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Model of chloride penetration into cracked concrete subject to drying–wetting cycles
Construction and Building Materials 36 (2012) 259–269
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Review
Model of chloride penetration into cracked concrete subject to drying–wetting cycles Hailong Ye, Nanguo Jin ⇑, Xianyu Jin, Chuanqing Fu Department of Civil Engineering, Zhejiang University, 388 Yuhangtang Road, Hangzhou 310058, PR China
h i g h l i g h t s " Moisture transport considers crack width, roughness, tortuosity and capillary pores. " Drag coefficient is applied to calculate the roughness. " Parameters of crack characteristic are studied in the cracked zone. " Simplified method to calculate profiles of chloride concentration in cracked zone.
a r t i c l e
i n f o
Article history: Received 3 March 2012 Received in revised form 12 May 2012 Accepted 15 May 2012 Available online 23 June 2012 Keywords: Drying and wetting cycle Chloride ions Crack Cracked zone Drag coefficient
a b s t r a c t Drying and wetting cycles condition is always identified as the most unfavorable environment condition for reinforce concrete structure subjected to chloride-induced deterioration processes. When a crack occurs, it accelerates the ingress of chloride ions and affects durability. In order to clarify the mechanism of deterioration subjected cyclic drying–wetting condition, an innovative model describing the transport of chloride ions in cracked concrete is elaborated, meanwhile the results of experimental investigation are also reported. The transporting mechanism of chloride ions in porous medium is supposed still valid for cracked concrete. The advection part is simulated by moisture transport in rough cracks, modeled as a flux based on Poiseuille law considering crack width, crack surface roughness, tortuosity and capillary pores at crack surface. In addition, the drag coefficient is first applied to calculate the influence of roughness on crack in present work. Finally, a simplified and modified Fick’s second law is proposed here to estimate the chloride ions profiles in cracked concrete. Ó 2012 Elsevier Ltd. All rights reserved.
Contents 1. 2.
3.
4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chloride permeate in sound concrete in drying–wetting cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Moisture diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Chloride ingress in unsaturated concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Calculation the degree of saturation in concrete subjected to drying and wetting cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of chloride penetrate in the cracked zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The geometric properties of cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Calculation of the roughness based on drag coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Calculation of the tortuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Further modeling considering the capillary pores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Effective crack widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parametric studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Details of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
⇑ Corresponding author. Tel./fax: +86 0577 88208766. E-mail addresses: [email protected] (H. Ye), [email protected] (N. Jin), [email protected] (X. Jin), [email protected] (C. Fu). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.05.027
260 260 260 260 261 261 262 262 263 263 264 264 264 264 265 265 265
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Result and discussion. Conclusion . . . . . . . . . Acknowledgements . . References . . . . . . . . .
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1. Introduction Chloride-induced corrosion of steel reinforcement is a major cause affecting the service life of reinforced concrete (RC) structure exposed to marine environments. Chloride ions ingress solely by pore diffusion is a rare occurrence when structure exposed to continuous drying–wetting cycles by tidal and splash action. For chloride ions transport in sound concrete, various model and analysis technology have been proposed including the condition of cyclic drying–wetting [1]. However, cracks cannot be avoided for a real structure. Effects of cracks on the diffusion and permeation on chloride behavior have also been investigated by many researchers. It is generally recognized that the cracks accelerate the ingress of chloride ions. Raharinaivo et al. made a comparison of diffusion coefficient for cracked and uncracked concrete, showing an increase in the diffusion coefficient for cracked concrete by one or two orders of magnitude [2]; Mangat and Gurusamy studied the influence of cracks on chloride diffusion of steel fiber reinforced concrete and concluded that crack widths larger than 500 lm had more pronounced influence on chloride intrusion while crack widths less than 200 lm appeared to have nearly no effect on chloride intrusion [3]; Gérard and Marchand analyzed the influence of traversing cracks on transport of ions in saturation concrete, and found that cracks tended to markedly modify the transport coefficient of the solid by a factor ranging from 2 to 10 [4]; Gowripalan et al. used the parameter of the ratio of crack width to cover to consider in relation to the diffusion process [5]; Boulfiza et al. used a simplified smeared approach to estimate the effect of cracks on chloride ingress using Fick’s law describing chloride ingress into cracked concrete [6]; Win et al. used a series of cracked reinforced concrete specimens, and found that the specimens having cracks showed rapid penetration of chloride ions [7]; Sßahmaran found that for crack widths less than 135 lm, the effect of crack widths on the effective diffusion coefficient of mortars was marginal, whereas for higher than 135 lm the coefficient increased rapidly based on V-shaped cracks [8]; Care investigated the cracks caused by temperature and the transport properties of chloride diffusion in cement paste, finding that chloride ingress increased with heat temperature [9]; Islam and Sazzad found that the surface chloride for sound part remains unchanged with the change of cracked widths [10]; Djerbi et al. investigated the influence of traversing cracks on chloride diffusion for OC, HPC and HPCSF concrete, finding that the coefficient in cracks was independent of material effects, when for crack widths smaller than 80 lm, the coefficient seemed constant [11]; Marsavina et al. used lot of artificial cracks with different widths and depths to research the chloride penetration, showing higher penetration of chloride at tip in comparison with the uncracked part of test specimens [12]. Park et al. proposed an equivalent chloride diffusivity considering a REV model with cracks, and found that the diffusion coefficient increased with the growing of crack widths [13,14]. However, most analyses were based on the condition of chloride penetration into saturated concrete, but in reality, concrete was often found in an unsaturated condition rather than a saturation condition especially when subjected to cyclic drying– wetting. In unsaturated condition, various models were proposed always at the same assumption that the chloride ingress process is a com-
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266 269 269 269
bination of diffusion and advection [1,15]. Diffusion denotes the ionic motion of low concentration while advection refers to the movement of chloride with moisture. When subjected to cyclic drying–wetting condition, the advection part dominates the rate of chloride penetration in wetting phase, while in drying phase, the concentration of solution at exposure surface increases due to the depleting of moisture [1]. So clarifying the moisture transport in cracked concrete is the key problem. Under service condition, cracks of concrete structures cannot be avoided during whole service life, what’s more, drying–wetting cycles condition is always identified as the most unfavorable environment condition for concrete deterioration processes [16]. In present work, a model describing the chloride penetration into sound and cracked concrete is proposed subjected to cyclic drying–wetting condition based on the works of Ababneh et al. and Maekawa et al. It is assumed that the mechanisms for drying phase and wetting phase, both for sound and cracked concrete, are the same (i.e. diffusion and advection) except for the moisture and chloride diffusion coefficients. 2. Chloride permeate in sound concrete in drying–wetting cycles 2.1. Moisture diffusion As the initially unsaturated concrete is put into contact with liquid water, the water intake is dominated by capillary absorption across the surface. Then, the relative humidity at surface is higher, resulting moisture transport driven by the difference of relative humidity between surface and inner. While saturated concrete surface is exposed to a lower humidity during drying phase, vapor diffusion process dominates because the evaporation process occurs at the gas–liquid interface. A nonlinear diffusion model for both two phases are present according to a modified Dacy’ s law [17]:
J w ¼ Dh rh
ð1Þ
where h is relative humidity, Dh is humidity diffusion coefficient, which depends on the relative humidity by timing factor with a reference value at isothermal conditions [18]:
Dh ¼ Dh;ref
pffiffiffiffiffiffih pffiffiffiffiffiffim i 0
ð2Þ
where Sw represents degree of saturation with condensed water and adsorbed water, whose relative content alters in drying–wetting cycles, the computing method of Sw will be introduced later. The rate of change of the mass of water we per unit volume of concrete is determined from the flux field by the relation [17,19]:
@we ¼ rJ w @t
ð3Þ
Eq. (3) is based on the hypothesis of dissipation decoupling, disregarding possible couplings between heat and moisture transport. 2.2. Chloride ingress in unsaturated concrete As said in introduction, the flux of chloride ions permeate in concrete subjected to drying–wetting cycles takes both diffusion and advection (i.e. related to the diffusion of moisture), the diffu-
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
sion model for the free chloride in unsaturated concrete is given by [15]:
@C fr @C fr @w rðDcl rC fr Þ þ l e C fr ¼ @t @C to @t
ð4Þ
where Cto is total concentration of ions in grams of total chloride per gram of concrete; Cfr is free concentration of ions in grams of free chloride per gram of concrete; Dcl is chloride diffusivity of concrete; oCfr/oCto is binding capacity of concrete; l is unit converter to grams of chloride per gram of concrete given by Eq. (5); owe/ot is rate of change of total water content given by Eq. (3). The l as explained, is the ratio of the volume of pore solution to the weight of concrete [15], according to its definition, this parameter can be rewritten as a function of the porosity and degree of saturation as follows:
l¼
1
qsol bsol
¼
wconc V conc qconc ¼ wsol V conc /Scg qsol
ð5Þ
where qsol is density of pore solution in grams of pore solution per liter; bsol is ratio of volume of pore solution to weight of concrete; wconc and wsol are weight of the concrete and solution; qconc and qsol are density of the concrete and pore solution; / is total porosity of concrete. Substituting Eq. (5) into Eq. (4):
@C fr @C fr q ¼ r½Dcl rðC fr Þ þ r Dh conc C fr rðhÞ @t @C to /Sw qsol
ð6Þ
From Eq. (6), it can be seen that the final concentration of ions is influenced by the water stored in pores, which change according to the relative humidity in drying–wetting cycles, in another word, the relationship between relative humidity and degree of saturation must be determined necessarily when calculating the concentration of chloride ions in concrete. 2.3. Calculation the degree of saturation in concrete subjected to drying and wetting cycles As previously mentioned, Eq. (6) can be rewritten as a function of relative humidity without in the form of degree of saturation because of the corresponding relationship between them. Then, it is necessary to describe the liquid water and water vapor in pores, whose relative content just corresponds to the definition of the degree of saturation. In drying–wetting cycles, the liquid and vapor water switch each other, making a serious influence on the concentration of ions dissolved in solution. In drying and wetting cycles, sorption hysteresis and its effect on moisture transport have to be investigated, because no oneto-one relations between different state variables can be obtained, e.g. between the relative humidity and water mass concentration, it is the relationship between humidity and degree of saturation as follows [20]:
Sw ¼ f ðh; drying—wetting historyÞ
ð7Þ
There is a relationship based on the Laplace and Kelvin equations among the relative humidity h, degree of saturation Sw and the capillary pressure pc in concrete pores which are treated to be cylindrical with different radii, and it can be expressed in Eq. (8). The detailed form can be found by the research of Maekawa et al.
pc ¼
qw RT Mw
ln h ¼
2c rs
liquid–vapor interface forms are completely filled with water, whereas larger pores are partially saturation with a layer of absorbed water, the thickness ta can be calculated by using the modified B.E.T. theory. Then, in the wetting phases, the degree of saturation can be gotten by combining condensed water, determined from rc derived from the moisture equilibrium equations and pore distribution function, and adsorbed water in the unsaturated pores. The degree of saturation Sw can be calculated by [1]:
Sw ¼ f wetting ðhÞ ¼
Z
1
½1 ðr ta =rÞ2 dV þ
rc
Z
rc
dV ¼ Sa þ Sc
However, in the drying phase, there is a difference in the time elapsed before reaching the equilibrium because of the inkbottle effect mentioned as a cause of the short-term drying–wetting hysteresis. It can be inferred that the moisture trapped by the inkbottle effect gradually disperse into adjacent connected pores at the rates corresponding to ambient temperature and relative humidity, when getting balance, the degree of saturation in the drying phase can be expressed as [1]:
Sw ¼ f drying ðhÞ ¼ Sc þ
Z
1
ðSc =VÞdV ¼ Sc ½1 ln Sc
ð10Þ
rc
3. Modeling of chloride penetrate in the cracked zone In the real environment, concrete structures always have cracks, which make moisture dissolved with chloride more easily penetrates and reaches the reinforcing steel and speeds up the initiation of steel corrosion especially in cyclic drying–wetting conditions. In present work, this mechanism is assumed still suitable to the issues of chloride penetrates in cracks subjected to cyclic drying– wetting conditions. To ensure the impact of cracks on chloride ions transport more accurately, it is necessary to quantify the relationship between the crack and the associated transport properties of moisture, since advection part (i.e. transport of chloride ions with moisture flux), is one of the major cause of chloride transport in cyclic drying– wetting conditions. Hence, emphasis is laid on the transport of moisture in cracks. In present research, the advection part may kcr belong to Navier–Stokes equations, which is suitable just at the condition that the facture morphology can be treated as a straight, smoothly and parallel plate crack. Water flow rate through a crack, Qi expressed in Eq. (11) is related to the cube of crack with w according to Poiseuille law regarded as Newtonian fluids [21]. However, as shown in Fig. 2, the crack widths often vary along the flow path, as well as the visible crack length and crack surface are full of macro-roughness (i.e. tortuosity) and micro-roughness (i.e. surface roughness of aggregates and cement paste, and crack branching) [16]. So it might be too impudent to use Navier–Stokes equations directly. Many researchers had put forward an empirical reduction factor n to consider the combination of these effects, and the permeability coefficient can be expressed in Eq. (12) based on the parallel palates-shape crack model, while the empirical factor n ranges from 0.01 to 0.1 for plain and fiber reinforced concrete reported by Edvardsen and Mohr, Picandet et al. [16,22].
ð8Þ
where qw is density of liquid, Mw is molecular mass, R is gas constant, T is temperature, rs is pore radius; c is surface tension of liquid water. From Eq. (9), it can be known that a certain group of pores whose radii is smaller than the specific radius rc at which a
ð9Þ
0
Fig. 1. Network of the crack element.
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Q¼
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
w3 b rp 12g
kcr ¼ n
ð11Þ
w2 12
ð12Þ
where w, b represent the crack width and crack length, g represents the dynamic viscosity of fluid, rp is the pressure gradient that drives the flow. Unfortunately, the value of n ranges unstability and are purely determined according to experiments. Even though, Louis [21] modified Eq. (12), and further suggested an equation to estimate the permeability of a parallel palate-shape crack value in laminar flow based on the roughness of crack surface, this equation is as well a simplified empirical equation based on tests. 3.1. The geometric properties of cracks Similarly to Carmeliet et al. [23], a network representation of crack void space pattern is set up representing the tunnel of moisture transport. As shown in Fig. 1, crack path can be cut into many crack segments, nodes is located at the center of each crack segments whose area is Si = f(wi, bi), while wi, bi represent the width and length. Firstly considering the flow in the i th element. However, the influence of chloride ions attack on concrete mostly occurs at the larger crack widths, when crack widths decrease to a critical value along the direction parallel to crack segments, the influence of the crack can be neglected. In order to simplify the model, it is assumed that the crack widths wi along the crack length are constant and equal to the middle value of the segment, in addition, the estimation of crack lengths bi depends on the relevant crack widths wi which will be discussed later. 3.2. Calculation of the roughness based on drag coefficient In present work, in order to calculate the reduction factor n, it is proposed of a new method based on two principles of equivalence comparing an associative circle pipe with same geometric roughness of crack surface along the transport path. Firstly, only considering the micro-roughness flow element i, according to DacyWeisbach equations, the frictional head loss along the path can be expressed as a function of the drag coefficient along the way ki, which has been researched mostly for circle pipes. It assumes that the volume of these paths is proportional to the volume of a smooth and linear tube with the same frictional head loss over the same distances as shown in Fig. 3, for a flow transport through a smooth and liner tube, according to Poiseuille [24]:
Q i;tu ¼
pC 4i Dpij 8g Lij
ð13Þ
Fig. 3. Transform of the roughness of a crack into a circle model.
hf ;tu ¼ ki
2 2 Lij J i;tu 1 C 3i Dpij ¼ ki 2 256g g Lij 2C i 2g
ð14Þ
where Qi,tu is the flow rate through a circle tube, Ji,tu is the average velocity of tube, Ci is the radius of the tube, hf,tu represents frictional head loss, ki represents the drag coefficient along the way. Similarly, the frictional head loss along the parallel plates-shape cracks can also be expressed as follows:
hf ;cr ¼ keq
2 Lij J i;cr 4C i;eq 2g
ð15Þ
where Ji,tu is the average velocity of crack, Ci,eq is the hydraulic radius. In order to get an equivalent permeability coefficient, a drag coefficient is introduced instead of the drag coefficient along the way, thus getting keq ¼ 1; Jeq i;cr ¼ ti J i;cr Eq. (15) can be rewritten as:
hf ;cr ¼ t2
2 ðwi þ bi Þ w3i Dpij 476bi g g2 Lij
ð16Þ
The first principle of equivalence assumes that the frictional head loss in circle and crack is the same, reaching hf,cr = hf,tu, thus the equivalent drag coefficient ti can be expressed as:
ti
3 ¼ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ki bi C 3i ðwi þ bi Þw3i
ð17Þ
The second principle of equivalence is that if the crack path and the circle tube are all smooth, then conductivity-equivalent must satisfy following conditions:
bi w3i Dpij pC 4i Dpij ¼ 12g Lij 8g Lij Getting C i ¼
ð18Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2w3 b ffi 4 i i ; then submitting it into Eq. (17) and repre3p
senting bi/wi with ji, Eq. (17) can be rewritten as:
ti
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 8 j3 ji 4 i ¼ ki 4 ji þ 1
ð19Þ
Next, the relationship between the drag coefficient ki and the roughness Ri must be bridged. Crack geometry satisfies the properties of so-called self-similarity or self-affinity with respect to scale changes. For simply, in order to quantize the roughness of crack geometry, a mathematical discipline introduced by Mandelbort is used here. the roughness of element i can be determined by calculating the average height of surface asperities with respect to its reference line as shown in Fig. 4 [25]: Fig. 2. Digitized profiles of an actual crack.
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Fig. 4. Method of quantifying roughness and tortuosity.
Ri ¼
n1 1X jzi hzij n i¼1
ð20Þ
where n is the ranges of the analyzed set of the measured values, hzi is the arithmetic average height. Assuming that the drag coefficient is dominated by the roughness of cracks, the Nicholas equation is applied here [24]:
Ri 2wi
ki ¼ ð2 lg
þ 1:74Þ2
si ¼
1 n1
1 1 þ s2i
i¼1
jziþ1 zi j ðxiþ1 xi Þ
ð24Þ
i
3.4. Further modeling considering the capillary pores
Tortuosity factor takes the wavy characteristic of crack profiles into consideration, because of the effective crack length larger than the nominal crack length. The tortuosity si is quantified by transforming the average slope into a pragmatic method as shown in Fig. 4 which can be expressed as follows [25]:
si ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3 ji 2 8 ji f ðSÞ s w i i 4 ji þ1 w2i ki;cr ðSÞ ¼ f ðSÞ si ti ¼ 12 12 2 lg Ri þ 1:74 2w
ð21Þ
3.3. Calculation of the tortuosity
n1 X
of saturation in Eq. (2). Finally the equivalent coefficient in Eq. (12) can be rewritten as:
ð22Þ
ð23Þ
In the unsaturation state, similarly to modified Dacy’ s law, the equivalent is modified by timing the factor of function of degree
Along the crack surface, there exist a lot of pores as shown in Fig. 5, thus, capillary forces of the crack surface cannot be completely neglected, it assumes that the capillary pores are modeled as tiny cylinder tubes with various radii. According to Hagen Poiseuille flow, the flux in one tube can be expressed as [1]:
J i;cap ¼
ki;r
g0
rP
ð25Þ
where ki,r is the permeability, which can be calculated by R r
2 ki;ca ¼ 18 0 c rdA based on the assumption that only the capillary pores filled with water can transport chloride ions. From Eq. (8), the pores whose radii r is smaller that specific radius are completely filled with water, whereas larger pores are partially saturation with a layer of absorbed water, which somewhat contribute to the chloride transport along the crack surface as well.
Fig. 5. Cross section of concrete [33] and model of capillary pores at crack surface.
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
The area of moisture at crack section is wibi, the total moisture associated with this area can be calculated by without considering the moisture absorbed in the pore spaces,
Ai;cr ðhÞ ¼ wi bi Li;cr ð1 /i;cap Þ
ð26Þ
The area of the pore space can approximately be expressed as:
Ai;ca ¼ bi Li;cr /i;cap
ð27Þ
where /i;cap is the porosity of capillary in concrete, thus the cross sectional area of moisture should add the area of pores whose radii is smaller than rc, getting:
Aeff i;cr ðhÞ ¼ wi bi Li;cr ð1 /i;cap þ /i;cap ðSc þ Sa ÞÞ
ð28Þ
Then, the area of capillary is
Aeff i;ca ðhÞ ¼ bi Li;cr /i;cap Sc
ð29Þ
The average coefficient of moisture transport in cracks can be formulated by averaging of cross section areas.
Deff i;h ðhÞ ¼
! ki;ca /i;cap Sc þ ki;cr ðSw Þ wi ð1 /i;cap þ /i;cap ðSc þ Sa ÞÞ ql RT g0 Ml h /i;cap Sc þ wi ð1 /i;cap þ /i;cap ðSc þ Sa ÞÞ ð30Þ
If idealizing the crack as another material with different transport properties, the chloride ions transport in the cracked concrete can be modeled as a two-dimensional transport issue, in addition, the diffusion part in the crack also increases by a ratio of w than in the sound concrete. The mass balance equation can be rewritten as:
! @C fr @C fr @ 2 C fr @ 2 C fr Dcl w þ ¼ @t @C to @x2 @y2 @ @h @ qcon @h þ Deff þ C D C fr h fr @x h @x @y /Sw qsol @y
ð31Þ
It can be seen that the equivalent coefficient involves crack widths, crack lengths and roughness, then if all three parameters are known, the coefficient of every element can be calculated, however, in the most case, only the crack widths and lengths at surface can be easily measured. If assuming that the crack lengths and roughness are invariant along the path, the effective crack widths can be calculated according to Akhavan et al. [21], however, there exists some distinction because of the early simplification of treating the crack lengths depend on the relevant crack widths. Based on the same conditions of fluid continuity (Q = Q1 = Q2 = = Qi) and the total pressure loss across the path is the sum of the loss of each element (DP = DP1 + + DPn). Finally the effective crack widths can be expressed as:
weff
s
J sdrying ¼ Dh ðh henv Þ
ð35Þ
where Dh is surface moisture transfer coefficient, hs is relative humidity at surface, henv is relative humidity in environment. Then, at the progress of drying phase, the content of chloride ions compared with concrete at concrete surface stay unchangeable
ðC sfr;drying Þiþ1 ¼ ðC sfr;wetting Þi
ð36Þ
5. Numerical solution In order to calculate the chloride profiles in cracked zone, the cracked concrete is regard as the heterogenetic material in two dimensions. The alternating-direction implicit (ADI) finite-difference method [15,26] is employed here for solving the chloride transport problem. The principle of the ADI method can be applied to solve nonlinear partial differential equations, using two difference equations over two successive time steps, each of duration Dt/2, particularly, Eq. (37) is implicit in the x-direction (the direction of the cracks) and explicit in the y direction, while Eq. (38) is implicit in the y-direction and explicit in the x direction for this two-dimensions finite difference discretization and 1 mm is chosen as the mesh size both for x and y directions here.
h i C i;j C ni;j ¼ ani;j Dni;j d2x C i;j þ dx Dni;j dx C i;j þ Dni;j d2y C ni;j þdy Dni;j dy C ni;j þ #ni;j Ani;j C ni;j Dt=2 ð37Þ
3.5. Effective crack widths
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 3 Pn 3 ð1=w i¼1 iÞ
where index i refers to the number of time of drying–wetting cycles. In drying phase, the relative humidity of concrete at surface become lower gradually because of the progress of evaporation, at which the chloride ions dissolved does not following vanish, on the contrary, on the course of wetting phase to drying phase, the concentration of chloride ions on the surface increase with the evaporation until to getting balance. At the exposed boundaries, the conditions are defined as fluxes of relative humidity.
ð32Þ
nþ1 h i C i;j C i;j nþ1 n n ¼ ani;j Dni;j d2x C i;j þ dx Dni;j dx C i;j þ Dni;j d2y C i;j þdy Dni;j dy C nþ1 i;j þ #i;j Ai;j C i;j Dt=2 ð38Þ
where indictor represents the intermediate value of these two steps, dx ; d2x and dy ; d2y represent the first- and second-order central difference operators in the x and y directions, a represents the binding capacity of concrete, D represents the diffusion coefficients according to the above proposed material model, different for cracked and sound concrete, A represents the moisture transport, which also can be solved by simplified ADI method, and the corresponding coefficients are based on the proposed model, # represents the parameter used for unit converter, different for cracked and sound concrete, while it is expressed in Eq. (5) for sound concrete. The stability of the ADI method is unconditionally stable and the solution converges within a discretization error of O[(Dt)2 + (Dx)2 + (Dy)2] discussed by Neumann [27].
4. Surface condition 6. Parametric studies In wetting phase, it assumes that the relative humidity is full when contact with water, and the concentrations of free chloride ions at the exposed surface get balance soon with the water. Then, at the exposed boundaries, the conditions are defined as fluxes of chloride ions, relative humidity.
ðC sfr;wetting Þi ¼ C env s
h ¼ henv
ð33Þ ð34Þ
Through the above proposed model, it is clear that the characteristics of cracks have a significant influence on the transport of chloride in cracks. In this section, the full parametric studies are carried out on how the characteristics of cracks affect the chloride profiles. According to the model, the crack width is the most important parameter to representation the effect of cracks on chloride transport, obviously, the larger crack width is, the more chloride ions are delivered by the advection effect and the higher
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
0.3 mm crack width 0.2 mm crack width 0.1 mm crack width
0.0 0
50
Roughness 0.1 Roughness 0.01 Roughness 0.001
0.0 0
100
50
K=10 K=5 K=1
0.0
0
50
100
Depth from exposed surface (mm)
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
Depth from exposed surface (mm)
100
Dry time: Wet time=3:1 Dry time: Wet time=20:1 Dry time: Wet time=1:infinity
0.0
0
50
100
Depth from exposed surface (mm)
Depth from exposed surface (mm)
Fig. 6. (a) Influence of crack width on the chloride profiles at cracked zone. (b) Influence of the roughness on the chloride profiles at cracked zone. (c) Influence of j on the chloride profiles at cracked zone. (d) Influence of interval of drying–wetting cycles on the chloride profiles at cracked zone.
concentrations are at the cracked zone. The influence of crack width on the chloride profiles at cracked zone is shown in Fig. 6a. It can be seen that the deeper of the depth, the concentrations of chloride tend to be smaller, but not everywhere. At the regions near the surface there exists an ascending stage where concentration of chloride ions is higher than the concentration nearby at some ranges, which as well increase its range since the crack width enlarger. The influence of the roughness on the chloride profiles is shown in Fig. 6b. From Eq. (24), it is obvious that the larger value of the roughness, it becomes tougher for moisture and chloride transport in the crack, while the smaller the roughness is, the smoother the process become. It also can be found that to some extent, the influence of roughness fades back. The influence of j as shown in Fig. 6c indicates that j is not very significant. As the model proposed, j represents the ratio of crack lengths to crack widths, in the real environment, the crack lengths seem to be much larger than crack widths, it might indicate that not all the crack lengths have an influence on the crack widths when considering the behavior of chloride ions at certain point. It is clear that the influence rang of crack lengths to the crack widths must exit a characteristic value, while larger than the very value, the influence can be ignored. The influence of interval of wetting–drying cycles on the chloride profiles is shown in Fig. 6d. It can be seen that in some ranges the longer time of drying time is, the part of
advection is stronger after switching into wetting phase, resulting the higher concentration of chloride in concrete.
Table 1 Mixing ingredient of concrete (kg/m3).
Table 2 Mineral ingredient of Portland cement I.
7. Experimental procedure 7.1. Material Table 1. lists the mix proportions of the concrete employed as test specimens. Ordinary Portland cement (ASTM Type I 32.5), added by 5% CaSO42H2O, the composition of which is shown in Table 2, was used in all mixtures. Crushed gravel with continuous grading rang form 5 mm to 20 mm was used as the coarse aggregate and maximum aggregates size was 20 mm. Natural sand with good particle grading and fineness modulus of 2.64 was used. Grounded furnace slag (S95), with fineness of 450 m2/kg are also added in the concrete. The steels of HRB 335 and HPB 235 in Chinese Standard (GB 50010-2010) [28] were used as the longitudinal bars and hooping in present experiment. 7.2. Details of specimens The specimen configuration is shown in Fig. 7. The specimens had a rectangular cross section of 150 mm 75 mm, and their total length was 400 mm. All specimens were subjected to the initial curing in the environment where RH was 100% and temperature was 20 ± 3 °C for 28 days after removal from the molds since 24 h. In order to induce the natural shape cracks, specimens were all prefabricated with cracks with 0.6 mm in width and 2 cm in depth by stainless steel at the middle span of the specimens. Afterward, the natural shape cracks were generated by loading. In order to control the crack width and reach the expected value at the surface, a new method of generating cracks is first introduced in present work. As shown in
w/c
Cement
Slag
Fine aggregate
Coarse aggregate
Water
Mineral ingredient
C3S
C2S
C3A
C4AF
CaSO42H2O
0.53
185
185
750
1112
188
Content (%)
55.5
19.1
6.5
10.1
5
266
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Fig. 7. Specimens and automatic test device for drying–wetting cycles.
Fig. 7, two screws separately located in the specimens when depositing concrete. After the load application, the crack widths were measured on the bottom surface of each specimen by microscope and the crack depths at the side surfaces were measured by micrometer. Thus, by adjusting the loading value of screws, the expected value of crack widths can be obtained. In addition, all surfaces except the bottom surface of specimens were coated with epoxy in order to control the penetration of chloride ions into concrete. Tables 3 and 4 show the example of distribution of cracks whose surface crack width is 0.1 mm and 0.2 mm. All specimens were subjected to accelerated penetration of chloride ions for 90 days through a cyclic drying–wetting test based on an automatic test device. During the test, each cycle of the drying–wetting test consisted in wetting for 6 h through immersion in a sodium chloride solution (5.0% NaCl) and drying for 18 h at 30% relative humidity. The environmental temperature kept constant at 50 °C to accelerate the transportation of chloride ions. The whole experimental procedure was carried out in a multi-function artificial environment simulation chamber. After the accelerated penetration of chloride ions, concrete samples were taken from each specimen for measuring the chloride ions content. The specimen was split open as shown in Fig. 8, and the special form of one specimen is shown in Fig. 9. To obtain the chloride penetration profiles, slices were taken from the specimen by cutting at depths of 15, 35, 55, 75, 95 and 115 mm from exposed surface. The measure of roughness and tortuosity were based on the procedure described in the model. In present work the cracked zone is defined as shown in Fig. 10. The thickness of the cracked zone was 5 mm on each side from the crack surface, based on the roughness of the crack surface and the average particle size of the aggregate, the same proposed by Kato et al. [29]. The select of cracked zone is discussed later in
Table 3 Distribution of crack for specimen 1. Crack morphology specimen 1
Crack depth (mm)
Crack width (mm)
115 95 75 55 35 15 0 (Reserved crack)
0 0.02 0.04 0.06 0.08 0.1 1.1
Table 4 Distribution of crack for specimen 2. Crack morphology specimen 2
Crack depth (mm)
Crack width (mm)
115 95 75 55 35 15 0 (Reserved crack)
0 0.04 0.1 0.1 0.2 0.2 1.2
detail. The concrete samples were taken from the cracked surface at interval of 5 mm on cracked zone and 10 mm for longer distance. The chloride ions content in each sample was determined based on Energy Dispersive Spectrometer (EDS) [30].
8. Result and discussion The chloride ions profiles in the crack surface of the silt specimens after 90 days are shown in Fig. 11. It can be seen that the experimental results of the chloride ions profiles were effectively evaluated by the proposed model. As many researchers found in chloride ions transport in sound concrete, there exists an ascent stage on chloride ions profiles because of the advection part, resulting the chloride concentration at skin layer higher than the concentration at surface, this stage is always regarded as convection zone [31]. However, whether this convection zone exists or not in crack concrete subjected to cyclic drying–wetting condition is still not clear so far. From the experimental results and calculated results, it can be seen that there exists an ascent point at the skin layer as well. It can be explain by the reason similar to that of sound concrete. During the cyclic drying–wetting condition, the
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Fig. 8. Split of a specimen.
Fig. 9. The figure of the crack.
it is difficult to make clear the mechanism of chloride ions transportation in cracked concrete based only on these experimental results. As shown, the crack has a complicated shape and the method of simplifying the crack characteristic also results in some errors both in modeling and experiment during these tests, in addition, because of concentration difference of every point among the cracked zone, every point along the crack surface serves as a new boundary for chloride ions transport vertically into cracked zone. From the point of selecting thickness of cracked zone, it can be seen that during the distance from crack surface to about 5 mm from each side, the concentration of chloride ions vary larger and stronger than that in deeper zone. It also can be seen that the chloride ions concentration vertically from the crack surface tend to be smaller than that of crack surface, and the higher concentration of chloride ions at crack surface, the concentration also tend higher than others at the same distance from crack surface. These phenomena can be explained by the assumption that the mechanism of chloride ions transport in cracked zone is approximately a process dominated by diffusion. It can be estimated that if the concentration of the chloride ions solution in the crack are uniform, the chloride ions content in cracked zone would not change along the depth from exposed surface. However, the aggregates embed in the specimens have an important influence on the chloride ions profiles during the tests by changing the ingress direction and capturing the place. Now it was impossible to clarify the trajectory accurately in cracked zone. By comparing the chloride ions profiles in different crack widths at surface, it also can be found that the chloride ions concentration tend to increase both in the vertically and parallel directions in cracked zone since crack widths increase. But this tendency is only obvious at the points near the exposed surface but gradually fades away at crack pit, it should be known that the crack widths along the cracked zone must have some narrow points, not the ‘V’ shape as many researches assumed. These narrow points may have a distinct influence on the chloride ions profiles by disorganizing the chloride ions transport. As disused above, the presentment of cracks at concrete covers increase the length of convection zone by a large amount, and it has a negative effect on the determining of initial corrosion time in cyclic drying–wetting condition. In order to simply the model and apply to into practical application pragmatically, similarly to FIB [32], the most commonly used equation called Fick’s second law equation is modified to simplify the calculation of chloride ions profiles in cracked concrete taking consideration of the influence of convection zone.
Cðx; tÞ ¼ C 0 þ ðC S;DX
Fig. 10. Definition of the cracked zone.
concentration at skin layer exist a state when concentration is higher than that of surface, resulting the diffusion of chloride ions at the skin layer, and as time going, it moves forward. The length of advection zone is related to the condition of marine environment, material properties and crack characteristics. One point should be emphasized, because of the effect of capillary pores, the effective coefficient also appears different. During the drying phase, the effective coefficient is less because of the ‘ink effect’ reflected on the capillary pore on the cracked zone. In order to clarify chloride ions transport through the crack, chloride ions profiles both in the vertically and parallel directions in the cracked zone and deeper zone form 5 mm to 30 mm of the slit specimens are plotted as well as shown in Fig. 12. However,
267
! x DX C 0 Þerfc pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dapp;C t
ð39Þ
where C is chloride ions concentration, C0 is initial chloride ions content, CS,DX is chloride ions concentration at DX, DX is depth of convection zone, Dapp,C is diffusion coefficient. Especially, based on the model and test results in present work, it is proposed the following equation shown in Eq. (40) to determine Dapp,C in cracked zone by assuming that the advection part dominates the process, the ratio of crack lengths and crack wide as 10 in Eq. (24), and the influence of capillary pores is expressed as a function of porosity of concrete and the degree of saturation as approximately 0.5 in Eq. (30). This equation takes width w, surface roughness expressed in the term of equivalent drag coefficient k, tortuosity s and porosity of concrete / into consideration. In addition, this equation should be modified consideration of the relative humidity, which can approximately be determined in the same way as in the sound zone.
pffiffiffi
Dapp;C ¼
sw2 k
6½1 þ /=ð1 0:5/Þ
ð40Þ
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
0.6
Experiment of 0.1mm surface crack width Calculation
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
268
0.5
0.4
0.3
0.2
0.1 0
20
40
60
80
100
120
Experiment of 0.2mm surface crack width Calculation
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10
140
0
20
Depth from exposed surface (mm)
40
60
80
100
120
140
Depth from exposed surface (mm)
0.55 0.50
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
Fig. 11. (a) Chloride ions profiles in cracked zone for specimen with a surface crack width of 0.1 mm. (b) Chloride ions profiles in cracked zone for specimen with a surface crack width of 0.2 mm.
0~15mm 15~35mm 35~55mm 55~75mm 75~95mm 95~115mm 115~135mm
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 -30
-20
-10
0
10
20
0.55 0.50
0~15mm 15~35mm 35~55mm 55~75mm 75~95mm 95~115mm 115~135mm
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 -30
30
-20
-10
0
10
20
30
Distance from the crack surface (mm)
Distance from the crack surface (mm)
0.6
proposed line test results
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
Fig. 12. (a) Chloride ions profiles for specimen with a surface crack width of 0.1 mm. (b) Chloride ions profiles for specimen with a surface crack width of 0.2 mm.
0.5 0.4 0.3 0.2 0.1 0
20
40
60
80
100
120
Depth from exposed surface (mm)
proposed line test results
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0
20
40
60
80
100
120
Depth from exposed surface (mm)
Fig. 13. (a) Comparison of the proposed line and test results in cracked zone for specimens with a surface crack width of 0.1 mm. (b) Comparison of the proposed line and test results in cracked zone for specimens with a surface crack width of 0.2 mm.
Another important task is the determine of the depth of convection zone DX, as in sound concrete, the depth of convection zone ranges from 5 to 15 mm, but in the cracked zone, it is estimated that it may locate in a range from approximately 20 to 35 mm based on experiment results. Therefore, it assumes that DX equals the neighbor of 20 mm in approximately 0.1 mm crack width, while about 35 mm
in approximately 0.2 mm crack width at the surface appearance. It is also proposed the following curve to determine the parameter used similarly in the cracked zone to calculated the chloride ions concentration. As the comparison of the proposed line and the test results shown in Fig. 13, it can be seen that the proposed parameter is safety for practical use.
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
9. Conclusion In present work, a model describing the chloride ions penetration into cracked concrete subjected to cyclic drying–wetting condition is studied. Following conclusions can be obtained: (1) The transport of chloride ions in cracked zone is elaborated. The mechanism of ions transport in porous medium is supposed still valid for cracked zone. The moisture transport in rough cracks is modeled as a flux based on Poiseuille law considering the crack width, roughness and tortuosity. In addition, the drag coefficient is first applied to calculate the roughness meanwhile the capillary pores of the crack surface are considered as well. (2) The crack width is the most important parameter to representation the effect of cracks on chloride ions transport. The larger crack widths, the smaller roughness and the smaller of the parameter j is, the higher chloride ions concentration tends in cracked zone. (3) The concentration of chloride ions in cracked zone tend to decrease with the increase of crack depths but not everywhere. Further, the larger the crack width is, the deeper the advection is. (4) The chloride ions concentration vertically from the crack surface tends to be smaller than that at surface, and the higher concentration of chloride ions at the crack surface, the concentration also tends be higher than others at the same distance from the crack surface. (5) Simplified parameters are proposed here to calculate the profiles of chloride concentration in cracked zone, especially, in cracked zone with 0.1 mm at appearance, the depths of convection zone might approximately be 20 mm, while for 0.2 mm, it might be 35 mm. In addition, the equation to estimate the diffusivity also proposed here considering the characteristics of cracks.
Acknowledgements The financial support from the National Basic Research Program (973 Program) (Grant No. 2009CB623204), of the People’s Republic of China and the National Natural Science Foundation (Grant Nos. 50838008 and 51178413) is gratefully acknowledged. References [1] Maekawa K, Ishida T, Kishi T. Multi-scale modeling of concrete performance. J Adv Concr Technol 2003;1(2):91–126. [2] Raharinaivo A, Brevet P, Grimaldi G, Pannier G. Relationships between concrete deterioration and reinforcing-steel corrosion. Durability Build Mater 1986;4(2):97–112. [3] Mangat P, Gurusamy K. Permissible crack widths in steel fibre reinforced marine concrete. Mater Struct 1987;20(5):338–47.
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Review
Model of chloride penetration into cracked concrete subject to drying–wetting cycles Hailong Ye, Nanguo Jin ⇑, Xianyu Jin, Chuanqing Fu Department of Civil Engineering, Zhejiang University, 388 Yuhangtang Road, Hangzhou 310058, PR China
h i g h l i g h t s " Moisture transport considers crack width, roughness, tortuosity and capillary pores. " Drag coefficient is applied to calculate the roughness. " Parameters of crack characteristic are studied in the cracked zone. " Simplified method to calculate profiles of chloride concentration in cracked zone.
a r t i c l e
i n f o
Article history: Received 3 March 2012 Received in revised form 12 May 2012 Accepted 15 May 2012 Available online 23 June 2012 Keywords: Drying and wetting cycle Chloride ions Crack Cracked zone Drag coefficient
a b s t r a c t Drying and wetting cycles condition is always identified as the most unfavorable environment condition for reinforce concrete structure subjected to chloride-induced deterioration processes. When a crack occurs, it accelerates the ingress of chloride ions and affects durability. In order to clarify the mechanism of deterioration subjected cyclic drying–wetting condition, an innovative model describing the transport of chloride ions in cracked concrete is elaborated, meanwhile the results of experimental investigation are also reported. The transporting mechanism of chloride ions in porous medium is supposed still valid for cracked concrete. The advection part is simulated by moisture transport in rough cracks, modeled as a flux based on Poiseuille law considering crack width, crack surface roughness, tortuosity and capillary pores at crack surface. In addition, the drag coefficient is first applied to calculate the influence of roughness on crack in present work. Finally, a simplified and modified Fick’s second law is proposed here to estimate the chloride ions profiles in cracked concrete. Ó 2012 Elsevier Ltd. All rights reserved.
Contents 1. 2.
3.
4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chloride permeate in sound concrete in drying–wetting cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Moisture diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Chloride ingress in unsaturated concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Calculation the degree of saturation in concrete subjected to drying and wetting cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling of chloride penetrate in the cracked zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The geometric properties of cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Calculation of the roughness based on drag coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Calculation of the tortuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Further modeling considering the capillary pores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Effective crack widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parametric studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Details of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
⇑ Corresponding author. Tel./fax: +86 0577 88208766. E-mail addresses: [email protected] (H. Ye), [email protected] (N. Jin), [email protected] (X. Jin), [email protected] (C. Fu). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.05.027
260 260 260 260 261 261 262 262 263 263 264 264 264 264 265 265 265
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Result and discussion. Conclusion . . . . . . . . . Acknowledgements . . References . . . . . . . . .
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1. Introduction Chloride-induced corrosion of steel reinforcement is a major cause affecting the service life of reinforced concrete (RC) structure exposed to marine environments. Chloride ions ingress solely by pore diffusion is a rare occurrence when structure exposed to continuous drying–wetting cycles by tidal and splash action. For chloride ions transport in sound concrete, various model and analysis technology have been proposed including the condition of cyclic drying–wetting [1]. However, cracks cannot be avoided for a real structure. Effects of cracks on the diffusion and permeation on chloride behavior have also been investigated by many researchers. It is generally recognized that the cracks accelerate the ingress of chloride ions. Raharinaivo et al. made a comparison of diffusion coefficient for cracked and uncracked concrete, showing an increase in the diffusion coefficient for cracked concrete by one or two orders of magnitude [2]; Mangat and Gurusamy studied the influence of cracks on chloride diffusion of steel fiber reinforced concrete and concluded that crack widths larger than 500 lm had more pronounced influence on chloride intrusion while crack widths less than 200 lm appeared to have nearly no effect on chloride intrusion [3]; Gérard and Marchand analyzed the influence of traversing cracks on transport of ions in saturation concrete, and found that cracks tended to markedly modify the transport coefficient of the solid by a factor ranging from 2 to 10 [4]; Gowripalan et al. used the parameter of the ratio of crack width to cover to consider in relation to the diffusion process [5]; Boulfiza et al. used a simplified smeared approach to estimate the effect of cracks on chloride ingress using Fick’s law describing chloride ingress into cracked concrete [6]; Win et al. used a series of cracked reinforced concrete specimens, and found that the specimens having cracks showed rapid penetration of chloride ions [7]; Sßahmaran found that for crack widths less than 135 lm, the effect of crack widths on the effective diffusion coefficient of mortars was marginal, whereas for higher than 135 lm the coefficient increased rapidly based on V-shaped cracks [8]; Care investigated the cracks caused by temperature and the transport properties of chloride diffusion in cement paste, finding that chloride ingress increased with heat temperature [9]; Islam and Sazzad found that the surface chloride for sound part remains unchanged with the change of cracked widths [10]; Djerbi et al. investigated the influence of traversing cracks on chloride diffusion for OC, HPC and HPCSF concrete, finding that the coefficient in cracks was independent of material effects, when for crack widths smaller than 80 lm, the coefficient seemed constant [11]; Marsavina et al. used lot of artificial cracks with different widths and depths to research the chloride penetration, showing higher penetration of chloride at tip in comparison with the uncracked part of test specimens [12]. Park et al. proposed an equivalent chloride diffusivity considering a REV model with cracks, and found that the diffusion coefficient increased with the growing of crack widths [13,14]. However, most analyses were based on the condition of chloride penetration into saturated concrete, but in reality, concrete was often found in an unsaturated condition rather than a saturation condition especially when subjected to cyclic drying– wetting. In unsaturated condition, various models were proposed always at the same assumption that the chloride ingress process is a com-
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266 269 269 269
bination of diffusion and advection [1,15]. Diffusion denotes the ionic motion of low concentration while advection refers to the movement of chloride with moisture. When subjected to cyclic drying–wetting condition, the advection part dominates the rate of chloride penetration in wetting phase, while in drying phase, the concentration of solution at exposure surface increases due to the depleting of moisture [1]. So clarifying the moisture transport in cracked concrete is the key problem. Under service condition, cracks of concrete structures cannot be avoided during whole service life, what’s more, drying–wetting cycles condition is always identified as the most unfavorable environment condition for concrete deterioration processes [16]. In present work, a model describing the chloride penetration into sound and cracked concrete is proposed subjected to cyclic drying–wetting condition based on the works of Ababneh et al. and Maekawa et al. It is assumed that the mechanisms for drying phase and wetting phase, both for sound and cracked concrete, are the same (i.e. diffusion and advection) except for the moisture and chloride diffusion coefficients. 2. Chloride permeate in sound concrete in drying–wetting cycles 2.1. Moisture diffusion As the initially unsaturated concrete is put into contact with liquid water, the water intake is dominated by capillary absorption across the surface. Then, the relative humidity at surface is higher, resulting moisture transport driven by the difference of relative humidity between surface and inner. While saturated concrete surface is exposed to a lower humidity during drying phase, vapor diffusion process dominates because the evaporation process occurs at the gas–liquid interface. A nonlinear diffusion model for both two phases are present according to a modified Dacy’ s law [17]:
J w ¼ Dh rh
ð1Þ
where h is relative humidity, Dh is humidity diffusion coefficient, which depends on the relative humidity by timing factor with a reference value at isothermal conditions [18]:
Dh ¼ Dh;ref
pffiffiffiffiffiffih pffiffiffiffiffiffim i 0
ð2Þ
where Sw represents degree of saturation with condensed water and adsorbed water, whose relative content alters in drying–wetting cycles, the computing method of Sw will be introduced later. The rate of change of the mass of water we per unit volume of concrete is determined from the flux field by the relation [17,19]:
@we ¼ rJ w @t
ð3Þ
Eq. (3) is based on the hypothesis of dissipation decoupling, disregarding possible couplings between heat and moisture transport. 2.2. Chloride ingress in unsaturated concrete As said in introduction, the flux of chloride ions permeate in concrete subjected to drying–wetting cycles takes both diffusion and advection (i.e. related to the diffusion of moisture), the diffu-
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sion model for the free chloride in unsaturated concrete is given by [15]:
@C fr @C fr @w rðDcl rC fr Þ þ l e C fr ¼ @t @C to @t
ð4Þ
where Cto is total concentration of ions in grams of total chloride per gram of concrete; Cfr is free concentration of ions in grams of free chloride per gram of concrete; Dcl is chloride diffusivity of concrete; oCfr/oCto is binding capacity of concrete; l is unit converter to grams of chloride per gram of concrete given by Eq. (5); owe/ot is rate of change of total water content given by Eq. (3). The l as explained, is the ratio of the volume of pore solution to the weight of concrete [15], according to its definition, this parameter can be rewritten as a function of the porosity and degree of saturation as follows:
l¼
1
qsol bsol
¼
wconc V conc qconc ¼ wsol V conc /Scg qsol
ð5Þ
where qsol is density of pore solution in grams of pore solution per liter; bsol is ratio of volume of pore solution to weight of concrete; wconc and wsol are weight of the concrete and solution; qconc and qsol are density of the concrete and pore solution; / is total porosity of concrete. Substituting Eq. (5) into Eq. (4):
@C fr @C fr q ¼ r½Dcl rðC fr Þ þ r Dh conc C fr rðhÞ @t @C to /Sw qsol
ð6Þ
From Eq. (6), it can be seen that the final concentration of ions is influenced by the water stored in pores, which change according to the relative humidity in drying–wetting cycles, in another word, the relationship between relative humidity and degree of saturation must be determined necessarily when calculating the concentration of chloride ions in concrete. 2.3. Calculation the degree of saturation in concrete subjected to drying and wetting cycles As previously mentioned, Eq. (6) can be rewritten as a function of relative humidity without in the form of degree of saturation because of the corresponding relationship between them. Then, it is necessary to describe the liquid water and water vapor in pores, whose relative content just corresponds to the definition of the degree of saturation. In drying–wetting cycles, the liquid and vapor water switch each other, making a serious influence on the concentration of ions dissolved in solution. In drying and wetting cycles, sorption hysteresis and its effect on moisture transport have to be investigated, because no oneto-one relations between different state variables can be obtained, e.g. between the relative humidity and water mass concentration, it is the relationship between humidity and degree of saturation as follows [20]:
Sw ¼ f ðh; drying—wetting historyÞ
ð7Þ
There is a relationship based on the Laplace and Kelvin equations among the relative humidity h, degree of saturation Sw and the capillary pressure pc in concrete pores which are treated to be cylindrical with different radii, and it can be expressed in Eq. (8). The detailed form can be found by the research of Maekawa et al.
pc ¼
qw RT Mw
ln h ¼
2c rs
liquid–vapor interface forms are completely filled with water, whereas larger pores are partially saturation with a layer of absorbed water, the thickness ta can be calculated by using the modified B.E.T. theory. Then, in the wetting phases, the degree of saturation can be gotten by combining condensed water, determined from rc derived from the moisture equilibrium equations and pore distribution function, and adsorbed water in the unsaturated pores. The degree of saturation Sw can be calculated by [1]:
Sw ¼ f wetting ðhÞ ¼
Z
1
½1 ðr ta =rÞ2 dV þ
rc
Z
rc
dV ¼ Sa þ Sc
However, in the drying phase, there is a difference in the time elapsed before reaching the equilibrium because of the inkbottle effect mentioned as a cause of the short-term drying–wetting hysteresis. It can be inferred that the moisture trapped by the inkbottle effect gradually disperse into adjacent connected pores at the rates corresponding to ambient temperature and relative humidity, when getting balance, the degree of saturation in the drying phase can be expressed as [1]:
Sw ¼ f drying ðhÞ ¼ Sc þ
Z
1
ðSc =VÞdV ¼ Sc ½1 ln Sc
ð10Þ
rc
3. Modeling of chloride penetrate in the cracked zone In the real environment, concrete structures always have cracks, which make moisture dissolved with chloride more easily penetrates and reaches the reinforcing steel and speeds up the initiation of steel corrosion especially in cyclic drying–wetting conditions. In present work, this mechanism is assumed still suitable to the issues of chloride penetrates in cracks subjected to cyclic drying– wetting conditions. To ensure the impact of cracks on chloride ions transport more accurately, it is necessary to quantify the relationship between the crack and the associated transport properties of moisture, since advection part (i.e. transport of chloride ions with moisture flux), is one of the major cause of chloride transport in cyclic drying– wetting conditions. Hence, emphasis is laid on the transport of moisture in cracks. In present research, the advection part may kcr belong to Navier–Stokes equations, which is suitable just at the condition that the facture morphology can be treated as a straight, smoothly and parallel plate crack. Water flow rate through a crack, Qi expressed in Eq. (11) is related to the cube of crack with w according to Poiseuille law regarded as Newtonian fluids [21]. However, as shown in Fig. 2, the crack widths often vary along the flow path, as well as the visible crack length and crack surface are full of macro-roughness (i.e. tortuosity) and micro-roughness (i.e. surface roughness of aggregates and cement paste, and crack branching) [16]. So it might be too impudent to use Navier–Stokes equations directly. Many researchers had put forward an empirical reduction factor n to consider the combination of these effects, and the permeability coefficient can be expressed in Eq. (12) based on the parallel palates-shape crack model, while the empirical factor n ranges from 0.01 to 0.1 for plain and fiber reinforced concrete reported by Edvardsen and Mohr, Picandet et al. [16,22].
ð8Þ
where qw is density of liquid, Mw is molecular mass, R is gas constant, T is temperature, rs is pore radius; c is surface tension of liquid water. From Eq. (9), it can be known that a certain group of pores whose radii is smaller than the specific radius rc at which a
ð9Þ
0
Fig. 1. Network of the crack element.
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Q¼
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
w3 b rp 12g
kcr ¼ n
ð11Þ
w2 12
ð12Þ
where w, b represent the crack width and crack length, g represents the dynamic viscosity of fluid, rp is the pressure gradient that drives the flow. Unfortunately, the value of n ranges unstability and are purely determined according to experiments. Even though, Louis [21] modified Eq. (12), and further suggested an equation to estimate the permeability of a parallel palate-shape crack value in laminar flow based on the roughness of crack surface, this equation is as well a simplified empirical equation based on tests. 3.1. The geometric properties of cracks Similarly to Carmeliet et al. [23], a network representation of crack void space pattern is set up representing the tunnel of moisture transport. As shown in Fig. 1, crack path can be cut into many crack segments, nodes is located at the center of each crack segments whose area is Si = f(wi, bi), while wi, bi represent the width and length. Firstly considering the flow in the i th element. However, the influence of chloride ions attack on concrete mostly occurs at the larger crack widths, when crack widths decrease to a critical value along the direction parallel to crack segments, the influence of the crack can be neglected. In order to simplify the model, it is assumed that the crack widths wi along the crack length are constant and equal to the middle value of the segment, in addition, the estimation of crack lengths bi depends on the relevant crack widths wi which will be discussed later. 3.2. Calculation of the roughness based on drag coefficient In present work, in order to calculate the reduction factor n, it is proposed of a new method based on two principles of equivalence comparing an associative circle pipe with same geometric roughness of crack surface along the transport path. Firstly, only considering the micro-roughness flow element i, according to DacyWeisbach equations, the frictional head loss along the path can be expressed as a function of the drag coefficient along the way ki, which has been researched mostly for circle pipes. It assumes that the volume of these paths is proportional to the volume of a smooth and linear tube with the same frictional head loss over the same distances as shown in Fig. 3, for a flow transport through a smooth and liner tube, according to Poiseuille [24]:
Q i;tu ¼
pC 4i Dpij 8g Lij
ð13Þ
Fig. 3. Transform of the roughness of a crack into a circle model.
hf ;tu ¼ ki
2 2 Lij J i;tu 1 C 3i Dpij ¼ ki 2 256g g Lij 2C i 2g
ð14Þ
where Qi,tu is the flow rate through a circle tube, Ji,tu is the average velocity of tube, Ci is the radius of the tube, hf,tu represents frictional head loss, ki represents the drag coefficient along the way. Similarly, the frictional head loss along the parallel plates-shape cracks can also be expressed as follows:
hf ;cr ¼ keq
2 Lij J i;cr 4C i;eq 2g
ð15Þ
where Ji,tu is the average velocity of crack, Ci,eq is the hydraulic radius. In order to get an equivalent permeability coefficient, a drag coefficient is introduced instead of the drag coefficient along the way, thus getting keq ¼ 1; Jeq i;cr ¼ ti J i;cr Eq. (15) can be rewritten as:
hf ;cr ¼ t2
2 ðwi þ bi Þ w3i Dpij 476bi g g2 Lij
ð16Þ
The first principle of equivalence assumes that the frictional head loss in circle and crack is the same, reaching hf,cr = hf,tu, thus the equivalent drag coefficient ti can be expressed as:
ti
3 ¼ 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ki bi C 3i ðwi þ bi Þw3i
ð17Þ
The second principle of equivalence is that if the crack path and the circle tube are all smooth, then conductivity-equivalent must satisfy following conditions:
bi w3i Dpij pC 4i Dpij ¼ 12g Lij 8g Lij Getting C i ¼
ð18Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2w3 b ffi 4 i i ; then submitting it into Eq. (17) and repre3p
senting bi/wi with ji, Eq. (17) can be rewritten as:
ti
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 8 j3 ji 4 i ¼ ki 4 ji þ 1
ð19Þ
Next, the relationship between the drag coefficient ki and the roughness Ri must be bridged. Crack geometry satisfies the properties of so-called self-similarity or self-affinity with respect to scale changes. For simply, in order to quantize the roughness of crack geometry, a mathematical discipline introduced by Mandelbort is used here. the roughness of element i can be determined by calculating the average height of surface asperities with respect to its reference line as shown in Fig. 4 [25]: Fig. 2. Digitized profiles of an actual crack.
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Fig. 4. Method of quantifying roughness and tortuosity.
Ri ¼
n1 1X jzi hzij n i¼1
ð20Þ
where n is the ranges of the analyzed set of the measured values, hzi is the arithmetic average height. Assuming that the drag coefficient is dominated by the roughness of cracks, the Nicholas equation is applied here [24]:
Ri 2wi
ki ¼ ð2 lg
þ 1:74Þ2
si ¼
1 n1
1 1 þ s2i
i¼1
jziþ1 zi j ðxiþ1 xi Þ
ð24Þ
i
3.4. Further modeling considering the capillary pores
Tortuosity factor takes the wavy characteristic of crack profiles into consideration, because of the effective crack length larger than the nominal crack length. The tortuosity si is quantified by transforming the average slope into a pragmatic method as shown in Fig. 4 which can be expressed as follows [25]:
si ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3 ji 2 8 ji f ðSÞ s w i i 4 ji þ1 w2i ki;cr ðSÞ ¼ f ðSÞ si ti ¼ 12 12 2 lg Ri þ 1:74 2w
ð21Þ
3.3. Calculation of the tortuosity
n1 X
of saturation in Eq. (2). Finally the equivalent coefficient in Eq. (12) can be rewritten as:
ð22Þ
ð23Þ
In the unsaturation state, similarly to modified Dacy’ s law, the equivalent is modified by timing the factor of function of degree
Along the crack surface, there exist a lot of pores as shown in Fig. 5, thus, capillary forces of the crack surface cannot be completely neglected, it assumes that the capillary pores are modeled as tiny cylinder tubes with various radii. According to Hagen Poiseuille flow, the flux in one tube can be expressed as [1]:
J i;cap ¼
ki;r
g0
rP
ð25Þ
where ki,r is the permeability, which can be calculated by R r
2 ki;ca ¼ 18 0 c rdA based on the assumption that only the capillary pores filled with water can transport chloride ions. From Eq. (8), the pores whose radii r is smaller that specific radius are completely filled with water, whereas larger pores are partially saturation with a layer of absorbed water, which somewhat contribute to the chloride transport along the crack surface as well.
Fig. 5. Cross section of concrete [33] and model of capillary pores at crack surface.
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
The area of moisture at crack section is wibi, the total moisture associated with this area can be calculated by without considering the moisture absorbed in the pore spaces,
Ai;cr ðhÞ ¼ wi bi Li;cr ð1 /i;cap Þ
ð26Þ
The area of the pore space can approximately be expressed as:
Ai;ca ¼ bi Li;cr /i;cap
ð27Þ
where /i;cap is the porosity of capillary in concrete, thus the cross sectional area of moisture should add the area of pores whose radii is smaller than rc, getting:
Aeff i;cr ðhÞ ¼ wi bi Li;cr ð1 /i;cap þ /i;cap ðSc þ Sa ÞÞ
ð28Þ
Then, the area of capillary is
Aeff i;ca ðhÞ ¼ bi Li;cr /i;cap Sc
ð29Þ
The average coefficient of moisture transport in cracks can be formulated by averaging of cross section areas.
Deff i;h ðhÞ ¼
! ki;ca /i;cap Sc þ ki;cr ðSw Þ wi ð1 /i;cap þ /i;cap ðSc þ Sa ÞÞ ql RT g0 Ml h /i;cap Sc þ wi ð1 /i;cap þ /i;cap ðSc þ Sa ÞÞ ð30Þ
If idealizing the crack as another material with different transport properties, the chloride ions transport in the cracked concrete can be modeled as a two-dimensional transport issue, in addition, the diffusion part in the crack also increases by a ratio of w than in the sound concrete. The mass balance equation can be rewritten as:
! @C fr @C fr @ 2 C fr @ 2 C fr Dcl w þ ¼ @t @C to @x2 @y2 @ @h @ qcon @h þ Deff þ C D C fr h fr @x h @x @y /Sw qsol @y
ð31Þ
It can be seen that the equivalent coefficient involves crack widths, crack lengths and roughness, then if all three parameters are known, the coefficient of every element can be calculated, however, in the most case, only the crack widths and lengths at surface can be easily measured. If assuming that the crack lengths and roughness are invariant along the path, the effective crack widths can be calculated according to Akhavan et al. [21], however, there exists some distinction because of the early simplification of treating the crack lengths depend on the relevant crack widths. Based on the same conditions of fluid continuity (Q = Q1 = Q2 = = Qi) and the total pressure loss across the path is the sum of the loss of each element (DP = DP1 + + DPn). Finally the effective crack widths can be expressed as:
weff
s
J sdrying ¼ Dh ðh henv Þ
ð35Þ
where Dh is surface moisture transfer coefficient, hs is relative humidity at surface, henv is relative humidity in environment. Then, at the progress of drying phase, the content of chloride ions compared with concrete at concrete surface stay unchangeable
ðC sfr;drying Þiþ1 ¼ ðC sfr;wetting Þi
ð36Þ
5. Numerical solution In order to calculate the chloride profiles in cracked zone, the cracked concrete is regard as the heterogenetic material in two dimensions. The alternating-direction implicit (ADI) finite-difference method [15,26] is employed here for solving the chloride transport problem. The principle of the ADI method can be applied to solve nonlinear partial differential equations, using two difference equations over two successive time steps, each of duration Dt/2, particularly, Eq. (37) is implicit in the x-direction (the direction of the cracks) and explicit in the y direction, while Eq. (38) is implicit in the y-direction and explicit in the x direction for this two-dimensions finite difference discretization and 1 mm is chosen as the mesh size both for x and y directions here.
h i C i;j C ni;j ¼ ani;j Dni;j d2x C i;j þ dx Dni;j dx C i;j þ Dni;j d2y C ni;j þdy Dni;j dy C ni;j þ #ni;j Ani;j C ni;j Dt=2 ð37Þ
3.5. Effective crack widths
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 3 Pn 3 ð1=w i¼1 iÞ
where index i refers to the number of time of drying–wetting cycles. In drying phase, the relative humidity of concrete at surface become lower gradually because of the progress of evaporation, at which the chloride ions dissolved does not following vanish, on the contrary, on the course of wetting phase to drying phase, the concentration of chloride ions on the surface increase with the evaporation until to getting balance. At the exposed boundaries, the conditions are defined as fluxes of relative humidity.
ð32Þ
nþ1 h i C i;j C i;j nþ1 n n ¼ ani;j Dni;j d2x C i;j þ dx Dni;j dx C i;j þ Dni;j d2y C i;j þdy Dni;j dy C nþ1 i;j þ #i;j Ai;j C i;j Dt=2 ð38Þ
where indictor represents the intermediate value of these two steps, dx ; d2x and dy ; d2y represent the first- and second-order central difference operators in the x and y directions, a represents the binding capacity of concrete, D represents the diffusion coefficients according to the above proposed material model, different for cracked and sound concrete, A represents the moisture transport, which also can be solved by simplified ADI method, and the corresponding coefficients are based on the proposed model, # represents the parameter used for unit converter, different for cracked and sound concrete, while it is expressed in Eq. (5) for sound concrete. The stability of the ADI method is unconditionally stable and the solution converges within a discretization error of O[(Dt)2 + (Dx)2 + (Dy)2] discussed by Neumann [27].
4. Surface condition 6. Parametric studies In wetting phase, it assumes that the relative humidity is full when contact with water, and the concentrations of free chloride ions at the exposed surface get balance soon with the water. Then, at the exposed boundaries, the conditions are defined as fluxes of chloride ions, relative humidity.
ðC sfr;wetting Þi ¼ C env s
h ¼ henv
ð33Þ ð34Þ
Through the above proposed model, it is clear that the characteristics of cracks have a significant influence on the transport of chloride in cracks. In this section, the full parametric studies are carried out on how the characteristics of cracks affect the chloride profiles. According to the model, the crack width is the most important parameter to representation the effect of cracks on chloride transport, obviously, the larger crack width is, the more chloride ions are delivered by the advection effect and the higher
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Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
0.3 mm crack width 0.2 mm crack width 0.1 mm crack width
0.0 0
50
Roughness 0.1 Roughness 0.01 Roughness 0.001
0.0 0
100
50
K=10 K=5 K=1
0.0
0
50
100
Depth from exposed surface (mm)
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
Depth from exposed surface (mm)
100
Dry time: Wet time=3:1 Dry time: Wet time=20:1 Dry time: Wet time=1:infinity
0.0
0
50
100
Depth from exposed surface (mm)
Depth from exposed surface (mm)
Fig. 6. (a) Influence of crack width on the chloride profiles at cracked zone. (b) Influence of the roughness on the chloride profiles at cracked zone. (c) Influence of j on the chloride profiles at cracked zone. (d) Influence of interval of drying–wetting cycles on the chloride profiles at cracked zone.
concentrations are at the cracked zone. The influence of crack width on the chloride profiles at cracked zone is shown in Fig. 6a. It can be seen that the deeper of the depth, the concentrations of chloride tend to be smaller, but not everywhere. At the regions near the surface there exists an ascending stage where concentration of chloride ions is higher than the concentration nearby at some ranges, which as well increase its range since the crack width enlarger. The influence of the roughness on the chloride profiles is shown in Fig. 6b. From Eq. (24), it is obvious that the larger value of the roughness, it becomes tougher for moisture and chloride transport in the crack, while the smaller the roughness is, the smoother the process become. It also can be found that to some extent, the influence of roughness fades back. The influence of j as shown in Fig. 6c indicates that j is not very significant. As the model proposed, j represents the ratio of crack lengths to crack widths, in the real environment, the crack lengths seem to be much larger than crack widths, it might indicate that not all the crack lengths have an influence on the crack widths when considering the behavior of chloride ions at certain point. It is clear that the influence rang of crack lengths to the crack widths must exit a characteristic value, while larger than the very value, the influence can be ignored. The influence of interval of wetting–drying cycles on the chloride profiles is shown in Fig. 6d. It can be seen that in some ranges the longer time of drying time is, the part of
advection is stronger after switching into wetting phase, resulting the higher concentration of chloride in concrete.
Table 1 Mixing ingredient of concrete (kg/m3).
Table 2 Mineral ingredient of Portland cement I.
7. Experimental procedure 7.1. Material Table 1. lists the mix proportions of the concrete employed as test specimens. Ordinary Portland cement (ASTM Type I 32.5), added by 5% CaSO42H2O, the composition of which is shown in Table 2, was used in all mixtures. Crushed gravel with continuous grading rang form 5 mm to 20 mm was used as the coarse aggregate and maximum aggregates size was 20 mm. Natural sand with good particle grading and fineness modulus of 2.64 was used. Grounded furnace slag (S95), with fineness of 450 m2/kg are also added in the concrete. The steels of HRB 335 and HPB 235 in Chinese Standard (GB 50010-2010) [28] were used as the longitudinal bars and hooping in present experiment. 7.2. Details of specimens The specimen configuration is shown in Fig. 7. The specimens had a rectangular cross section of 150 mm 75 mm, and their total length was 400 mm. All specimens were subjected to the initial curing in the environment where RH was 100% and temperature was 20 ± 3 °C for 28 days after removal from the molds since 24 h. In order to induce the natural shape cracks, specimens were all prefabricated with cracks with 0.6 mm in width and 2 cm in depth by stainless steel at the middle span of the specimens. Afterward, the natural shape cracks were generated by loading. In order to control the crack width and reach the expected value at the surface, a new method of generating cracks is first introduced in present work. As shown in
w/c
Cement
Slag
Fine aggregate
Coarse aggregate
Water
Mineral ingredient
C3S
C2S
C3A
C4AF
CaSO42H2O
0.53
185
185
750
1112
188
Content (%)
55.5
19.1
6.5
10.1
5
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H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Fig. 7. Specimens and automatic test device for drying–wetting cycles.
Fig. 7, two screws separately located in the specimens when depositing concrete. After the load application, the crack widths were measured on the bottom surface of each specimen by microscope and the crack depths at the side surfaces were measured by micrometer. Thus, by adjusting the loading value of screws, the expected value of crack widths can be obtained. In addition, all surfaces except the bottom surface of specimens were coated with epoxy in order to control the penetration of chloride ions into concrete. Tables 3 and 4 show the example of distribution of cracks whose surface crack width is 0.1 mm and 0.2 mm. All specimens were subjected to accelerated penetration of chloride ions for 90 days through a cyclic drying–wetting test based on an automatic test device. During the test, each cycle of the drying–wetting test consisted in wetting for 6 h through immersion in a sodium chloride solution (5.0% NaCl) and drying for 18 h at 30% relative humidity. The environmental temperature kept constant at 50 °C to accelerate the transportation of chloride ions. The whole experimental procedure was carried out in a multi-function artificial environment simulation chamber. After the accelerated penetration of chloride ions, concrete samples were taken from each specimen for measuring the chloride ions content. The specimen was split open as shown in Fig. 8, and the special form of one specimen is shown in Fig. 9. To obtain the chloride penetration profiles, slices were taken from the specimen by cutting at depths of 15, 35, 55, 75, 95 and 115 mm from exposed surface. The measure of roughness and tortuosity were based on the procedure described in the model. In present work the cracked zone is defined as shown in Fig. 10. The thickness of the cracked zone was 5 mm on each side from the crack surface, based on the roughness of the crack surface and the average particle size of the aggregate, the same proposed by Kato et al. [29]. The select of cracked zone is discussed later in
Table 3 Distribution of crack for specimen 1. Crack morphology specimen 1
Crack depth (mm)
Crack width (mm)
115 95 75 55 35 15 0 (Reserved crack)
0 0.02 0.04 0.06 0.08 0.1 1.1
Table 4 Distribution of crack for specimen 2. Crack morphology specimen 2
Crack depth (mm)
Crack width (mm)
115 95 75 55 35 15 0 (Reserved crack)
0 0.04 0.1 0.1 0.2 0.2 1.2
detail. The concrete samples were taken from the cracked surface at interval of 5 mm on cracked zone and 10 mm for longer distance. The chloride ions content in each sample was determined based on Energy Dispersive Spectrometer (EDS) [30].
8. Result and discussion The chloride ions profiles in the crack surface of the silt specimens after 90 days are shown in Fig. 11. It can be seen that the experimental results of the chloride ions profiles were effectively evaluated by the proposed model. As many researchers found in chloride ions transport in sound concrete, there exists an ascent stage on chloride ions profiles because of the advection part, resulting the chloride concentration at skin layer higher than the concentration at surface, this stage is always regarded as convection zone [31]. However, whether this convection zone exists or not in crack concrete subjected to cyclic drying–wetting condition is still not clear so far. From the experimental results and calculated results, it can be seen that there exists an ascent point at the skin layer as well. It can be explain by the reason similar to that of sound concrete. During the cyclic drying–wetting condition, the
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
Fig. 8. Split of a specimen.
Fig. 9. The figure of the crack.
it is difficult to make clear the mechanism of chloride ions transportation in cracked concrete based only on these experimental results. As shown, the crack has a complicated shape and the method of simplifying the crack characteristic also results in some errors both in modeling and experiment during these tests, in addition, because of concentration difference of every point among the cracked zone, every point along the crack surface serves as a new boundary for chloride ions transport vertically into cracked zone. From the point of selecting thickness of cracked zone, it can be seen that during the distance from crack surface to about 5 mm from each side, the concentration of chloride ions vary larger and stronger than that in deeper zone. It also can be seen that the chloride ions concentration vertically from the crack surface tend to be smaller than that of crack surface, and the higher concentration of chloride ions at crack surface, the concentration also tend higher than others at the same distance from crack surface. These phenomena can be explained by the assumption that the mechanism of chloride ions transport in cracked zone is approximately a process dominated by diffusion. It can be estimated that if the concentration of the chloride ions solution in the crack are uniform, the chloride ions content in cracked zone would not change along the depth from exposed surface. However, the aggregates embed in the specimens have an important influence on the chloride ions profiles during the tests by changing the ingress direction and capturing the place. Now it was impossible to clarify the trajectory accurately in cracked zone. By comparing the chloride ions profiles in different crack widths at surface, it also can be found that the chloride ions concentration tend to increase both in the vertically and parallel directions in cracked zone since crack widths increase. But this tendency is only obvious at the points near the exposed surface but gradually fades away at crack pit, it should be known that the crack widths along the cracked zone must have some narrow points, not the ‘V’ shape as many researches assumed. These narrow points may have a distinct influence on the chloride ions profiles by disorganizing the chloride ions transport. As disused above, the presentment of cracks at concrete covers increase the length of convection zone by a large amount, and it has a negative effect on the determining of initial corrosion time in cyclic drying–wetting condition. In order to simply the model and apply to into practical application pragmatically, similarly to FIB [32], the most commonly used equation called Fick’s second law equation is modified to simplify the calculation of chloride ions profiles in cracked concrete taking consideration of the influence of convection zone.
Cðx; tÞ ¼ C 0 þ ðC S;DX
Fig. 10. Definition of the cracked zone.
concentration at skin layer exist a state when concentration is higher than that of surface, resulting the diffusion of chloride ions at the skin layer, and as time going, it moves forward. The length of advection zone is related to the condition of marine environment, material properties and crack characteristics. One point should be emphasized, because of the effect of capillary pores, the effective coefficient also appears different. During the drying phase, the effective coefficient is less because of the ‘ink effect’ reflected on the capillary pore on the cracked zone. In order to clarify chloride ions transport through the crack, chloride ions profiles both in the vertically and parallel directions in the cracked zone and deeper zone form 5 mm to 30 mm of the slit specimens are plotted as well as shown in Fig. 12. However,
267
! x DX C 0 Þerfc pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dapp;C t
ð39Þ
where C is chloride ions concentration, C0 is initial chloride ions content, CS,DX is chloride ions concentration at DX, DX is depth of convection zone, Dapp,C is diffusion coefficient. Especially, based on the model and test results in present work, it is proposed the following equation shown in Eq. (40) to determine Dapp,C in cracked zone by assuming that the advection part dominates the process, the ratio of crack lengths and crack wide as 10 in Eq. (24), and the influence of capillary pores is expressed as a function of porosity of concrete and the degree of saturation as approximately 0.5 in Eq. (30). This equation takes width w, surface roughness expressed in the term of equivalent drag coefficient k, tortuosity s and porosity of concrete / into consideration. In addition, this equation should be modified consideration of the relative humidity, which can approximately be determined in the same way as in the sound zone.
pffiffiffi
Dapp;C ¼
sw2 k
6½1 þ /=ð1 0:5/Þ
ð40Þ
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
0.6
Experiment of 0.1mm surface crack width Calculation
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
268
0.5
0.4
0.3
0.2
0.1 0
20
40
60
80
100
120
Experiment of 0.2mm surface crack width Calculation
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10
140
0
20
Depth from exposed surface (mm)
40
60
80
100
120
140
Depth from exposed surface (mm)
0.55 0.50
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
Fig. 11. (a) Chloride ions profiles in cracked zone for specimen with a surface crack width of 0.1 mm. (b) Chloride ions profiles in cracked zone for specimen with a surface crack width of 0.2 mm.
0~15mm 15~35mm 35~55mm 55~75mm 75~95mm 95~115mm 115~135mm
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 -30
-20
-10
0
10
20
0.55 0.50
0~15mm 15~35mm 35~55mm 55~75mm 75~95mm 95~115mm 115~135mm
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 -30
30
-20
-10
0
10
20
30
Distance from the crack surface (mm)
Distance from the crack surface (mm)
0.6
proposed line test results
Chloride Content (%wt. of concrete)
Chloride Content (%wt. of concrete)
Fig. 12. (a) Chloride ions profiles for specimen with a surface crack width of 0.1 mm. (b) Chloride ions profiles for specimen with a surface crack width of 0.2 mm.
0.5 0.4 0.3 0.2 0.1 0
20
40
60
80
100
120
Depth from exposed surface (mm)
proposed line test results
0.55 0.50 0.45 0.40 0.35 0.30 0.25 0
20
40
60
80
100
120
Depth from exposed surface (mm)
Fig. 13. (a) Comparison of the proposed line and test results in cracked zone for specimens with a surface crack width of 0.1 mm. (b) Comparison of the proposed line and test results in cracked zone for specimens with a surface crack width of 0.2 mm.
Another important task is the determine of the depth of convection zone DX, as in sound concrete, the depth of convection zone ranges from 5 to 15 mm, but in the cracked zone, it is estimated that it may locate in a range from approximately 20 to 35 mm based on experiment results. Therefore, it assumes that DX equals the neighbor of 20 mm in approximately 0.1 mm crack width, while about 35 mm
in approximately 0.2 mm crack width at the surface appearance. It is also proposed the following curve to determine the parameter used similarly in the cracked zone to calculated the chloride ions concentration. As the comparison of the proposed line and the test results shown in Fig. 13, it can be seen that the proposed parameter is safety for practical use.
H. Ye et al. / Construction and Building Materials 36 (2012) 259–269
9. Conclusion In present work, a model describing the chloride ions penetration into cracked concrete subjected to cyclic drying–wetting condition is studied. Following conclusions can be obtained: (1) The transport of chloride ions in cracked zone is elaborated. The mechanism of ions transport in porous medium is supposed still valid for cracked zone. The moisture transport in rough cracks is modeled as a flux based on Poiseuille law considering the crack width, roughness and tortuosity. In addition, the drag coefficient is first applied to calculate the roughness meanwhile the capillary pores of the crack surface are considered as well. (2) The crack width is the most important parameter to representation the effect of cracks on chloride ions transport. The larger crack widths, the smaller roughness and the smaller of the parameter j is, the higher chloride ions concentration tends in cracked zone. (3) The concentration of chloride ions in cracked zone tend to decrease with the increase of crack depths but not everywhere. Further, the larger the crack width is, the deeper the advection is. (4) The chloride ions concentration vertically from the crack surface tends to be smaller than that at surface, and the higher concentration of chloride ions at the crack surface, the concentration also tends be higher than others at the same distance from the crack surface. (5) Simplified parameters are proposed here to calculate the profiles of chloride concentration in cracked zone, especially, in cracked zone with 0.1 mm at appearance, the depths of convection zone might approximately be 20 mm, while for 0.2 mm, it might be 35 mm. In addition, the equation to estimate the diffusivity also proposed here considering the characteristics of cracks.
Acknowledgements The financial support from the National Basic Research Program (973 Program) (Grant No. 2009CB623204), of the People’s Republic of China and the National Natural Science Foundation (Grant Nos. 50838008 and 51178413) is gratefully acknowledged. References [1] Maekawa K, Ishida T, Kishi T. Multi-scale modeling of concrete performance. J Adv Concr Technol 2003;1(2):91–126. [2] Raharinaivo A, Brevet P, Grimaldi G, Pannier G. Relationships between concrete deterioration and reinforcing-steel corrosion. Durability Build Mater 1986;4(2):97–112. [3] Mangat P, Gurusamy K. Permissible crack widths in steel fibre reinforced marine concrete. Mater Struct 1987;20(5):338–47.
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