with the peak value of the total chloride content being translated away from the exposed surface progressively with the wet/dry cycling. Assuming isothermal conditions, the water content can differ for the same relative humidity, depending on the dynamic moisture loading history [21e28]. For a constant air temperature and constant relative humidity, equilibrium conditions can be obtained in a concrete specimen, i.e. the pore pressure in the specimen becomes approximately the same as the RH of the ambient air. The relation between relative humidity and the experimentally measured water content is given by isotherm curves [22]. Usually, in the experiments for determination of isotherm curves, very small or thin specimens are used in order to quickly reach the equilibrium state, for a step-like increase or decrease of the ambient RH [25]. The main adsorption curve is obtained by plotting the values of the water content for each increase of RH, starting from a completely dry specimen. Similarly, starting from a fully saturated specimen and incrementally decreasing RH of the ambient air, the main desorption curve can be obtained. Depending on the loading history, porous materials such as concrete, have different moisture values for the same RH. Therefore, at a constant temperature the sorption curves differ from each other, with the position of the desorption curve always being above the adsorption isotherm (see Fig. 1). The hysteretic behaviour of cementitious material for cases where the conditions change periodically from drying to wetting is described with the so-called scanning curves (see Fig. 1). They are positioned between the main adsorption and desorption curves, which are acting as envelopes [22,23,25]. Here, the main adsorption and desorption curves are given as input data, coming from experimental tests or empirical
formulae. In the incremental transient FE analysis, the model explicitly computes the distribution of relative humidity in the concrete and then determines the moisture content based on the sorption curves. By knowing the position on the isotherm curve from a previous time step, a new distribution of the relative pore pressure can be calculated for the current boundary conditions. Subsequently, the corresponding water content is established from a new location on the isotherm curve. In the case of static moisture conditions, i.e. constant wetting or drying, the analysis is following the main adsorption or desorption curve. This is valid only under the assumption that the initial starting condition (previously wetted or dried concrete) corresponds to the boundary environmental condition (wetting or drying). In the case of dynamic moisture conditions, i.e. with the change from wetting to drying or vice versa, the scanning curves need to be employed. In the present work they are calculated empirically, using the formulae proposed by Pedersen [23]. They differ from the main adsorption or desorption curves and behave in a loop-like manner depending on the number of wetting and drying cycles. The implemented hysteretic behaviour of concrete [16] was verified by simulating several experimental tests [25,29]. The recently improved CHTM model, which includes hysteretic moisture behaviour, is here used to compute the distribution of chloride in concrete under cyclic wetting-drying conditions. In the first part of the paper, the hysteretic model, based on the above mentioned theoretical background and its implementation into the FE code are discussed. The model is then employed in a transient numerical finite element study of concrete specimens exposed to chloride attack in a dynamic moisture environment. The chloride profiles along the concrete cover depth are computed and compared with the experimental results of Polder & Peelen [19].
2. Hysteretic moisture model for concrete 2.1. Moisture transport During wetting and drying cycles, it is assumed that concrete is exposed only to changes in relative humidity, so the macroscopic pressure-driven liquid flow is not considered. Transport of moisture through the concrete is described as a vapour transport, which means that for nonsaturated concrete at uniform temperature, the moisture flux jw,mass(kg/m2s) may be expressed as [30]:
jw;mass ¼ dv ðhÞpv;sat Vh
(1)
where h is the relative humidity (dimensionless), dv(h) is the water vapour permeability (s) and pv,sat is the water vapour saturation pressure (Pa). The mass conservation condition reads:
rw
vqw ðhÞ vqw ðhÞ vh ¼ rw ¼ V$ dv ðhÞpv;sat Vh vt vh vt
(2)
where t is the time (s), rw is the density of water (kg of water/m3 of water), qw is the volume fraction of pore water (m3 of water/m3 of w concrete) and rw vq ≡x is the moisture capacity (derivative of the vh sorption isotherm). For nonsaturated concrete at temperatures that are slightly different from the reference temperature of 25 C, dv(h) can be approximately described by the expression [31]:
dv ðhÞ ¼ a0 f1 ðhÞ Fig. 1. Illustration of typical adsorption, desorption and scanning curves for a cementitious material.
(3)
in which a0 is the reference permeability at 25 C and for mature paste the values of a0 are in the range of a0 ¼ 10101014(s).
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
Function f1(h) reflects the moisture transport within the adsorbed water layers, and according to [31e33]:
1a 4 1h 1 þ 1h
r qwa ðhÞ ¼ 0:01$ con $ua ðhÞ rw (8)
1 0:18 $ln
where a z 0.05 and hc z 0.75 at 25 C. Temperature dependence of water vapour saturation pressure is given by the expression [34]:
17:08085,t
or
(4)
c
psat;v ¼ 610:8,e 234:175þt
lnðhÞ 10:214
1 1:13 $ln
ua ðhÞ ¼ 4:79$e f1 ðhÞ ¼ a þ
75
(5)
where t is the temperature in ºC. The solution of differential Eq. (2) gives the distribution of relative humidity, not the content of water in the capillary pores. As the transport of moisture in concrete is a slow process, the different phases of water in the pores of concrete remain almost in thermodynamic equilibrium at any time [32] and the content of free water can be determined using sorption isotherms that relate the relative humidity and content of evaporable water at constant temperature. However, it is known that for materials such as concrete, there are different curves that relate the relative humidity and free water in the pores during adsorption (wetting) and desorption (drying). In other words, the hysteresis loops are observed during wetting and drying cycles. Thus, in order to calculate the concrete moisture distribution more accurately, a proper mathematical model of this hysteresis phenomenon in cycles of wetting and drying is required. For a hysteretic material, such as concrete, the moisture capacity is determined by the slope of scanning curves that are always between the main adsorption and desorption isotherms. In the present model, scanning curves are defined by simple empirical model proposed by Pedersen [23]:
ud ðhÞ ¼ 4:76$e
1lnðhÞ 4:85
or
r qwd ðhÞ ¼ 0:01$ con $ud ðhÞ rw (9)
where ua and ud are the moisture content (mass percent related to dry mass of concrete) for adsorption and desorption, respectively, qwa and qwd are the volume fractions of water for adsorption and desorption, respectively, and rcon ¼ 2300 kg/m3 is the assumed concrete density. The water content during scanning between the main adsorption and desorption isotherms is determined by the following relation:
qw ¼ qw0 þ xhys $Dh
(10)
in which qw0 is the initial volume fraction of water and Dh is the difference between humidity at the time jþ1 and j at spatial point i. Note that Dh depends on the chosen time step. The hysteresis model, with the current volume fraction of pore water, in which the volume fraction of pore water is calculated from the main adsorption and desorption curves and the derivative of the sorption isotherm at a given relative humidity, has the advantage of being applicable to different sorption isotherms, i.e. for different types of concrete and different temperatures. This makes the model relatively easy to implement into a numerical framework such as the finite element method.
2.2. Chloride convective diffusion
xhys;a ¼
xhys;d ¼
0:1ðqw qwa Þ2 xd þ ðqw qwd Þ2 xa ðqwd qwa Þ2
ðqw qwa Þ2 xd þ 0:1ðqw qwd Þ2 xa ðqwd qwa Þ2
(6)
(7)
where xhys is the moisture capacity (slope of scanning curve) at a given relative humidity, qwa and qwd are the volume fraction of pore water calculated from the main adsorption and desorption curves corresponding to the current relative humidity, xa ¼ rw vqvhwa is the moisture capacity at given relative humidity calculated from main adsorption isotherm, xd ¼ rw vqvhwd is the moisture capacity at given relative humidity calculated from the main desorption isotherm, qw is current volume fraction of pore water. When the most recent moisture history shows that in the previous two time steps the water content was increasing, the function defined by Eq. (6) is used in Eq. (2). However, the function defined by Eq. (7) is used in Eq. (2) if in the previous two time steps the water content was decreasing. The two previous time steps in the numerical procedure are considered in order to assure more reliable definition of moisture history. The main adsorption and desorption isotherms for concrete (for example w/c ¼ 0.48) are described by the following expressions [22]:
The macroscopic flux of free chloride ions in partially-saturated concrete exposed to drying and wetting consists of two parts: one is due to molecular diffusion of chlorides in pore water, the other one is connected with the flow of pore water due to capillary suction and diffusion, bringing with itself dissolved chloride ions. Therefore transport of chloride ions through concrete, before depassivation of steel, will be considered as convective diffusion. The macroscopic flux of chloride jc,diff, due to diffusion is expressed as:
jc;diff ¼ Dc ðqw ; TÞVCc
(11)
where Cc is the concentration of free chloride dissolved in pore 3 water (kg Cl/m of pore solution), T is the temperature and Dc(qw,T) is the effective chloride diffusion coefficient (m2/s). The macroscopic flux of chloride jc,conv, due to moisture transport is expressed as:
jc;conv ¼ Cc vw
(12)
in which vw is the mean water velocity. Thus, the total mass flux of chloride ions, jc, takes the form:
jc ¼ Cc vw Dc ðqw ; TÞVCc
(13)
Due to the fact that part of the chlorides can be bound into cement hydration products, the balance equation for chloride ions in concrete is given by Ref. [35]:
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
76
vðqw Cc Þ vC ¼ V$ðqw jc Þ cb vt vt
(14)
3 where Ccb is the concentration of bound chloride (kg Cl/m of concrete). Inserting Eq. (13) in the balance Eq. (14) it follows:
h i vC vðqw Cc Þ ¼ V$ Cc jw;vol qw Dc ðqw ; TÞVCc cb vt vt
3. Numerical implementation
(15)
where jw;vol ¼ qw vw ¼ dvrðhÞ pv;sat Vh denotes specific water discharge w (¼volume of pore water per unit area of concrete per unit time). Under the assumption that rw ¼ const, the mass balance for pore water, without source or sink terms, can be written as
vqw ¼ V$jw;vol vt
(16)
By combining Eq. (15) and balance Eq. (16) the equation that describes transport of chloride ions through concrete before depassivation of reinforcement is obtained as:
qw
vCc ¼ vt
humidity on pore water content in concrete according to the adsorption isotherms used in our model [22], U is the activation energy of the chloride diffusion process (44.6 kJ/mol for w/c ¼ 0.5), R is the universal gas constant [8.31 103 kJ/(mol$K)] and Tref ¼ 296 K.
dv ðhÞ vC pv;sat Vh$V Cc þ V$ðqw Dc ðqw ; TÞVCc Þ cb rw vt
By using the finite elements to solve the above partial differential equations, the strong form has to be rewritten into a weak form. The weak form of the system of partial differential equations is formulated by employing the Galerkin weighted residual method as follows [36]:
½Ch
vh þ ½Kh fhg ¼ ½Fh vt
o n o C_ c þ P1Ccb C_ cb þ KCc fCc g ¼ f0g n o P2Ccb C_ cb þ P2Cc fCc g þ P3Ccb fCcb g ¼ f0g
P1Cc
n
Z ½Ch ¼
In the computations presented here, it is taken into account that there is a linear relation between concentration of bound and free chloride at a constant temperature (linear binding isotherm) and also that there is a limit value for bound chloride [18]. As chloride binding does not immediately reach the equilibrium, the binding rate should be considered and therefore the mathematical model of chloride binding mechanism by hardened cement paste is given as a linear non-equilibrium isotherm:
Dc ðqw ; TÞ ¼ Dc;ref 1 þ
½Kh ¼
#1
ð1 hc Þ4
"
U $exp R
1 1 Tref T
!#
(19) where Dc,ref is the reference value of Dc evaluated at standard conditions, hc ¼ 0.75, h(w) expresses dependence of relative
Z
T
qw ½N ½NdU;
P2Cc ¼
U
Z
T
U
P1Ccb ¼
Z
bhum ½NT ½NdG
bhum henv ½NT ½NdG
qw Dc ðqw ; TÞ½VN ½VNdU
KCc ¼
Z G
G
Z
P1Cc ¼ Z
U
vw T ½N ½NdU vh
dv ðhÞpv;sat ½VNT ½VNdU þ ½Fh ¼
ð1 hðwÞÞ4
Z U
(18)
where kr is the binding rate coefficient and a ¼ 0.7 is the slope of linear adsorption isotherm. In other words, it is assumed that equilibrium is not reached instantaneously, but rather that it is approached at a certain rate, which in general is proportional to the difference between the already-adsorbed chloride content and the chloride content in the equilibrium state. It is well known that the amount of bound chloride ions depends on many factors such as cement content, water-cement ratio, porosity, etc., and these can be included in the coefficients kr and a. For three different types of concrete, characterized by three different water-cement ratio, the values of kr are given in Table 2 and a is chosen as a constant for all types of concrete according to Saetta et al. [18]. The dependence of chloride diffusion coefficient on water content and temperature is expressed as [18]:
"
(21)
with
(17)
vCcb ¼ kr ðaCc Ccb Þ vt
(20)
U T
½N ½NdU; U
P3Ccb ¼
Z
Z
kr a½NT ½NdU
U
dv ðhÞ pv;sat Vh$½NT ½VNdU rw
P2Ccb ¼
Z
½NT ½NdU
U
kr ½NT ½NdU
U
(22) where [N] is the column matrix of shape functions, which relates relative humidity h in concrete, mass concentration of free chloride in pore water Cc, mass concentration of bound chloride in concrete Ccb with their nodal values and G is the concrete surface exposed to environmental conditions. Eqs (20) and (21) are solved by using the direct integration method of an implicit type. 4. Numerical example: distribution of chlorides in nonsaturated concrete Exposure of concrete to cyclic wetting and drying conditions has strong implications on the transport processes important for the
Table 1 Investigated concrete mixes with Portland cement [19]. Cement type
w/c
Cem. cont. (kg/m3)
Agg. cont. (kg/m3)
Water cont. (kg/m3)
CEM I Port. cement
0.42 0.47 0.57
338 318 287
1780 1790 1827
143 149 163
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
77
corrosion-induced damage. Among others, transport of chloride ions, responsible for depassivation of steel, is strongly influenced by the hysteretic moisture behaviour of concrete. In such cases, distribution of chloride ions along the concrete depth exhibits characteristic “peak” profile shapes. Due to a pronounced moisture gradient, convective transport of ions is activated and at certain times it is even more dominant than diffusion. Therefore, experiments in which the chloride profiles were measured in concrete under cyclic moisture conditions represent a valuable example to validate the numerical hysteretic moisture model. To verify the here presented model, the experiments carried out by Polder & Peelen [19] are numerically simulated. 4.1. Experimental tests of Polder & Peelen (2002)
Fig. 2. Change of relative humidity at a constant temperature of 20 C for one exposure cycle in experiments by Polder & Peelen [19].
In experiments carried out by Polder & Peelen [19], the tested concrete specimens were prisms with dimensions of 100 100 300 mm, prepared with 12 different concrete mixes. In
Fig. 3. The geometry of the finite element model and discretization (all in mm).
(a)
(b)
(c)
Fig. 4. Main adsorption and desorption curves (input) from the experiments by (a) Baroghel-Bouny [25] and (b,c) Hansen [22] used for concrete specimens with the w/c ¼ 0.43, 0.48 and 0.55, respectively.
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
78
Table 2 Used model parameters for the numerical analysis of the relative humidity and chloride ions distribution. Hysteretic moisture model for concrete Sorption isotherms
w/c ¼ 0.43 [25] 11
w/c ¼ 0.48 [22]
w/c ¼ 0.55 [22]
Water vapour permeability, dv (s) [31] Surface humidity transf. coeff., bhum (m/s) [37] Transport of chloride ions
2.0 10 8.0 107
7.0 10 1.0 106
1.0 1010 2.0 106
Referent diffusion coeff., Dc,ref (m2/s) [18] Chloride binding rate coeff., kr (s1) [38]
2.0 1011 1.5 107
6.0 1011 5.0 107
2.0 1010 2.0 106
the present paper only 3 mixes with the Portland cement are investigated (see Table 1). After casting, the specimens were first kept in a fog room for six days and afterwards stored in a climate room with a constant temperature of 20 C and constant humidity of 80% for 21 days. In the experiments, only one surface of each specimen was exposed to 26 weekly cycles. One cycle consisted of 24 h exposure to 3% NaCl solution, which corresponds to a chloride concentration of 18 kg/ m3, followed by drying for 6 days under a temperature of 20 C and relative humidity of 50%. One cycle with the corresponding exposure conditions at the specimen surface is shown in Fig. 2. Chloride penetration profiles were obtained after the finished exposure of 26 wetting and drying cycles. The total chloride amount, as the percentage of the cement content, was measured in drilled increments of 2 mm up to the depth of 20 mm. The 2 mm thick concrete samples were firstly ground off and then dissolved in a hot nitric acid. Subsequently, the chloride amount was determined by using the Volhard's titration [19]. 4.2. Numerical model It is assumed that the transport processes are occurring only in
Fig. 5. Experimental and numerical results of the chloride profiles obtained after 26 weekly cycles of wetting and drying.
11
one direction, along the depth of the concrete specimen. Therefore, only one column of eight node solid elements is modelled with the total depth of 100 mm. Due to numerical stability reasons, which arise from the pronounced convective transport of chloride ions, a very fine mesh of 0.1 mm is chosen for the segment from the exposed surface up to 5 mm of depth. For the depths up to 20 mm and finally 100 mm, the element size is taken as 0.5 mm and 1 mm, respectively (Fig. 3). Due to a challenging and demanding task of simulating 26 weekly cycles, the time step size is chosen to be 3600 s, which corresponds to a total of 4368 time steps. In this way a total computational time of approximately 20 min is obtained on single processor work station, resulting in an optimized time frame needed for the calibration of the model parameters. As mentioned in the previous section, three types of concrete specimens corresponding to the w/c of 0.43, 0.48 and 0.55 are investigated. Corresponding parameters needed for the analysis of relative humidity, i.e. moisture content distribution and transport of chloride ions in the concrete specimen are given in Tab 2. The calibration is performed for all three w/c ratios separately in such a way as to obtain the best possible agreement with the experimentally measured total chloride distribution along the specimen depth of 20 mm. Information about the parameter values in the literature, which take into account the cyclic drying and wetting
Fig. 6. Numerical results of the chloride profiles obtained after 26 weekly cycles of wetting and drying for concrete specimen with w/c ¼ 0.48, with and without accounting for the hysteretic moisture behaviour of concrete.
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
conditions and the corresponding concrete quality, are relatively scarce. For instance, the values of 2.43e4.17 107, obtained by Akita et el. [37], are used as an approximate starting reference for the calibration of the surface humidity transfer coefficient, while the water vapour permeability is kept in the recommended range of 1010e1014 s [31]. Data for the main adsorption and desorption isotherms were not measured in the experiments of Polder & Peelen [19]. Therefore, the sorption curves are chosen from the literature to approximately correspond with the investigated w/c. The curves for w/c ¼ 0.43, measured by Baroghel-Bouny [25], are used as alternative curves in the case of the specimen with w/c ¼ 0.42 (Fig. 4a). Similarly, the curves obtained by Hansen [22] for w/c of 0.48 and 0.55 are taken as input curves for the w/c ratio of 0.47 and 0.57, respectively (Fig. 4b and c). The initial water content is determined from the corresponding desorption curve, with the assumption that the initial relative humidity in the specimen is constant and equal to 80%. The initial chloride concentrations are taken as 0 kg/m3 in all three examples. The temperature is kept constant throughout the analysis and equal to 20 C. The boundary conditions are defined for the nodes on the exposed surface (see Fig. 3). Same as in the experiment, a total number of 26 weekly cycles with wetting and drying phases are simulated. The varying values of the relative humidity and chloride concentration for one cycle are taken as shown in Fig. 2. For calculating the distribution of relative humidity, the mixed
79
boundary conditions are used:
n$jw;vol ¼ bhum ðhs henv Þ
(23)
where n is the normal to the exposed surface, bhum is the surface moisture transfer coefficient (m/s, see Table 2), hs is the value of the relative moisture at the exposed concrete surface and henv is the given relative moisture of the ambient air. Since the transport process of chloride ions is treated as a symmetric problem, the same boundary conditions for both exposed surfaces of the specimen were specified. In the wetting phase of the cycle, during which time the specimen is exposed to the 3% NaCl solution, the Dirichlet boundary conditions are used for the calculation of the transport of chlorides. During the drying period, since chlorides will remain in the concrete, the exposed surfaces is treated as an insulated area with the flux of chloride ions being equal to zero at the exposed boundary nodes. 4.3. Comparison between numerically and experimentally obtained chloride profiles The distribution of the total chloride content along the specimen depth for all three investigated concrete mixes is shown in Fig. 5. The total chloride amount, containing the free chloride ions originally expressed in kg/m3 of pore solution and bound chloride ions in kg/m3 of concrete, is presented as a percentage of the cement content. The corresponding values of the cement amount in the
Fig. 7. Numerical results for the relative humidity and chloride profiles obtained after the wetting phase for the 5th, 10th, 15th and 26th cycle in the case of concrete specimens with w/c ¼ 0.48 (a) and 0.43 (b).
80
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
mixes are given in Table 1. It can be seen that the experimental and numerical results give a characteristic peak of the chloride ions near the exposed surface. The numerical results indicate that the exposure to wetting and drying conditions is one of the governing mechanisms for obtaining characteristic peak shapes of the chloride profiles along the concrete depth. The increase of the w/c ratio is followed not only by a
more prominent translation of the profile peak, but also leads to higher values of the total chloride content along the whole depth. This is accounted for by increasing the model parameters for the moisture and chloride transport with the w/c ratio (see Table 2). It should be noted that the position of the peaks in the case of the numerical results is closer to the concrete surface compared to the experimental data (see Fig. 5). The current analysis does not
Fig. 8. Chloride profiles and the corresponding RH distribution after the first wetting phase, for different initial RH conditions (w/c ¼ 0.48).
Fig. 9. Numerical results for the relative humidity and chloride profiles (total chloride content) obtained after the drying phase for the 5th, 10th, 15th and 26th cycle in the case of concrete specimens with w/c ¼ 0.48 (a) and 0.43 (b).
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
account for the carbonation processes occurring in the concrete, which can have a significant influence on the reduction of chloride binding properties of the hydration products. In the experiments [19] it was observed that the carbonation depths varied from 2 to 3 mm, which can reduce the bound chloride content in affected region and could therefore push the peaks further away from the exposed surface. To investigate the influence of the hysteretic moisture behaviour on the shape of the chloride profiles, the distribution of the total chloride content for the specimen with w/c ¼ 0.48 is calculated for two cases: with and without accounting for the hysteretic moisture response of concrete (see Fig. 6). For the latter case, the model parameters with the boundary and initial conditions are assumed the same as specified above (Table 2). Furthermore, the moisture content is obtained from the calculated relative humidity by using
81
only the desorption isotherm curve (Fig. 4b). Hence, the hysteretic moisture model and the subsequent scanning curves are not accounted for. It can be seen (Fig. 6) that the hysteretic moisture behaviour has a strong influence on predicting the characteristic peak profile obtained in the experiments of [19]. In the case without the hysteresis, a relatively small maximum in the chloride content is calculated near the surface, but the overall shape corresponds more to a diffusive dominant environment in which the chloride amount slowly decreases along the depth from the exposed surface. 4.4. Influence of the wetting phase on the chloride distribution To investigate the influence of the wetting conditions on the distribution of chlorides in the concrete specimen, the relative
Fig. 10. Numerical results obtained during the drying phase of the 15th cycle (w/c ¼ 0.48) for: (a) relative humidity, (b) distribution of chloride up to 20 mm depth and (c) zoom of chloride distribution close to the concrete surface, up to 5 mm depth, (d) bound chloride distribution, up to 5 mm depth.
82
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
Fig. 11. Dependence of the reduction factor f1(h) on the relative humidity.
humidity and the corresponding chloride profiles are shown in Fig. 7 for w/c ¼ 0.48 and 0.43, respectively. The profiles are presented for the 5th, 10th, 15th and 26th cycles after the finished wetting phase and up to the depth of 20 mm. It can be seen that during the wetting phase, the chlorides are pushed inwards in both cases. Subsequently, they are added to the existing amount of chlorides near the exposed surface from the
previous cycles, which leads to an appearance of a peak in the profile shape for the type w/c ¼ 0.48 (Fig. 7a). Also the influence of the relative humidity gradient becomes higher in later cycles, causing larger amounts of ions to be transported inside the specimen. In the case of w/c ¼ 0.43, the appearance of the peak is not pronounced during the wetting phase at later cycles, which can be explained by the lower value of the water vapour permeability corresponding to a lower porosity of this concrete. From Eq. (17) it can be seen that the values of the permeability coefficient strongly influence the convective part of the chloride transport and therefore the ingress of chlorides. The distribution of ions in the drying phase is also a significant factor for the appearance of the peak during the wetting stage and it is addressed below. The extent of the chloride ingress during the wetting phase is highly dependent on the convective component of the transport process. To illustrate the influence of the relative humidity gradient, the chloride profiles for w/c ¼ 0.48 are shown in Fig. 8 after first 24 h of exposure to the NaCl solution. Five different initial relative humidities (60%, 70%, 80%, 90% and 99% RH) are assumed and the corresponding RH distribution after wetting is plotted in Fig. 8. It can be seen that in the case of the dominant diffusive conditions, i.e. for the initial RH of 99%, chlorides are penetrating the concrete in a less progressive manner compared to the lower initial humidities. With the increase of the relative humidity gradient, the
Fig. 12. Scanning curves during the 26 weekly cycles of wetting and drying at the depth of 10 mm for specimen types w/c ¼ 0.43 (a), 0.48 (b) and 0.55 (c).
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
chlorides are entering through the cover in a much larger extent and the profile shapes are changing from a concave to an S shape curve. However, note that according to the used mathematical model for the transport of chlorides all of the curves are still of “Fickian” type. 4.5. Influence of the drying phase on the chloride distribution The relative humidity and the corresponding chloride profiles after the finished drying phase for the 5th, 10th, 15th and 26th cycle are shown in Fig. 9. The profiles are plotted up to the depth of 20 mm and for w/c ratios of 0.48 and 0.43. During drying of the exposed surface, which lasts for 6 days, the relative humidity is decreasing. The decreasing value is dependent on the surface humidity transfer coefficient and the water vapour permeability. To better understand the influence of drying on the chloride distribution, the relative humidity and chloride profiles are shown in Fig. 10 for different stages of the drying phase during the 15th cycle (w/c ¼ 0.48). From Fig. 10c can be seen that at the beginning of the drying phase, the chlorides are still being transported inside the concrete specimen with the relative humidity gradient oriented away from the exposed surface (Fig. 10a). The RH in the boundary nodes is slowly decreasing with the ongoing drying. From 5 to 50 h after the start of drying, it can be observed that the gradient of RH, which is now oriented towards the surface, is relatively small as the values
83
along the depth are following the decreasing trend on the surface. The diffusion contribution to the total chloride flux has a stronger influence during this period and in this way the chlorides and the profile peak are pushed even further inside (see Fig. 10c). The relative humidity gradient, which is oriented towards the exposed surface is increasing in the later phase of drying, but in the concrete region with the values of RH being under 80% (Fig. 10a). The relatively low RH affects not only the convective, but also the diffusive flux as shown in Eqs. (17) and (19). By plotting the values of the reduction factor f1(h), it can be seen that the value of the vapour permeability and the chloride diffusion coefficient starts to significantly decrease after 90% RH and reaches only the half of its value at 75% relative humidity (Fig. 11). Hence, the total flux of chloride ions near the surface is reduced compared to the diffusion dominated flux at higher depths. Comparing the relative humidity distribution at the end of the drying phases for w/c ¼ 0.48 and 0.43 (see Fig. 9), it can be seen that the value of 75% RH (near the exposed surface) is reached more rapidly with the increase of w/c. Therefore, the migration of the chloride ions in this region is more pronounced if w/c is higher. 4.6. Shape of the scanning curves during the wetting and drying cycles To illustrate the change of relative humidity and water content during 26 weekly cycles of wetting and drying, the scanning curves
Fig. 13. Scanning curves during the 26 weekly cycles of wetting and drying at the depth of 20 mm for specimen types w/c ¼ 0.43 (a), 0.48 (b) and 0.55 (c).
84
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
for all three w/c ratios are plotted in Fig. 12. The water content is expressed in terms of mass water content and the numerical data is taken for the depth of 10 mm from the exposed surface. Fig. 12 illustrates that for the concrete mixes with a lower w/c ratio, the scanning curves are covering a relatively small RH range. As the values of water vapour permeability and surface humidity coefficient increase with a higher w/c ratio, the scanning curves are stretched over a larger relative humidity range. Furthermore, the corresponding total change of the mass water content during 26 cycles is also increasing with the w/c ratio. It can be concluded that with a higher w/c ratio, the moisture hysteretic behaviour of concrete (the scanning loops) is more pronounced with the exposure to cyclic wetting and drying conditions. Therefore, the position of the chloride profile peak gets translated further away from the surface as the w/c ratio increases (Fig. 5). It is important to note that the scanning loops become less prominent with the depth (see Fig. 13) as the interior of the concrete experiences lower swings in RH than that near the exposed surface. 5. Summary and conclusions The hysteretic moisture model for concrete, as a further extension of the 3D chemo-hygro-thermo-mechanical model, is validated on the example of chloride distribution in concrete exposed to wetting-drying conditions. The experimental tests performed by Polder & Peelen [19] are simulated and additional parametric studies are carried out. It is shown that the predicted chloride profiles for three different concrete mixtures show a good agreement with the experimental data and therefore validate, the model which realistically accounts for the convective flux of chloride ions due to the moisture transport. The profiles are characterized with a peak, which is translated away from the exposed concrete surface. The appearance of the profile maximum of the total chloride content is explained by a complex interaction of the cyclic wetting and drying conditions and can be realistically predicted by taking the hysteretic moisture behaviour of concrete into account. It is shown that the hysteretic behaviour is more pronounced with the increase of w/c ratio, which results in a more prominent translation of the profile peak further away from the surface and in the higher values of the total chloride content along the specimen depth. Acknowledgement The authors are grateful for the financial supports of “Deutsche Forschung Gemeinschaft” (DFG, Grant Nr. 601984) and “Croatian Science Foundation” (HRZZ, Grant Nr. 9068). References [1] Z.P. Ba zant, Physical model for steel corrosion in concrete sea structures e theory, J. Struct. Div-ASCE 105 (6) (1979) 1137e1153. [2] C.L. Page, N.R. Short, A. El Tarras, Diffusion of chloride ions in hardened cement pastes, Cem. Concr. Res. 11 (3) (1981) 395e406. [3] K. Tutti, Corrosion of Steel in Concrete, Technical report. Stockholm, 1993. [4] G. Balabani c, N. Bi cani c, A. Ðurekovi c, The influence of w/c ratio, concrete cover thickness and degree of water saturation on the corrosion rate of reinforcing steel in concrete, Cem. Concr. Res. 26 (5) (1996) 761e769. [5] G. Balabani c, N. Bi cani c, A. Durekovi c, Mathematical modeling of electrochemical steel corrosion in concrete, J. Eng. Mech. 122 (12) (1996) 1113e1122. [6] T. Zhang, O.E. Gjørv, Diffusion behavior of chloride ions in concrete, Cem. Concr. Res. 26 (6) (1996) 907e917. [7] C. Andrade, J.M. Díez, C. Alonso, Mathematical modeling of a concrete surface “skin effect” on diffusion in chloride contaminated media, Adv. Cem. Based
Mater. 6 (2) (1997) 39e44. rez, Service Life Modelling of RC Highway Structures Exposed to [8] B. Martín e Pe Chlorides, Dissertation. Toronto, 1999. [9] M.D.A. Thomas, P.B. Bamforth, Modelling chloride diffusion in concrete, Cem. Concr. Res. 29 (4) (1999) 487e495. [10] G. Glass, N.R. Buenfeld, The influence of chloride binding on the chloride induced corrosion risk in reinforced concrete, Corros. Sci. 42 (2) (2000) 329e344. [11] T. Ishida, P.O. Iqbal, H.T.L. Anh, Modeling of chloride diffusivity coupled with non-linear binding capacity in sound and cracked concrete, Cem. Concr. Res. 39 (10) (2009) 913e923. [12] L. Marsavina, K. Audenaert, G de Schutter, N. Faur, D. Marsavina, Experimental and numerical determination of the chloride penetration in cracked concrete, Constr. Build. Mater. 23 (1) (2009) 264e274. [13] J. O zbolt, G. Balabani c, G. Periski c, M. Kuster, Modelling the effect of damage on transport processes in concrete, Constr. Build. Mater. 24 (9) (2010) 1638e1648. [14] J. O zbolt, G. Balabani c, M. Kuster, 3D Numerical modelling of steel corrosion in concrete structures, Corros. Sci. 53 (12) (2011) 4166e4177. [15] J. O zbolt, F. Orsani c, G. Balabani c, Modeling pull-out resistance of corroded reinforcement in concrete: coupled three-dimensional finite element model, Cem. Concr. Comp. 46 (2014) 41e45. [16] F. Orsani c, Chemo-hygro-thermo-mechanical Model for Simulation of Corrosion Induced Damage in Reinforced Concrete, 2015. Dissertation. Stuttgart. [17] A. Ababneh, F. Benboudjema, Y. Xi, Chloride penetration in non saturated concrete, J. Mater. Civ. Eng. 15 (2) (2013) 183e191. [18] A. Saetta, R. Scotta, R. Vitaliani, Analysis of chloride diffusion into partially saturated concrete, ACI Mater. J. 90 (5) (1993) 441e451. [19] R.B. Polder, W.H.A. Peelen, Characterisation of chloride transport and reinforcement corrosion in concrete under cyclic wetting and drying by electrical resistivity, Cem. Concr. Comp. 24 (5) (2002) 427e435. [20] M. Fenaux, Modelling of Chloride Transport in Non-saturated Concrete, From microscale to macroscale, Universidad Politecnica de Madrid, 2013. [21] Y. Mualem, A conceptual model of hysteresis, Water Resour. Res. 10 (3) (1974) 514e520. [22] K.K. Hansen, Sorption Isotherms e a Catalogue, Technical University Denmark, 1986. [23] C.R. Pedersen, Combined Heat and Moisture Transfer in Building Constructions, Dissertation, Lyngby, Denmark, 1990. [24] Y. Xi, Z.P. Ba zant, H.M. Jennings, Moisture diffusion in cementitious materials adsorption isotherms, Adv. Cem. Based Mater 1 (6) (1994) 248e257. [25] V. Baroghel-Bouny, Water vapour sorption experiments on hardened cementitious materials: part I: essential tool for analysis of hygral behaviour and its relation to pore structure, Cem. Concr. Res. 37 (3) (2007) 414e437. [26] H. Ranaivomanana, J. Verdier, A. Sellier, X. Bourbon, Toward a better comprehension and modeling of hysteresis cycles in the water sorptiondesorption process for cement based materials, Cem. Concr. Res. 41 (8) (2011) 817e827. [27] M.Z. Ba zant, Z.P. Ba zant, Theory of sorption hysteresis in nanoporous solids: part II molecular condensation, J. Mech. Phys. Solids 60 (9) (2012) 1660e1675. [28] H. Derluyn, D. Derome, J. Carmeliet, E. Stora, R. Barbarulo, Hysteretic moisture behavior of concrete: modeling and analysis, Cem. Concr. Res. 42 (2012) 1379e1388. [29] W. Dong, Y.O.H. Murakami, S. Suzuki, T. Tsutsumi, Influence of both stirrup spacing and anchorage performance on residual strength of corroded RC beams, J. Adv. Concr. Technol. 9 (3) (2011) 261e275. [30] J. Kwiatkowski, M. Woloszyn, J.J. Roux, Modelling of hysteresis influence on mass transfer in building materials, Build. Environ. 44 (2009) 633e642. [31] Z.P. Ba zant, M.F. Kaplan, Concrete at High Temperatures, Longman Group Limited, 1996. [32] Z.P. Ba zant, L.J. Najjar, Nonlinear water diffusion in nonsaturated concrete, Mater. Struct. 5 (1972) 3e20. [33] Z.P. Ba zant, Pore pressure, uplift and failure analysis of concrete dam, in: Proc. Symp. On Criteria & Assumptions for Numerical Analysis of Dam, University of Wales, Swansea, 1975, pp. 781e808. [34] F. Ochs, W. Heidemann, H. Müller-Steinhagen, Effective thermal conductivity of moistened insulation materials as a function of temperature, Int. J. Heat. Mass Transf. 5 (2008) 539e552. [35] J. Bear, Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Boston, 1991. [36] T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Ltd, 2001. [37] H. Akita, T. Fujiwara, Y. Ozaka, A practical procedure for the analysis of moisture transfer within concrete due to drying, Mag. Concr. Res. 49 (179) (1997) 129e137. [38] L. Tang, L.O. Nilsson, in: K. Scrivener, J.F. Young (Eds.), Accelerated Methods for Chloride Diffusivity and Their Application in Prediction for Chloride Penetration. Mechanisms of Chemical Degradation of Cement-based System, E &F N Spon, 1996.
Contents lists available at ScienceDirect
Cement and Concrete Composites journal homepage: www.elsevier.com/locate/cemconcomp
Modeling influence of hysteretic moisture behavior on distribution of chlorides in concrete Josko O zbolt a, b, *, Filip Orsani c a, Gojko Balabani cb a b
Institute of Construction Materials, University of Stuttgart, Pfaffenwaldring 4, 70560 Stuttgart, Germany Faculty of Civil Engineering, University of Rijeka, Radmile Matejcic 3, 51000 Rijeka, Croatia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 July 2015 Received in revised form 28 December 2015 Accepted 1 January 2016 Available online 9 January 2016
Aggressive environmental conditions, such as exposure to the sea climate or use of de-icing salts, can have a considerable influence on the durability of reinforced concrete structures due to corrosioninduced damage of reinforcement. Recently, the coupled 3D chemo-hygro-thermo-mechanical (CHTM) model for simulation of processes related to the chloride induced corrosion of steel reinforcement in concrete was developed. In the model, it is assumed that for wetting and drying of concrete, the transport of water is controlled by a single sorption curve. However, it is well known that concrete exhibits a hysteretic moisture behaviour, which significantly influences the distribution of moisture and chlorides. To account for the hysteretic moisture behaviour of concrete and for simulating a more realistic time and space distribution of moisture, the CHTM model was further improved. The proposed hysteretic model is implemented into a 3D finite element code and it is validated using a numerical example, which shows reasonably good agreement with the available test results. Similar to what is observed in the experimental tests, it is shown that due to the wetting and drying of the concrete surface, the peak concentration of chloride moves progressively deeper into the concrete specimen. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Reinforced concrete Corrosion Chemo-hygro-thermo-mechanical model Hysteretic moisture behaviour Microplane model Finite elements
1. Introduction Corrosion-induced damage of concrete is a major problem for durability of reinforced concrete (RC) elements exposed to severe climate conditions. The onset of corrosion can be caused by carbonation of the concrete cover or by reaching a critical concentration of free chloride ions in the vicinity of the reinforcement bar. Here presented work is focussing on the processes related to the chloride-induced corrosion. Generally, computation of corrosion current density depends upon the following physical, electrochemical and mechanical processes: (1) Transport of capillary water, oxygen and chloride through the concrete cover; (2) Immobilization of chloride in the concrete; (3) Drying and wetting of concrete and related hysteretic property of concrete; (4) Transport of OH ions through electrolyte in concrete pores; (5) Cathodic and anodic polarization; (6) Transport of corrosion products; (7) Creep of concrete around reinforcement bars and (8) Damage of
* Corresponding author. Institute of Construction Materials, University of Stuttgart, Pfaffenwaldring 4, 70560 Stuttgart, Germany. E-mail addresses: [email protected] (J. O zbolt), fi[email protected] (F. Orsani c), [email protected] (G. Balabani c). http://dx.doi.org/10.1016/j.cemconcomp.2016.01.004 0958-9465/© 2016 Elsevier Ltd. All rights reserved.
concrete due to mechanical and non-mechanical actions [1]. In partially saturated concrete, the main physical phenomenon governing transport of chlorides in concrete are diffusion in the capillary pore water and convection due to flow of the pore water as a consequence of capillary suction and diffusion. A moisture hysteretic behaviour typical for cementitious material affects the transport of water and consequently the ingress of chlorides. For more accurate prediction of chloride's distribution in the concrete cover, binding of chlorides on the pore walls also needs to be taken into account. Currently, there are a number of models in the literature that are able to simulate processes before and after depassivation of reinforcement in concrete [1e15]. A recently developed 3D chemohygro-thermo mechanical model [13e15] is able to simulate nonmechanical and mechanical processes and their interaction before and after depassivation of steel reinforcement. The model is implemented into a 3D finite element (FE) code and the results have shown that it is able to realistically replicate relevant processes before [13] and after depassivation of reinforcement [14,15]. However, in the original model, the hysteretic behaviour of concrete, which is relevant for the distribution of moisture and chlorides and consequently also for the processes related to the active
74
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
corrosion phase, was not included. Therefore, the model was recently extended to account for this hysteretic moisture response of concrete [16]. It is well known that the transport processes important for the onset and corrosion progression of steel reinforcement in concrete are highly dependent on the pore moisture content. With the change of the environmental relative humidity (RH), the relative pore pressure is correspondingly changing and hence, affecting the water content in the concrete pores. Unsaturated conditions in concrete specimens have a strong influence on the distribution of chlorides. Convective flux of chloride ions due to moisture transport becomes more dominant compared to a slower, diffusive flux. Hence, the ingress of chlorides is more pronounced compared to the chloride attack under saturated conditions [17]. Furthermore, exposure to wetting and drying cycles can result in characteristic “peak” chloride profiles along the concrete depth [18e20], with the peak value of the total chloride content being translated away from the exposed surface progressively with the wet/dry cycling. Assuming isothermal conditions, the water content can differ for the same relative humidity, depending on the dynamic moisture loading history [21e28]. For a constant air temperature and constant relative humidity, equilibrium conditions can be obtained in a concrete specimen, i.e. the pore pressure in the specimen becomes approximately the same as the RH of the ambient air. The relation between relative humidity and the experimentally measured water content is given by isotherm curves [22]. Usually, in the experiments for determination of isotherm curves, very small or thin specimens are used in order to quickly reach the equilibrium state, for a step-like increase or decrease of the ambient RH [25]. The main adsorption curve is obtained by plotting the values of the water content for each increase of RH, starting from a completely dry specimen. Similarly, starting from a fully saturated specimen and incrementally decreasing RH of the ambient air, the main desorption curve can be obtained. Depending on the loading history, porous materials such as concrete, have different moisture values for the same RH. Therefore, at a constant temperature the sorption curves differ from each other, with the position of the desorption curve always being above the adsorption isotherm (see Fig. 1). The hysteretic behaviour of cementitious material for cases where the conditions change periodically from drying to wetting is described with the so-called scanning curves (see Fig. 1). They are positioned between the main adsorption and desorption curves, which are acting as envelopes [22,23,25]. Here, the main adsorption and desorption curves are given as input data, coming from experimental tests or empirical
formulae. In the incremental transient FE analysis, the model explicitly computes the distribution of relative humidity in the concrete and then determines the moisture content based on the sorption curves. By knowing the position on the isotherm curve from a previous time step, a new distribution of the relative pore pressure can be calculated for the current boundary conditions. Subsequently, the corresponding water content is established from a new location on the isotherm curve. In the case of static moisture conditions, i.e. constant wetting or drying, the analysis is following the main adsorption or desorption curve. This is valid only under the assumption that the initial starting condition (previously wetted or dried concrete) corresponds to the boundary environmental condition (wetting or drying). In the case of dynamic moisture conditions, i.e. with the change from wetting to drying or vice versa, the scanning curves need to be employed. In the present work they are calculated empirically, using the formulae proposed by Pedersen [23]. They differ from the main adsorption or desorption curves and behave in a loop-like manner depending on the number of wetting and drying cycles. The implemented hysteretic behaviour of concrete [16] was verified by simulating several experimental tests [25,29]. The recently improved CHTM model, which includes hysteretic moisture behaviour, is here used to compute the distribution of chloride in concrete under cyclic wetting-drying conditions. In the first part of the paper, the hysteretic model, based on the above mentioned theoretical background and its implementation into the FE code are discussed. The model is then employed in a transient numerical finite element study of concrete specimens exposed to chloride attack in a dynamic moisture environment. The chloride profiles along the concrete cover depth are computed and compared with the experimental results of Polder & Peelen [19].
2. Hysteretic moisture model for concrete 2.1. Moisture transport During wetting and drying cycles, it is assumed that concrete is exposed only to changes in relative humidity, so the macroscopic pressure-driven liquid flow is not considered. Transport of moisture through the concrete is described as a vapour transport, which means that for nonsaturated concrete at uniform temperature, the moisture flux jw,mass(kg/m2s) may be expressed as [30]:
jw;mass ¼ dv ðhÞpv;sat Vh
(1)
where h is the relative humidity (dimensionless), dv(h) is the water vapour permeability (s) and pv,sat is the water vapour saturation pressure (Pa). The mass conservation condition reads:
rw
vqw ðhÞ vqw ðhÞ vh ¼ rw ¼ V$ dv ðhÞpv;sat Vh vt vh vt
(2)
where t is the time (s), rw is the density of water (kg of water/m3 of water), qw is the volume fraction of pore water (m3 of water/m3 of w concrete) and rw vq ≡x is the moisture capacity (derivative of the vh sorption isotherm). For nonsaturated concrete at temperatures that are slightly different from the reference temperature of 25 C, dv(h) can be approximately described by the expression [31]:
dv ðhÞ ¼ a0 f1 ðhÞ Fig. 1. Illustration of typical adsorption, desorption and scanning curves for a cementitious material.
(3)
in which a0 is the reference permeability at 25 C and for mature paste the values of a0 are in the range of a0 ¼ 10101014(s).
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
Function f1(h) reflects the moisture transport within the adsorbed water layers, and according to [31e33]:
1a 4 1h 1 þ 1h
r qwa ðhÞ ¼ 0:01$ con $ua ðhÞ rw (8)
1 0:18 $ln
where a z 0.05 and hc z 0.75 at 25 C. Temperature dependence of water vapour saturation pressure is given by the expression [34]:
17:08085,t
or
(4)
c
psat;v ¼ 610:8,e 234:175þt
lnðhÞ 10:214
1 1:13 $ln
ua ðhÞ ¼ 4:79$e f1 ðhÞ ¼ a þ
75
(5)
where t is the temperature in ºC. The solution of differential Eq. (2) gives the distribution of relative humidity, not the content of water in the capillary pores. As the transport of moisture in concrete is a slow process, the different phases of water in the pores of concrete remain almost in thermodynamic equilibrium at any time [32] and the content of free water can be determined using sorption isotherms that relate the relative humidity and content of evaporable water at constant temperature. However, it is known that for materials such as concrete, there are different curves that relate the relative humidity and free water in the pores during adsorption (wetting) and desorption (drying). In other words, the hysteresis loops are observed during wetting and drying cycles. Thus, in order to calculate the concrete moisture distribution more accurately, a proper mathematical model of this hysteresis phenomenon in cycles of wetting and drying is required. For a hysteretic material, such as concrete, the moisture capacity is determined by the slope of scanning curves that are always between the main adsorption and desorption isotherms. In the present model, scanning curves are defined by simple empirical model proposed by Pedersen [23]:
ud ðhÞ ¼ 4:76$e
1lnðhÞ 4:85
or
r qwd ðhÞ ¼ 0:01$ con $ud ðhÞ rw (9)
where ua and ud are the moisture content (mass percent related to dry mass of concrete) for adsorption and desorption, respectively, qwa and qwd are the volume fractions of water for adsorption and desorption, respectively, and rcon ¼ 2300 kg/m3 is the assumed concrete density. The water content during scanning between the main adsorption and desorption isotherms is determined by the following relation:
qw ¼ qw0 þ xhys $Dh
(10)
in which qw0 is the initial volume fraction of water and Dh is the difference between humidity at the time jþ1 and j at spatial point i. Note that Dh depends on the chosen time step. The hysteresis model, with the current volume fraction of pore water, in which the volume fraction of pore water is calculated from the main adsorption and desorption curves and the derivative of the sorption isotherm at a given relative humidity, has the advantage of being applicable to different sorption isotherms, i.e. for different types of concrete and different temperatures. This makes the model relatively easy to implement into a numerical framework such as the finite element method.
2.2. Chloride convective diffusion
xhys;a ¼
xhys;d ¼
0:1ðqw qwa Þ2 xd þ ðqw qwd Þ2 xa ðqwd qwa Þ2
ðqw qwa Þ2 xd þ 0:1ðqw qwd Þ2 xa ðqwd qwa Þ2
(6)
(7)
where xhys is the moisture capacity (slope of scanning curve) at a given relative humidity, qwa and qwd are the volume fraction of pore water calculated from the main adsorption and desorption curves corresponding to the current relative humidity, xa ¼ rw vqvhwa is the moisture capacity at given relative humidity calculated from main adsorption isotherm, xd ¼ rw vqvhwd is the moisture capacity at given relative humidity calculated from the main desorption isotherm, qw is current volume fraction of pore water. When the most recent moisture history shows that in the previous two time steps the water content was increasing, the function defined by Eq. (6) is used in Eq. (2). However, the function defined by Eq. (7) is used in Eq. (2) if in the previous two time steps the water content was decreasing. The two previous time steps in the numerical procedure are considered in order to assure more reliable definition of moisture history. The main adsorption and desorption isotherms for concrete (for example w/c ¼ 0.48) are described by the following expressions [22]:
The macroscopic flux of free chloride ions in partially-saturated concrete exposed to drying and wetting consists of two parts: one is due to molecular diffusion of chlorides in pore water, the other one is connected with the flow of pore water due to capillary suction and diffusion, bringing with itself dissolved chloride ions. Therefore transport of chloride ions through concrete, before depassivation of steel, will be considered as convective diffusion. The macroscopic flux of chloride jc,diff, due to diffusion is expressed as:
jc;diff ¼ Dc ðqw ; TÞVCc
(11)
where Cc is the concentration of free chloride dissolved in pore 3 water (kg Cl/m of pore solution), T is the temperature and Dc(qw,T) is the effective chloride diffusion coefficient (m2/s). The macroscopic flux of chloride jc,conv, due to moisture transport is expressed as:
jc;conv ¼ Cc vw
(12)
in which vw is the mean water velocity. Thus, the total mass flux of chloride ions, jc, takes the form:
jc ¼ Cc vw Dc ðqw ; TÞVCc
(13)
Due to the fact that part of the chlorides can be bound into cement hydration products, the balance equation for chloride ions in concrete is given by Ref. [35]:
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
76
vðqw Cc Þ vC ¼ V$ðqw jc Þ cb vt vt
(14)
3 where Ccb is the concentration of bound chloride (kg Cl/m of concrete). Inserting Eq. (13) in the balance Eq. (14) it follows:
h i vC vðqw Cc Þ ¼ V$ Cc jw;vol qw Dc ðqw ; TÞVCc cb vt vt
3. Numerical implementation
(15)
where jw;vol ¼ qw vw ¼ dvrðhÞ pv;sat Vh denotes specific water discharge w (¼volume of pore water per unit area of concrete per unit time). Under the assumption that rw ¼ const, the mass balance for pore water, without source or sink terms, can be written as
vqw ¼ V$jw;vol vt
(16)
By combining Eq. (15) and balance Eq. (16) the equation that describes transport of chloride ions through concrete before depassivation of reinforcement is obtained as:
qw
vCc ¼ vt
humidity on pore water content in concrete according to the adsorption isotherms used in our model [22], U is the activation energy of the chloride diffusion process (44.6 kJ/mol for w/c ¼ 0.5), R is the universal gas constant [8.31 103 kJ/(mol$K)] and Tref ¼ 296 K.
dv ðhÞ vC pv;sat Vh$V Cc þ V$ðqw Dc ðqw ; TÞVCc Þ cb rw vt
By using the finite elements to solve the above partial differential equations, the strong form has to be rewritten into a weak form. The weak form of the system of partial differential equations is formulated by employing the Galerkin weighted residual method as follows [36]:
½Ch
vh þ ½Kh fhg ¼ ½Fh vt
o n o C_ c þ P1Ccb C_ cb þ KCc fCc g ¼ f0g n o P2Ccb C_ cb þ P2Cc fCc g þ P3Ccb fCcb g ¼ f0g
P1Cc
n
Z ½Ch ¼
In the computations presented here, it is taken into account that there is a linear relation between concentration of bound and free chloride at a constant temperature (linear binding isotherm) and also that there is a limit value for bound chloride [18]. As chloride binding does not immediately reach the equilibrium, the binding rate should be considered and therefore the mathematical model of chloride binding mechanism by hardened cement paste is given as a linear non-equilibrium isotherm:
Dc ðqw ; TÞ ¼ Dc;ref 1 þ
½Kh ¼
#1
ð1 hc Þ4
"
U $exp R
1 1 Tref T
!#
(19) where Dc,ref is the reference value of Dc evaluated at standard conditions, hc ¼ 0.75, h(w) expresses dependence of relative
Z
T
qw ½N ½NdU;
P2Cc ¼
U
Z
T
U
P1Ccb ¼
Z
bhum ½NT ½NdG
bhum henv ½NT ½NdG
qw Dc ðqw ; TÞ½VN ½VNdU
KCc ¼
Z G
G
Z
P1Cc ¼ Z
U
vw T ½N ½NdU vh
dv ðhÞpv;sat ½VNT ½VNdU þ ½Fh ¼
ð1 hðwÞÞ4
Z U
(18)
where kr is the binding rate coefficient and a ¼ 0.7 is the slope of linear adsorption isotherm. In other words, it is assumed that equilibrium is not reached instantaneously, but rather that it is approached at a certain rate, which in general is proportional to the difference between the already-adsorbed chloride content and the chloride content in the equilibrium state. It is well known that the amount of bound chloride ions depends on many factors such as cement content, water-cement ratio, porosity, etc., and these can be included in the coefficients kr and a. For three different types of concrete, characterized by three different water-cement ratio, the values of kr are given in Table 2 and a is chosen as a constant for all types of concrete according to Saetta et al. [18]. The dependence of chloride diffusion coefficient on water content and temperature is expressed as [18]:
"
(21)
with
(17)
vCcb ¼ kr ðaCc Ccb Þ vt
(20)
U T
½N ½NdU; U
P3Ccb ¼
Z
Z
kr a½NT ½NdU
U
dv ðhÞ pv;sat Vh$½NT ½VNdU rw
P2Ccb ¼
Z
½NT ½NdU
U
kr ½NT ½NdU
U
(22) where [N] is the column matrix of shape functions, which relates relative humidity h in concrete, mass concentration of free chloride in pore water Cc, mass concentration of bound chloride in concrete Ccb with their nodal values and G is the concrete surface exposed to environmental conditions. Eqs (20) and (21) are solved by using the direct integration method of an implicit type. 4. Numerical example: distribution of chlorides in nonsaturated concrete Exposure of concrete to cyclic wetting and drying conditions has strong implications on the transport processes important for the
Table 1 Investigated concrete mixes with Portland cement [19]. Cement type
w/c
Cem. cont. (kg/m3)
Agg. cont. (kg/m3)
Water cont. (kg/m3)
CEM I Port. cement
0.42 0.47 0.57
338 318 287
1780 1790 1827
143 149 163
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
77
corrosion-induced damage. Among others, transport of chloride ions, responsible for depassivation of steel, is strongly influenced by the hysteretic moisture behaviour of concrete. In such cases, distribution of chloride ions along the concrete depth exhibits characteristic “peak” profile shapes. Due to a pronounced moisture gradient, convective transport of ions is activated and at certain times it is even more dominant than diffusion. Therefore, experiments in which the chloride profiles were measured in concrete under cyclic moisture conditions represent a valuable example to validate the numerical hysteretic moisture model. To verify the here presented model, the experiments carried out by Polder & Peelen [19] are numerically simulated. 4.1. Experimental tests of Polder & Peelen (2002)
Fig. 2. Change of relative humidity at a constant temperature of 20 C for one exposure cycle in experiments by Polder & Peelen [19].
In experiments carried out by Polder & Peelen [19], the tested concrete specimens were prisms with dimensions of 100 100 300 mm, prepared with 12 different concrete mixes. In
Fig. 3. The geometry of the finite element model and discretization (all in mm).
(a)
(b)
(c)
Fig. 4. Main adsorption and desorption curves (input) from the experiments by (a) Baroghel-Bouny [25] and (b,c) Hansen [22] used for concrete specimens with the w/c ¼ 0.43, 0.48 and 0.55, respectively.
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
78
Table 2 Used model parameters for the numerical analysis of the relative humidity and chloride ions distribution. Hysteretic moisture model for concrete Sorption isotherms
w/c ¼ 0.43 [25] 11
w/c ¼ 0.48 [22]
w/c ¼ 0.55 [22]
Water vapour permeability, dv (s) [31] Surface humidity transf. coeff., bhum (m/s) [37] Transport of chloride ions
2.0 10 8.0 107
7.0 10 1.0 106
1.0 1010 2.0 106
Referent diffusion coeff., Dc,ref (m2/s) [18] Chloride binding rate coeff., kr (s1) [38]
2.0 1011 1.5 107
6.0 1011 5.0 107
2.0 1010 2.0 106
the present paper only 3 mixes with the Portland cement are investigated (see Table 1). After casting, the specimens were first kept in a fog room for six days and afterwards stored in a climate room with a constant temperature of 20 C and constant humidity of 80% for 21 days. In the experiments, only one surface of each specimen was exposed to 26 weekly cycles. One cycle consisted of 24 h exposure to 3% NaCl solution, which corresponds to a chloride concentration of 18 kg/ m3, followed by drying for 6 days under a temperature of 20 C and relative humidity of 50%. One cycle with the corresponding exposure conditions at the specimen surface is shown in Fig. 2. Chloride penetration profiles were obtained after the finished exposure of 26 wetting and drying cycles. The total chloride amount, as the percentage of the cement content, was measured in drilled increments of 2 mm up to the depth of 20 mm. The 2 mm thick concrete samples were firstly ground off and then dissolved in a hot nitric acid. Subsequently, the chloride amount was determined by using the Volhard's titration [19]. 4.2. Numerical model It is assumed that the transport processes are occurring only in
Fig. 5. Experimental and numerical results of the chloride profiles obtained after 26 weekly cycles of wetting and drying.
11
one direction, along the depth of the concrete specimen. Therefore, only one column of eight node solid elements is modelled with the total depth of 100 mm. Due to numerical stability reasons, which arise from the pronounced convective transport of chloride ions, a very fine mesh of 0.1 mm is chosen for the segment from the exposed surface up to 5 mm of depth. For the depths up to 20 mm and finally 100 mm, the element size is taken as 0.5 mm and 1 mm, respectively (Fig. 3). Due to a challenging and demanding task of simulating 26 weekly cycles, the time step size is chosen to be 3600 s, which corresponds to a total of 4368 time steps. In this way a total computational time of approximately 20 min is obtained on single processor work station, resulting in an optimized time frame needed for the calibration of the model parameters. As mentioned in the previous section, three types of concrete specimens corresponding to the w/c of 0.43, 0.48 and 0.55 are investigated. Corresponding parameters needed for the analysis of relative humidity, i.e. moisture content distribution and transport of chloride ions in the concrete specimen are given in Tab 2. The calibration is performed for all three w/c ratios separately in such a way as to obtain the best possible agreement with the experimentally measured total chloride distribution along the specimen depth of 20 mm. Information about the parameter values in the literature, which take into account the cyclic drying and wetting
Fig. 6. Numerical results of the chloride profiles obtained after 26 weekly cycles of wetting and drying for concrete specimen with w/c ¼ 0.48, with and without accounting for the hysteretic moisture behaviour of concrete.
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
conditions and the corresponding concrete quality, are relatively scarce. For instance, the values of 2.43e4.17 107, obtained by Akita et el. [37], are used as an approximate starting reference for the calibration of the surface humidity transfer coefficient, while the water vapour permeability is kept in the recommended range of 1010e1014 s [31]. Data for the main adsorption and desorption isotherms were not measured in the experiments of Polder & Peelen [19]. Therefore, the sorption curves are chosen from the literature to approximately correspond with the investigated w/c. The curves for w/c ¼ 0.43, measured by Baroghel-Bouny [25], are used as alternative curves in the case of the specimen with w/c ¼ 0.42 (Fig. 4a). Similarly, the curves obtained by Hansen [22] for w/c of 0.48 and 0.55 are taken as input curves for the w/c ratio of 0.47 and 0.57, respectively (Fig. 4b and c). The initial water content is determined from the corresponding desorption curve, with the assumption that the initial relative humidity in the specimen is constant and equal to 80%. The initial chloride concentrations are taken as 0 kg/m3 in all three examples. The temperature is kept constant throughout the analysis and equal to 20 C. The boundary conditions are defined for the nodes on the exposed surface (see Fig. 3). Same as in the experiment, a total number of 26 weekly cycles with wetting and drying phases are simulated. The varying values of the relative humidity and chloride concentration for one cycle are taken as shown in Fig. 2. For calculating the distribution of relative humidity, the mixed
79
boundary conditions are used:
n$jw;vol ¼ bhum ðhs henv Þ
(23)
where n is the normal to the exposed surface, bhum is the surface moisture transfer coefficient (m/s, see Table 2), hs is the value of the relative moisture at the exposed concrete surface and henv is the given relative moisture of the ambient air. Since the transport process of chloride ions is treated as a symmetric problem, the same boundary conditions for both exposed surfaces of the specimen were specified. In the wetting phase of the cycle, during which time the specimen is exposed to the 3% NaCl solution, the Dirichlet boundary conditions are used for the calculation of the transport of chlorides. During the drying period, since chlorides will remain in the concrete, the exposed surfaces is treated as an insulated area with the flux of chloride ions being equal to zero at the exposed boundary nodes. 4.3. Comparison between numerically and experimentally obtained chloride profiles The distribution of the total chloride content along the specimen depth for all three investigated concrete mixes is shown in Fig. 5. The total chloride amount, containing the free chloride ions originally expressed in kg/m3 of pore solution and bound chloride ions in kg/m3 of concrete, is presented as a percentage of the cement content. The corresponding values of the cement amount in the
Fig. 7. Numerical results for the relative humidity and chloride profiles obtained after the wetting phase for the 5th, 10th, 15th and 26th cycle in the case of concrete specimens with w/c ¼ 0.48 (a) and 0.43 (b).
80
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
mixes are given in Table 1. It can be seen that the experimental and numerical results give a characteristic peak of the chloride ions near the exposed surface. The numerical results indicate that the exposure to wetting and drying conditions is one of the governing mechanisms for obtaining characteristic peak shapes of the chloride profiles along the concrete depth. The increase of the w/c ratio is followed not only by a
more prominent translation of the profile peak, but also leads to higher values of the total chloride content along the whole depth. This is accounted for by increasing the model parameters for the moisture and chloride transport with the w/c ratio (see Table 2). It should be noted that the position of the peaks in the case of the numerical results is closer to the concrete surface compared to the experimental data (see Fig. 5). The current analysis does not
Fig. 8. Chloride profiles and the corresponding RH distribution after the first wetting phase, for different initial RH conditions (w/c ¼ 0.48).
Fig. 9. Numerical results for the relative humidity and chloride profiles (total chloride content) obtained after the drying phase for the 5th, 10th, 15th and 26th cycle in the case of concrete specimens with w/c ¼ 0.48 (a) and 0.43 (b).
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
account for the carbonation processes occurring in the concrete, which can have a significant influence on the reduction of chloride binding properties of the hydration products. In the experiments [19] it was observed that the carbonation depths varied from 2 to 3 mm, which can reduce the bound chloride content in affected region and could therefore push the peaks further away from the exposed surface. To investigate the influence of the hysteretic moisture behaviour on the shape of the chloride profiles, the distribution of the total chloride content for the specimen with w/c ¼ 0.48 is calculated for two cases: with and without accounting for the hysteretic moisture response of concrete (see Fig. 6). For the latter case, the model parameters with the boundary and initial conditions are assumed the same as specified above (Table 2). Furthermore, the moisture content is obtained from the calculated relative humidity by using
81
only the desorption isotherm curve (Fig. 4b). Hence, the hysteretic moisture model and the subsequent scanning curves are not accounted for. It can be seen (Fig. 6) that the hysteretic moisture behaviour has a strong influence on predicting the characteristic peak profile obtained in the experiments of [19]. In the case without the hysteresis, a relatively small maximum in the chloride content is calculated near the surface, but the overall shape corresponds more to a diffusive dominant environment in which the chloride amount slowly decreases along the depth from the exposed surface. 4.4. Influence of the wetting phase on the chloride distribution To investigate the influence of the wetting conditions on the distribution of chlorides in the concrete specimen, the relative
Fig. 10. Numerical results obtained during the drying phase of the 15th cycle (w/c ¼ 0.48) for: (a) relative humidity, (b) distribution of chloride up to 20 mm depth and (c) zoom of chloride distribution close to the concrete surface, up to 5 mm depth, (d) bound chloride distribution, up to 5 mm depth.
82
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
Fig. 11. Dependence of the reduction factor f1(h) on the relative humidity.
humidity and the corresponding chloride profiles are shown in Fig. 7 for w/c ¼ 0.48 and 0.43, respectively. The profiles are presented for the 5th, 10th, 15th and 26th cycles after the finished wetting phase and up to the depth of 20 mm. It can be seen that during the wetting phase, the chlorides are pushed inwards in both cases. Subsequently, they are added to the existing amount of chlorides near the exposed surface from the
previous cycles, which leads to an appearance of a peak in the profile shape for the type w/c ¼ 0.48 (Fig. 7a). Also the influence of the relative humidity gradient becomes higher in later cycles, causing larger amounts of ions to be transported inside the specimen. In the case of w/c ¼ 0.43, the appearance of the peak is not pronounced during the wetting phase at later cycles, which can be explained by the lower value of the water vapour permeability corresponding to a lower porosity of this concrete. From Eq. (17) it can be seen that the values of the permeability coefficient strongly influence the convective part of the chloride transport and therefore the ingress of chlorides. The distribution of ions in the drying phase is also a significant factor for the appearance of the peak during the wetting stage and it is addressed below. The extent of the chloride ingress during the wetting phase is highly dependent on the convective component of the transport process. To illustrate the influence of the relative humidity gradient, the chloride profiles for w/c ¼ 0.48 are shown in Fig. 8 after first 24 h of exposure to the NaCl solution. Five different initial relative humidities (60%, 70%, 80%, 90% and 99% RH) are assumed and the corresponding RH distribution after wetting is plotted in Fig. 8. It can be seen that in the case of the dominant diffusive conditions, i.e. for the initial RH of 99%, chlorides are penetrating the concrete in a less progressive manner compared to the lower initial humidities. With the increase of the relative humidity gradient, the
Fig. 12. Scanning curves during the 26 weekly cycles of wetting and drying at the depth of 10 mm for specimen types w/c ¼ 0.43 (a), 0.48 (b) and 0.55 (c).
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
chlorides are entering through the cover in a much larger extent and the profile shapes are changing from a concave to an S shape curve. However, note that according to the used mathematical model for the transport of chlorides all of the curves are still of “Fickian” type. 4.5. Influence of the drying phase on the chloride distribution The relative humidity and the corresponding chloride profiles after the finished drying phase for the 5th, 10th, 15th and 26th cycle are shown in Fig. 9. The profiles are plotted up to the depth of 20 mm and for w/c ratios of 0.48 and 0.43. During drying of the exposed surface, which lasts for 6 days, the relative humidity is decreasing. The decreasing value is dependent on the surface humidity transfer coefficient and the water vapour permeability. To better understand the influence of drying on the chloride distribution, the relative humidity and chloride profiles are shown in Fig. 10 for different stages of the drying phase during the 15th cycle (w/c ¼ 0.48). From Fig. 10c can be seen that at the beginning of the drying phase, the chlorides are still being transported inside the concrete specimen with the relative humidity gradient oriented away from the exposed surface (Fig. 10a). The RH in the boundary nodes is slowly decreasing with the ongoing drying. From 5 to 50 h after the start of drying, it can be observed that the gradient of RH, which is now oriented towards the surface, is relatively small as the values
83
along the depth are following the decreasing trend on the surface. The diffusion contribution to the total chloride flux has a stronger influence during this period and in this way the chlorides and the profile peak are pushed even further inside (see Fig. 10c). The relative humidity gradient, which is oriented towards the exposed surface is increasing in the later phase of drying, but in the concrete region with the values of RH being under 80% (Fig. 10a). The relatively low RH affects not only the convective, but also the diffusive flux as shown in Eqs. (17) and (19). By plotting the values of the reduction factor f1(h), it can be seen that the value of the vapour permeability and the chloride diffusion coefficient starts to significantly decrease after 90% RH and reaches only the half of its value at 75% relative humidity (Fig. 11). Hence, the total flux of chloride ions near the surface is reduced compared to the diffusion dominated flux at higher depths. Comparing the relative humidity distribution at the end of the drying phases for w/c ¼ 0.48 and 0.43 (see Fig. 9), it can be seen that the value of 75% RH (near the exposed surface) is reached more rapidly with the increase of w/c. Therefore, the migration of the chloride ions in this region is more pronounced if w/c is higher. 4.6. Shape of the scanning curves during the wetting and drying cycles To illustrate the change of relative humidity and water content during 26 weekly cycles of wetting and drying, the scanning curves
Fig. 13. Scanning curves during the 26 weekly cycles of wetting and drying at the depth of 20 mm for specimen types w/c ¼ 0.43 (a), 0.48 (b) and 0.55 (c).
84
J. Ozbolt et al. / Cement and Concrete Composites 67 (2016) 73e84
for all three w/c ratios are plotted in Fig. 12. The water content is expressed in terms of mass water content and the numerical data is taken for the depth of 10 mm from the exposed surface. Fig. 12 illustrates that for the concrete mixes with a lower w/c ratio, the scanning curves are covering a relatively small RH range. As the values of water vapour permeability and surface humidity coefficient increase with a higher w/c ratio, the scanning curves are stretched over a larger relative humidity range. Furthermore, the corresponding total change of the mass water content during 26 cycles is also increasing with the w/c ratio. It can be concluded that with a higher w/c ratio, the moisture hysteretic behaviour of concrete (the scanning loops) is more pronounced with the exposure to cyclic wetting and drying conditions. Therefore, the position of the chloride profile peak gets translated further away from the surface as the w/c ratio increases (Fig. 5). It is important to note that the scanning loops become less prominent with the depth (see Fig. 13) as the interior of the concrete experiences lower swings in RH than that near the exposed surface. 5. Summary and conclusions The hysteretic moisture model for concrete, as a further extension of the 3D chemo-hygro-thermo-mechanical model, is validated on the example of chloride distribution in concrete exposed to wetting-drying conditions. The experimental tests performed by Polder & Peelen [19] are simulated and additional parametric studies are carried out. It is shown that the predicted chloride profiles for three different concrete mixtures show a good agreement with the experimental data and therefore validate, the model which realistically accounts for the convective flux of chloride ions due to the moisture transport. The profiles are characterized with a peak, which is translated away from the exposed concrete surface. The appearance of the profile maximum of the total chloride content is explained by a complex interaction of the cyclic wetting and drying conditions and can be realistically predicted by taking the hysteretic moisture behaviour of concrete into account. It is shown that the hysteretic behaviour is more pronounced with the increase of w/c ratio, which results in a more prominent translation of the profile peak further away from the surface and in the higher values of the total chloride content along the specimen depth. Acknowledgement The authors are grateful for the financial supports of “Deutsche Forschung Gemeinschaft” (DFG, Grant Nr. 601984) and “Croatian Science Foundation” (HRZZ, Grant Nr. 9068). References [1] Z.P. Ba zant, Physical model for steel corrosion in concrete sea structures e theory, J. Struct. Div-ASCE 105 (6) (1979) 1137e1153. [2] C.L. Page, N.R. Short, A. El Tarras, Diffusion of chloride ions in hardened cement pastes, Cem. Concr. Res. 11 (3) (1981) 395e406. [3] K. Tutti, Corrosion of Steel in Concrete, Technical report. Stockholm, 1993. [4] G. Balabani c, N. Bi cani c, A. Ðurekovi c, The influence of w/c ratio, concrete cover thickness and degree of water saturation on the corrosion rate of reinforcing steel in concrete, Cem. Concr. Res. 26 (5) (1996) 761e769. [5] G. Balabani c, N. Bi cani c, A. Durekovi c, Mathematical modeling of electrochemical steel corrosion in concrete, J. Eng. Mech. 122 (12) (1996) 1113e1122. [6] T. Zhang, O.E. Gjørv, Diffusion behavior of chloride ions in concrete, Cem. Concr. Res. 26 (6) (1996) 907e917. [7] C. Andrade, J.M. Díez, C. Alonso, Mathematical modeling of a concrete surface “skin effect” on diffusion in chloride contaminated media, Adv. Cem. Based
Mater. 6 (2) (1997) 39e44. rez, Service Life Modelling of RC Highway Structures Exposed to [8] B. Martín e Pe Chlorides, Dissertation. Toronto, 1999. [9] M.D.A. Thomas, P.B. Bamforth, Modelling chloride diffusion in concrete, Cem. Concr. Res. 29 (4) (1999) 487e495. [10] G. Glass, N.R. Buenfeld, The influence of chloride binding on the chloride induced corrosion risk in reinforced concrete, Corros. Sci. 42 (2) (2000) 329e344. [11] T. Ishida, P.O. Iqbal, H.T.L. Anh, Modeling of chloride diffusivity coupled with non-linear binding capacity in sound and cracked concrete, Cem. Concr. Res. 39 (10) (2009) 913e923. [12] L. Marsavina, K. Audenaert, G de Schutter, N. Faur, D. Marsavina, Experimental and numerical determination of the chloride penetration in cracked concrete, Constr. Build. Mater. 23 (1) (2009) 264e274. [13] J. O zbolt, G. Balabani c, G. Periski c, M. Kuster, Modelling the effect of damage on transport processes in concrete, Constr. Build. Mater. 24 (9) (2010) 1638e1648. [14] J. O zbolt, G. Balabani c, M. Kuster, 3D Numerical modelling of steel corrosion in concrete structures, Corros. Sci. 53 (12) (2011) 4166e4177. [15] J. O zbolt, F. Orsani c, G. Balabani c, Modeling pull-out resistance of corroded reinforcement in concrete: coupled three-dimensional finite element model, Cem. Concr. Comp. 46 (2014) 41e45. [16] F. Orsani c, Chemo-hygro-thermo-mechanical Model for Simulation of Corrosion Induced Damage in Reinforced Concrete, 2015. Dissertation. Stuttgart. [17] A. Ababneh, F. Benboudjema, Y. Xi, Chloride penetration in non saturated concrete, J. Mater. Civ. Eng. 15 (2) (2013) 183e191. [18] A. Saetta, R. Scotta, R. Vitaliani, Analysis of chloride diffusion into partially saturated concrete, ACI Mater. J. 90 (5) (1993) 441e451. [19] R.B. Polder, W.H.A. Peelen, Characterisation of chloride transport and reinforcement corrosion in concrete under cyclic wetting and drying by electrical resistivity, Cem. Concr. Comp. 24 (5) (2002) 427e435. [20] M. Fenaux, Modelling of Chloride Transport in Non-saturated Concrete, From microscale to macroscale, Universidad Politecnica de Madrid, 2013. [21] Y. Mualem, A conceptual model of hysteresis, Water Resour. Res. 10 (3) (1974) 514e520. [22] K.K. Hansen, Sorption Isotherms e a Catalogue, Technical University Denmark, 1986. [23] C.R. Pedersen, Combined Heat and Moisture Transfer in Building Constructions, Dissertation, Lyngby, Denmark, 1990. [24] Y. Xi, Z.P. Ba zant, H.M. Jennings, Moisture diffusion in cementitious materials adsorption isotherms, Adv. Cem. Based Mater 1 (6) (1994) 248e257. [25] V. Baroghel-Bouny, Water vapour sorption experiments on hardened cementitious materials: part I: essential tool for analysis of hygral behaviour and its relation to pore structure, Cem. Concr. Res. 37 (3) (2007) 414e437. [26] H. Ranaivomanana, J. Verdier, A. Sellier, X. Bourbon, Toward a better comprehension and modeling of hysteresis cycles in the water sorptiondesorption process for cement based materials, Cem. Concr. Res. 41 (8) (2011) 817e827. [27] M.Z. Ba zant, Z.P. Ba zant, Theory of sorption hysteresis in nanoporous solids: part II molecular condensation, J. Mech. Phys. Solids 60 (9) (2012) 1660e1675. [28] H. Derluyn, D. Derome, J. Carmeliet, E. Stora, R. Barbarulo, Hysteretic moisture behavior of concrete: modeling and analysis, Cem. Concr. Res. 42 (2012) 1379e1388. [29] W. Dong, Y.O.H. Murakami, S. Suzuki, T. Tsutsumi, Influence of both stirrup spacing and anchorage performance on residual strength of corroded RC beams, J. Adv. Concr. Technol. 9 (3) (2011) 261e275. [30] J. Kwiatkowski, M. Woloszyn, J.J. Roux, Modelling of hysteresis influence on mass transfer in building materials, Build. Environ. 44 (2009) 633e642. [31] Z.P. Ba zant, M.F. Kaplan, Concrete at High Temperatures, Longman Group Limited, 1996. [32] Z.P. Ba zant, L.J. Najjar, Nonlinear water diffusion in nonsaturated concrete, Mater. Struct. 5 (1972) 3e20. [33] Z.P. Ba zant, Pore pressure, uplift and failure analysis of concrete dam, in: Proc. Symp. On Criteria & Assumptions for Numerical Analysis of Dam, University of Wales, Swansea, 1975, pp. 781e808. [34] F. Ochs, W. Heidemann, H. Müller-Steinhagen, Effective thermal conductivity of moistened insulation materials as a function of temperature, Int. J. Heat. Mass Transf. 5 (2008) 539e552. [35] J. Bear, Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Boston, 1991. [36] T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Ltd, 2001. [37] H. Akita, T. Fujiwara, Y. Ozaka, A practical procedure for the analysis of moisture transfer within concrete due to drying, Mag. Concr. Res. 49 (179) (1997) 129e137. [38] L. Tang, L.O. Nilsson, in: K. Scrivener, J.F. Young (Eds.), Accelerated Methods for Chloride Diffusivity and Their Application in Prediction for Chloride Penetration. Mechanisms of Chemical Degradation of Cement-based System, E &F N Spon, 1996.