Crack-altered durability properties and performance of structural concretes

Cement and Concrete Research 124 (2019) 105811

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Crack-altered durability properties and performance of structural concretes a,⁎

T

b,1

Kefei Li , Le Li a b

Civil Engineering Department, Tsinghua University, Beijing 100084, People's Republic of China Department of Civil Engineering and Architecture, East China Jiaotong University, Nanchang 330013, Jiangxi, People's Republic of China

A R T I C LE I N FO

A B S T R A C T

Keywords: Concrete Cracking Transport Properties Specification

This paper reviews the cracking-altered transport properties and durability performance of structural concretes. To this purpose, this paper addresses firstly the mass exchange and transfer in single cracks, including the ion adsorption and dissolution processes of fracture surface. Then, the effect of multi-cracks is investigated successively through the geometry description, theoretical modeling of transport properties and the relevant experiments. Afterwards, the crack control with respect to durability is reviewed via experimental study and technical specifications. From this comprehensive review, it is shown that (1) the liquid flow is determinant for the mass ex0change and transfer in crack and further study should be aimed at the drying-wetting actions, (2) the transport through multi-cracks highlights the connectivity and the different roles of crack opening; (3) rational crack control on loading cracks should extend to concrete cover quality and thickness in terms of external actions.

1. Introduction: cracking issue in durability context Concretes are brittle and liable to cracking. Accordingly structural concretes in service bear cracking from different causes, including the early-age thermal shrinkage, the autogenous shrinkage, long-term drying shrinkages, and/or mechanical loadings. Once cracking occurs, the concrete material actually provides, in addition to its internal pore network, new transport paths for the external aggressive agents, such as liquid water, CO2 and aqueous ions. From this point of view, the longterm durability of structural concretes can be affected. That is why limiting the cracking extent, “crack control” in engineering terms, is a crucial issue to assure the durability performance of structural concretes. Actually, the crack issue with respect to durability has long time been the concern of engineering and research communities. The earliest recorded investigation can be traced back as early as 1930s [1], and the target was to study the corrosion of reinforcement steel in a hollow reinforced concrete (RC) pile bearing flexural cracks. Since then, the study on the crack-altered properties and durability processes has accumulated for concrete materials. However, the mechanical and nonmechanical cracks were investigated separately: mechanical cracks were created in specimens, under compressive or tensile loadings, the permeability was measured on the mechanically damaged specimens, and the permeability change was related to the applied stress level [2–4]; liquid or gas permeability was measured on concrete specimens

subject to different (drying) shrinkage schemes and qualitative conclusions were made concerning the specific treatment [5,6]. Most exhaustive review on the relevant results can be found in ([7], Chapter 6). Note that, before concrete materials, the crack-altered transport properties have been extensively studied for geotechnical materials such as rocks, with very systematic modeling and description of the cracks [8–10]. The earlier studies on the mechanical cracks attempted to address the alteration of transport properties directly by loading stress levels, and usually no further attempts were made to quantify the crack characteristics in these studies. Several studied showed that the stress level is not the relevant indicator. After measuring the air permeability on mortar specimens under tri-axial stress conditions, Meziani and Skoczylas [11] found three typical phases under stress application: below 0.2fc level (fc for compressive strength), the aggregates and cement paste matrix are mechanically compacted, leading to a slight decrease of permeability; during the 0.2–0.7fc interval, the permeability is nearly constant regardless of the stress change; as compression reaches 0.7fc, cracks appear (around the aggregates) and the corresponding permeability increases, mainly due to the cracking. Actually, a simple poromechanical calculation can help to clarify the pure mechanical contribution. Assume concrete a porous medium, and its constitutive equation writes [12],

⁎

Corresponding author. E-mail address: [email protected] (K. Li). 1 Contributing author. https://doi.org/10.1016/j.cemconres.2019.105811 Received 31 January 2019; Received in revised form 25 June 2019; Accepted 8 July 2019 0008-8846/ © 2019 Elsevier Ltd. All rights reserved.

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Table 1 Fractal dimension values D of fracture surfaces before and after flow tests (LR stands for length ratio algorithm for 2D evaluation of short side of surface, CC stands for the cubic covering algorithm for 3D evaluation of rough surface [19]). No.

Material composition

C1 C2 C3 C4 C5 C6

σ − σ0 =

Cement paste, w/c = 0.40 Mortar, w/c = 0.6, fine sands Mortar, w/c = 0.5, standard sands Mortar, w/c = 0.6, standard sands Concrete, w/c = 0.6, fine/coarse aggregates Concrete, w/c = 0.6, without fine aggregates

K K (ϕ − ϕ0) − ⎛ + b⎞ (p − p0 ) b ⎝ bN ⎠

Fractals before flow 2D (LR)

3D (CC)

2D (LR)

3D (CC)

1.050 1.057 1.083 1.097 1.105 1.199

2.027 2.070 2.097 2.111 2.127 2.131

1.017 1.033 1.061 1.077 1.122 1.163

2.023 2.058 2.088 2.089 2.111 2.113

Winslow [17] showed that the cracks of cement-based materials have fractal properties, meaning the actual crack length L (2D), or the area of the rough surface A (3D), depends actually on the observation scale Δr, i.e.,

(1)

in which σ, σ0 are stress and reference stress (MPa), ϕ, ϕ0 are porosity and reference porosity (−), p,p0 are pressure and reference pressure in pores (MPa), K, N are the bulk modulus and Biot's tangent modulus (MPa) and b the Biot's coefficient (−). For drained case (p ≡ p0), taking the values for mortar materials, K = 3000–15,000 MPa, b = 0.04–0.35 [12], the stress level needed to make perceptional change to the microstructure, e.g. change of 1% in porosity, can be calculated as,

p ≡ p0 : Δσ =

K (ϕ − ϕ0) = 85.7~3750MPa b

Fractals after flow

L (Δr ) ∝ (Δr )1 − D , A (Δr ) ∝ (Δr )2 − D

(3)

in which the exponent D is referred to as the fractal dimension of crack, adopting 1.0 ≤ D ≤ 2.0 for crack length and 2.0 ≤ D ≤ 3.0 for fracture rough surface. Actually, for cement-based materials, the fractal property in Eq. (3) can only hold for a certain range of dimension scale, e.g. this dimension holds for a range of 100 μm–1 mm for fractured mortar surfaces [18]. In this range, the fractal dimension can be regarded as a pertinent indicator for the surface roughness, and the typical roughness values are reported in Table 1 for several typical cement-based materials. The 3D surface geometry of these materials was measured by a large-scale laser profilometer [19] with the vertical resolution ± 7 μm (Fig. 1). From the fractal values (before flow tests), one can see that the fractal dimension values are sensitive to the material composition, and with the addition of aggregates the fractal dimension increases, meaning the fracture surfaces become rougher.

(2)

From the obtained magnitude, one can see clearly that un-realistic high stress, at least 85 MPa, is needed for this change. Since the service stress in structural concretes is far lower than this value, the pure influence of stress on the alteration of transport properties can be neglected. In other terms, if ever the transport properties are changed under loadings, this change must be attributed to the cracking, which is the pertinent indicator for change of transport properties and processes. On the basis of the above state-of-the-art, the paper attempts to give a comprehensive review on the impact of cracks on the durability performance of structural concretes. No difference will be made between mechanical and non-mechanical cracks, only single crack and group of cracks are treated differently for their role in the mass exchange and transfer through concrete materials; and both fundamental research and engineering practice are covered. Following these lines, this paper is arranged as follows: the mass exchange and transfer in single cracks is discussed in Section 2, the transport through multicracks (crack groups) is addressed in Part 3, afterwards the crack control in view of long-term durability is discussed in Part 4, and the conclusions and perspectives on the crack issues are given in Part 5.

2.2. Mass transfer: liquid flow The simplest law accounting for the flow between two parallel surfaces is the steady laminar flow of Newtonian fluids, HagenPoiseuille's law, depicting a cubic law between the flow rate (m2/s) and the pressure gradient ΔP/L (Pa/m),

Q=

1 w 3 ΔP 12 μ L

(4)

in which w stands for the surface distance (m) and μ for the fluid viscosity (Pa s). Note that the flow rate in Eq. (4) refers to the volumetric liquid flow along unit length perpendicular to the crack opening w and flow direction. This law constitutes the basis of the permeability of single cracks [20], involving the only geometry parameter of distance between surfaces w. We refer to this distance as the crack opening width in the following. A revised version introduced the surface roughness into the law [21],

2. Mass exchange and transfer in single cracks 2.1. Geometry description The geometry description of a single crack, from mechanical or nonmechanical cause, is fundamental for the mass exchange and transfer. From tri-dimensional (3D) view, a single crack is constituted of two fractured surfaces, rough in nature, and the space enclosed and limited by the two surfaces, filled by one or several fluid phases. Thus, the two basic parameters of crack geometry are the surface geometry and the crack opening aperture. Note that, most cracks, in laboratory or in-field, are observed in bi-dimensional (2D) context. Ammouche et al. [13] used the length to width ratio and the packing density index to describe the random geometry of these 2D cracks. For the roughness characterization of fractured surface, several methods have been proposed in the literature so far. The local height difference, with respect to a certain baseline, was first used to characterize the surface roughness [14,15], bearing the unit of length. The crack roughness is also represented by the tortuosity, ratio between the actual length (surface) and the projection length (surface) [16].

−1

Q = g (Δh) ×

w 3 ΔP 2Δh ⎞ ⎤ with g(Δh) = ⎡1 + 6.0 ⎛ ⎢ μ L Dh ⎠ ⎥ ⎝ ⎣ ⎦ ⎜

⎟

(5)

Here Δh refers to the average surface height change (m) and Dh to the hydraulic radius (m), ratio between the flow section area and the wetting perimeter. From experimental point of view, the cubic law is often pre-assumed and the measured flow rate is calibrated on the law in Eq. (4), resulting in an equivalent crack opening we and flow rate coefficient α,

we =

3

12

QLμ w 3 and α = ⎛ e ⎞ ΔP ⎝w⎠

(6)

Using this assumption, the equivalent crack width was investigated for different cement-based materials [20,22]. Some important 2

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Fig. 1. Illustration of topology of rough surface of cement-based materials: fractured surface of mortar specimen (C2) (left), and reconstituted surface from laser profilometer scanning (right).

depends much on the external pressure levels and the “sealing effect” from cement secondary hydration should not be over-estimated. From the obtained results, the transient flow behaviors for cracks with opening larger than 0.2 mm should be taken as safety margin rather than acquired performance to be exploited in service.

observations were made from these works using different devices: first, the equivalent width is usually far less than the real geometry crack width, usually one magnitude below the geometry width, and this difference is far more important for smaller openings; second, the liquid (water) flow observed in experiments adopted an apparent transient behavior, having much larger flow at the first few hours and then decreasing substantially with time. The observations from Edvardsen [23] confirmed that the water flow in cracks with 0.1–0.2 mm opening stops after flow in 14 d, and cracks of 0.3 mm opening underwent substantial decrease of 80% compared to its initial flow rate. These two observations are fundamental for judging the impact of cracks on the liquid flow across concrete elements (parts) containing cracks. The difference between the equivalent opening and the geometry one is due to the surface roughness: the surface roughness generally provides local friction to water flow, thus decreases the total flow rate; the smaller is the crack geometry opening, the more important is the friction effect. However, the surface roughness can also cause the surface channeling: higher surface roughness leaves more flow channels, resulting in larger flow rate [22]. For the transient behavior of liquid/water flow in the cracks, it was believed that the secondary hydration of the rough surface would be responsible due to the newly formed hydrates between the surfaces [23]. However, the flow tests through a same crack with different infiltrating liquids, water and alcohol, do not support this argument: the flow does decrease with time, but for both water and alcohol, cf. Fig. 2. Since alcohol does not react with any cement minerals, the transient behavior of alcohol is more related to the intrinsic characteristics of liquid flow across rough surface. Actually, the friction can well account for this decrease: the rough surface generates friction with the flowing liquid particles, dissipating the kinetic energy associated with the flow into thermal energy and deviating the flow from a classical Newtonian flow; on the macroscopic scale, the flow rate is decreased, even to zero if the friction is important. But this transient effect depends much on the external driving force such as the pressure gradient on the liquid flow. Actually, most important “sealing effect” were always obtained from the flow under relatively low external pressures [19]. Correct estimation of flow in cracks is of utmost importance for both functional and durability performance. On the basis of the available knowledge, the crack flow should take into account both the crack opening and the surface roughness, and the surface roughness acts through the crack opening, cf. Eq. (5). The transient behavior of crack flow is beneficial for structural functionality and durability, but it

2.3. Mass exchange: leaching and binding As liquid flows across cracks, mass exchange and transfer can occur between the flowing liquid and the rough surface. As the flowing liquid refers to water or aqueous solution, the mass exchange and transfer can include several processes: (1) the adsorption of aqueous ions in the flowing solution by the surface, (2) dissolution of the solid phases on the fracture surface, and (3) the direct chemical reactions between water and the exposed (reactive) surface. For cement-based materials, the first process refers to the ion-exchange between the aqueous solution and the fracture surface. Depending on the ion species, this adsorption can have impact on different problematics: in radioactive waste disposal the adsorption of radionuclides in the crack will determine the long-term safety of the disposal works [24], and the adsorption of chloride ions will affect the electrochemical stability of the embedded reinforcement bars [25,26]. The second process involves the dissolution of the most soluble phase of cement hydrates, Portlandite (Ca(OH)2) or CH, and the subsequent transport of dissolved ions out to the environment. The long-term effect of this dissolution is to weaken the material solid matrix, named as “leaching” in durability context. The third process includes all secondary reactions between the flowing water and the exposed cement grains on the fracture surfaces. This section addresses the surface adsorption of chloride ions and the surface leaching of CH, and no secondary hydration issue is treated. To facilitate the discussion, a think model is illustrated in Fig. 3 for the surface adsorption and leaching. The general equation of the mass exchange and transport writes [27],

∂cm ∂ 2c m ∂c + m→ = DL −u m ∂t ∂x 2 ∂x

(7)

in which the term cm refers to the concentration, at time t and distance x from the inlet of liquid, of the adsorption ion (Cl−) or dissolved ion (Ca2+) in kg/m3, DL is the diffusivity of the solute (m2/s), m→ is the adsorption or dissolution rate in kg/(m3s), and u is the flow rate (m/s). 3

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Fig. 2. Flow tests across a mortar crack with different openings and different infiltrating liquids: the transient flow (left) and the flow rate normalized by its initial value during first 10 min (right). The C3 mortar in Table 1 was used, different openings were created by inserting spacer between the two facture surfaces and the “opening closed” case corresponds to two fractured surfaces mechanically tightened with no spacer inserted [22]. The external pressure gradient is kept as 1.8 kPa across 30 mm.

adsorption parameters are given in Table 2. In the same table are given the parameters regressed from the static adsorption in NaCl solution (0.5 mol/L). Compared to the static adsorption, the adsorption in flowing condition has lower initial adsorption rate and lower adsorption capacity. The underlying reason for lower adsorption rate and adsorption capacity in flowing condition can be attributed to multi-layer adsorption of chlorides on the fracture surface. At the surface, the chlorides exchange with OH− groups. Since the surface is usually negatively charged, the cations, such as Na+, K+, are also adsorbed. This creates a positively charged layer, which in turn attracts negative ions (Cl−) to build up, forming a multi-layer adsorption structure. Certainly, the outer layer Cl− is less stable than the inner layer ones. This multi-layer structure accommodates the chlorides, thus determines the Cl− adsorption capacity. Under flowing condition, the loose structure of outer layer is much affected by the liquid flow, and the thickness of adsorption layer is decreased, thus lowering the adsorption capacity. This mechanism is illustrated in Fig. 4(left). As for the adsorption kinetics, the values in Table 2 show that chlorides will take much longer time, 20 times longer for hardened cement paste (C1) and 6 times longer for mortar (C3), to reach the adsorption sites, migrating across the flowing solution than in the static solution. Accordingly, the flow perturbs both the adsorption sites (capacity) and kinetics of the fracture surface. This study reveals that the fracture surface has ion adsorption capacity but notably affected/decreased by the liquid flow in the crack. In practice, this would imply that the adsorption capacity of aggressive ions of fracture surfaces as the solution flows through the cracks should not be regarded as a reliable property against the penetration of external aggressive ions, because this capacity is much affected by the flow rate.

2.3.1. Adsorption The adsorption of chloride ions by cement-based materials includes the chemical binding, forming Friedel's salts [28], and the physical binding, through ion exchanges between Cl− ions in solution and the OH− groups on cement hydrates, calcium silicate hydrates (CSH) [29]. We assume that the chloride adsorption on the fracture surface follows the same mechanisms. Since the majority part of external chlorides is fixed through physical binding, only physical binding is considered herein. To quantify the chloride adsorption from the flowing water solution, we define first the adsorption kinetics in a static solution as a reference, i.e., ad m→ =

0 rad [1 − exp(−kad t )] kad

(8)

ad m→

stands for the adsorption quantity on concrete surface in which (kg/m2), kad is the adsorption rate (s−1), and rad0 is the initial adsorption rate (kg/(m2s)). Note that rad0/kad gives the final (equilibrium) quantity. Applying this adsorption rate as the source term in the mass conservation of chloride ions in Eq. (7) gives,

∂cm 2 0 ∂c + rad exp(−kad t ) = −u m ∂t bw ∂x

with m→ =

ad 2 dm→ b w dt

(9)

Here the term u defines the flow rate of chloride-contained aqueous solution (m/s) and bw the crack opening (m). Actually Eq. (9) depicts also how the basic properties, adsorption rate kad and capacity rad0/kad, can be calibrated through experiments: one needs to measure the chloride concentration cm, the chloride concentration gradient dcm/dx, and the flowing rate u, and the two properties can be calibrated by a time-series recording. Following this line, experiments were performed for chloride adsorption of fracture surface using NaCl solution (0.5 mol/ L) as the infiltrating aqueous solution. The regressed chloride

Fig. 3. Interaction between flow water and fracture surfaces in concrete crack: adsorption and dissolution processes (left), and 1D think model for the mass exchange and transfer (right). 4

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Table 2 Regressed adsorption and leaching parameters for fracture surfaces of cement-based materials (n.a. for not available). Material

Parameter −

−7

0

2

Initial Cl adsorption rate rad (10 kg/(m s)) Final Cl− adsorption capacity rad0/kad (10−3 kg/m2) Leaching rate of Portlandite kdis (10−9 kg/(m2 s)) Initial Cl− adsorption rate rad0 (10−7 kg/(m2 s)) Final Cl− adsorption capacity rad0/kad (10−3 kg/m2) Leaching rate of Portlandite kdis (10−9 kg/(m2 s)) Initial Cl− adsorption rate rad0 (10−7 kg/(m2 s)) Final Cl− adsorption capacity rad0/kad (10−3 kg/m2) Leaching rate of Portlandite kdis (10−9 kg/(m2 s))

C1 cement paste

C3 mortar, w/c = 0.5

C4 mortar w/c = 0.6

⎜

[Ca2 +][OH−]2 ⎞ K CH ⎠ ⎟

(10)

in which kd is the dissolution rate of CH (kg/(m2s)) and KCH is the dissolution coefficient, equal to 10–5.18 at 25 °C. Note that this law assumes a spherical geometry for the CH crystals [30]. Using this term as dissolution source in Eq. (7), the mass conservation of calcium ions, Ca2+, writes,

∂c (Ca2 +) 2 4c 3 (Ca2 +) ⎞ ∂c (Ca2 +) + k d ln ⎛ = −u ∂t bw ∂x ⎝ K CH ⎠ ⎜

⎟

with m→ =

2 dis m→ bw (11)

2+

Crack flow

10.515 64.6 0.0216 5.046 41.6 0.088 5.260 41.8 0.123

0.494 7.34 0.374 n.a. n.a. 0.130 0.856 5.28 n.a.

the driving force. With flow in crack, this diffusion layer is much perturbed and decreased in thickness, and the free water front comes closer to the surface, increasing the driving force for leaching. Note that the driving force is increased to a much higher level for cement paste than for mortar after the regressed values in Table 2. Another related issue is the possible geometry change by the CH dissolution on the rough surface. The values of fractal dimension are given in Table 1 for the fracture surfaces before and after crack flow tests, consisting 7 cycles of flow and including 36 h of water flow and 12 h of air-drying in each cycle. From these values the fractal dimension seems to decrease but to a very limited range, possibly related to the CH dissolution and carbonation. Since surface leaching is assumed to be detrimental for cement-based materials, the crack flow will enhance this process. In practice, this would imply that the leaching process should not be underestimated once the crack is subject to infiltrating water.

2.3.2. Leaching The surface leaching refers to the dissolution of the exposed CH on the fracture surface of crack by the flowing water. Again the dissolution of CH in static water is chosen as the reference to deduce the impact of crack flow. The static leaching in pure water, without Ca2+ and OH− ions initially, can be expressed through the following kinetic law, dis m→ = k d ln ⎛ ⎝

Static case

3. Mass transport in multi-cracks

2+

in which c(Ca ) denotes the aqueous concentration of Ca ions. Here the pertinent parameter is the dissolution rate kd (kg/(m2s)), and this parameter adopts different values for static leaching and flow leaching. Actually, Eq. (11) shows also how to calibrate this quantity: one needs to measure the Ca2+ concentration change with time, the gradient of Ca2+ concentration, and the flow rate u; and then the leaching parameter kd can be calibrated following a time-series recording of Ca2+ concentration at the outlet of crack. Such experiments were performed on the hardened cement paste (C1) and the mortar (C3), and the regressed values are given in Table 2. From these values, one can see that the flow promotes the surface leaching rate, 17 times for the cement paste and 1.5 times for the mortar. The underlying mechanism of the impact of flow on leaching is illustrated in Fig. 4(right). The leaching of calcium involves actually dissolution of CH in pores and the subsequent diffusion of Ca2+ ions out to the water environment in the crack. In static case, the diffusion layer exists between the surface and the free bulk water with cCa = 0. And the driving force for this dissolution-diffusion process is actually the cCa gradient across this diffusion layer. The larger is this layer the weaker is

Static water (boundary)

Bulk solution

Flowing water (boundary) Surface Ion diffusion Leached zone

Material

Adsorption layers

Static water

Reduce layer

Flowing water

Diffusion layer

Bulk solution

Na+ H2O

Conceptually, the geometry description of each single crack in the crack group can constitute the complete description of the total network. With respect to mass transport, however, multi-cracks introduce two different features: the interference among the single cracks, and the local connectivity of cracks. Since we are interested finally in the altered transport properties, these characteristics are important to be

Leaching depth

Friedel salt

3.1. Geometry description of crack networks

Reduce diffusion layer

Cl-

Though single crack behaviors can address a lot of engineering problems as to the mass exchange and transfer processes, cracks normally exist in groups and impact on the transport and durability processes of concrete through networks. This section, on the basis of the foregoing discussion on single cracks, continues to treat the topics related to multi-crack and crack networks. Accordingly, this section addresses firstly the geometry description of multi-cracks, the theoretical modeling is then elaborated for the altered transport properties, and the relevant experimental works are reviewed lastly.

Dissolution front

Material

Solid surface (impermeable)

Intact material Solid Ca2+

Fig. 4. Illustration of impact of flow on the mass exchange in crack: flow-reduced surface adsorption (left) and flow-enhanced surface leaching (right). 5

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clustering can be defined as,

taken into account. So far, the transport related geometry characteristics of crack networks can be classified into average-based and percolation-based quantities. The average-based quantities address the crack network as a whole while the percolation-based quantities describe the interaction and correlation among the cracks. These two types of geometry characteristics are given below.

f=

1 A

n

∑ xi

and ρ =

i=1

1 A

n

x

2

∑ ⎛ 2i ⎞ i=1

⎝

⎠

(12)

The geometry quantities associated with these cracks can include the crack length distribution dist(xi) and the opening aperture of crack apt(xi). Globally, the orientation of all cracks in the group can be described through an orientation factor ω,

1 ζ (θ) = A

n

∑ i=1

x 2 ⎛ i ⎞ cos2 θi ⎝2⎠

3.2. Crack-altered transport properties: theoretical modeling The theoretical modeling of transport properties of porous materials incorporating multi-cracks resorts to two theories, complementary one to the other. The effective medium theory (EMT) assumes the same role in transport for each single crack, regards cracks as inclusions in the material (homogeneous) matrix, and uses homogenization techniques to express the altered transport properties in terms of the crack characteristics [35]. During the last decades, the EMT has found successful applications in multiple fields [36]. However, the inherent assumption that each single crack contributes equally to the transport distorts the reality: as the cracks form spanning clusters the isolated cracks contribute no longer to the transport process, i.e. the connectivity of cracks and the interaction among cracks are missed in EMT. The percolation theory, on the other hand, addresses the role of crack clustering in the transport, especially as the spanning cluster is formed. Actually, the two theories address the two extremes for crack-incorporating materials: the EMT for evenly distributed but weakly connected multi-cracks, and the percolation theory for highly clustered cracks approaching spanning state. Both theories should be adapted to account for the intermediate cases. In the following, some 2D results from the two theories are briefly reviewed and the further adaptation of both theories to the cases in between is out of the scope of this paper, and the interested readers can refer to recent publications from the same authors.

ζ (θ)min with 0 ≤ θ ≤ 360∘ and ω = ζ (θ)max (13)

in which θi is the angle between crack xi and permeation direction. The factor ω = 1.0 as all cracks are distribution evenly in all directions and ω = 0 as all cracks are aligned to one direction. From average viewpoint, the crack connectivity, ϕ, can also be defined, m

ϕ=1−

n

∑ xj 2 / ∑ xi2 j=1

i=1

(14)

in which the cracks xj=1,m are assumed isolated. The parameters Ls, ω, ϕ constitute a complete set of averaged-based description of crack networks, and Fig. 5 illustrates the crack network on the cross section of a concrete specimen damaged under axially compressive loading [31,32]. 3.1.2. Percolation-based quantities The clustering of cracks in a network will have large impact on the transport properties of porous materials incorporating cracks [33]. The percolation-based quantities focus on the description of the clustering of cracks. If two cracks, spaced by distance r, belong to the same crack cluster, they are correlated to this distance r. Then, the correlation function C(r) depicts the probability of two cracks within distance r belonging to the same crack cluster, scaling to distance r through a fractal dimension Dc,

C (r ) ∝ r Dc

(16)

in which ξ is the size of crack cluster, Ld is the domain size, P is the likelihood of a crack being connected to the main crack cluster, ρ is the average-based crack density defined in Eq. (12). This connectivity states the fact that if all cracks belong to one cluster, P = 1.0, and this cluster spans across the domain, ξ = Ld, then the connectivity f = 1.0. Fig. 5 illustrates a 2D random crack network with the corresponding percolation-based quantities. Again neither average-based nor percolationbased parameters are defined for pure geometry interests: and these quantities are to account for the transport properties altered by the multi-cracks.

3.1.1. Average-based quantities As the domain A contains n cracks with the crack lengths xi(i=1,n). The crack specific length Ls and the crack density ρ can be defined as,

Ls =

ξ [P (ρ, Ld )](3 − Dc ) Ld

3.2.1. EMT model Let us define a representative elementary volume (REV) of porous medium incorporating crack inclusions, cf. Fig. 6. The basic assumption is that each crack inclusion affects the local flow field in an elliptical domain, having the (same) aspect ratio γD = b/(a + b) and the opening ratio γC = bw/a. The domain size is related to the crack density ρ

(15)

On the basis of this definition, the connectivity due to the crack

Fig. 5. Averaged-based and percolation-based quantities for crack networks: cracks created by axially compressive loading in HVFC5 specimen [31] having Ls = 0.087, ρ = 0.195, ϕ = 0.576, ω = 0.189 (left), and crack network generated numerically with ρ = 0.75, Dc = 1.62, f = 0.367 [34]. 6

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Ellipcal atmosphere b

a+b

2a

Solid porous matrix Fig. 6. Homogenization of crack inclusions in porous matrix through EMT.

Fig. 7. Relative permeability of porous medium containing multi-cracks in terms of crack density and opening ratio: EMT-IDD model results (left), and percolation results with keff0 = 11.57kmγC0.2, ν = −0.924 for scaling law in Eq. (21).

through [37],

⎛ kIDD ⎞ = ⎝ k m ⎠2D ⎜

1+ (γD )2D =

4 πρ

2 1+

−1 4 πρ

⎜

⎟

(19)

(20)

in which the term τ refers to the tortuosity of the crack cluster, f is the percolated based connectivity, defined in Eq. (16), ρ and γC are respectively the crack density and the crack opening ratio used in Eq. (19). Rearranging the terms provides the following scaling law,

π

π

π

1 − 2 ρ (1 + γC)[−γC + (1 + γC ) γD ]

keff = keff (k m ; f , τ , ρ, γC)

1 + 2 ρ (1 − γD ) 1 − 2 γD ρ

π

1 + 2 ρ (1 + γC )[γC + (1 + γC )(1 − γD )]

3.2.2. Percolation model As spanning cluster forms, the permeability of cracked medium is to increase greatly (percolation) and the permeability will depend only on the spanning cluster. The scaling law for permeability writes,

(17)

An interaction direct derivative (IDD) method [38] from EMT is adopted here to address the permeability of the porous medium incorporating crack inclusions. The mathematic details are available in [39], and the results for two cases are given below. As the crack permeability is much larger than the matrix permeability, i.e. ki ≫ km, the permeability of the cracked porous medium writes,

⎛ kIDD ⎞ = ⎝ k m ⎠2D

⎟

0 keff = keff (k m , γC ) |ρ − ρc |ν

(18)

(21)

in which ρc is the crack density at percolation, and the exponent ν lumps the connectivity and the cluster tortuosity, details see [34]. Finite element method was used to compute the permeability in finite domain containing random crack networks with different percolation characteristics, and to calibrate the law in Eq. (21), see Fig. 7.

with km for the permeability of matrix. The assumption ki ≫ km is usually acceptable for permeability of cement-based materials since for a crack with width 1 μm the corresponding intrinsic permeability is scaled to 10−12 m2 while the value for cement-based matrix (water permeability) is usually lower than 10−16 m2. That is why in Eq. (18) there is no information on the crack opening. Eq. (18) predicts a gradual increase of permeability with the crack density, which confirms the average nature of this solution. In other terms, this EMT-IDD result fails to capture the percolation, and applies to evenly distributed multicracks which are far from the spanning cluster stage. As the crack opening gets finer or transport properties other than permeability are concerned, the crack opening can have a role. Always from the EMTIDD method, the result considering the crack opening writes [37],

3.3. Crack-altered transport properties: experiments Besides the theoretical results, experimental results are available to address the crack-altered transport properties. The results in this category are rather numerous and only the studies with pertinent characterization on the crack geometry are reviewed. Zhou et al. [31,32,40] performed systematic experiments on the concrete damage, the crack 7

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4. Crack control for durability

characterization and the altered transport properties. Cylinder concrete specimens were loaded axially in cyclic compression to generate damage and then the concrete discs were sawed out from the middle part of cylinder to extract the crack geometry (Fig. 5 left). Afterwards, the altered transport properties were measured on these disc specimens. The experimental details can be found in [31]. The main results of this investigation are that the crack length distribution on the cross section follows a lognormal distribution, the crack density is correlated to the averaged-based crack connectivity, and the altered gas permeability has a clear correlation with the crack density. The cracks generated is by nature 3D, and the correlation study could only be done through 2D cross section, which largely limits the correlation validity. To solve this problem, another study turned to pure 2D context and investigated the geometry characteristics and the altered transport properties both in 2D context [41]. The pellet specimens were prepared and split manually to obtain different crack patterns. Then the pellets were reassembled into an integral specimen and treated by epoxy-resin, leaving a 2D crack pattern in thickness direction. Afterwards, these pellet specimens were subject to electrical resistivity and drying tests. Using the measured water vapor sorption isotherms, the liquid water permeability was calibrated from the drying tests. On the basis of these measured/regressed transport properties, the relation between the crack geometry and the altered properties were addressed, cf. Fig. 8. The main observation is that liquid permeability and electrical conductivity scale to the opening ratio not through the same exponent, 1.7 for liquid permeability and 0.45 for electrical conductivity, meaning the crack path resists more the electrical current than the liquid flow.

This section reviews the research results and technical regulations on concrete crack control in view of the long-term durability and service life. Crack control is closely related to the foregoing fundamental knowledge on the mass exchange and transport in single and multicracks, but more oriented to engineering practice. Several topics are addressed herein: the impact of cracks on steel corrosion, the specifications on crack width, and the crack control for durability of concrete infrastructures. 4.1. Impact of cracks on corrosion In the context of reinforced concrete, the crack issue has been raised due to the concern on its promoting nature for the corrosion processes of embedded steel. The relevant corrosion experiments on cracked concrete structures have been performed and the results from François et al. [42–46] constitute the most complete datasets on the long-term observation of steel corrosion in cracked and un-cracked zone. From the published data, the authors concluded that the initial mechanical cracks, within 0.3 mm, have little relevance to the corrosion-induced deterioration over a 26-year span of marine salts exposure, and the steel-concrete interface quality, due to the pouring direction and the top bleeding, have more impact on the long-term corrosion process of the reinforcement bars [43]. Otieno et al. [47–49] tested the corrosion stability of embedded steel in RC beams containing cracks under bending loadings. The salt ponds were constructed on the crack opening and the salt solution penetrates to the steel surface. The specimens were subject to drying-wetting cycles and the corrosion behaviors of the embedded steel bars were recorded. The main finding is that the corrosion stability of embedded steel depends much on the ratio between crack opening width and the concrete cover thickness. From the same experiments, though behaving similarly, the specimens exposed in natural conditions have much lower corrosion current than the laboratory specimens. Again, in laboratory accelerated conditions, Gantous et al. [43] carbonated cracked mortar specimens, then exposed the crack-bearing specimens to drying-wetting cycles, and recorded the corrosion behaviors of the embedded steel bars. They found that the corrosion at the crack bottom occurred but limited the subsequent steel corrosion along the steel surface so that the corrosion development length tends to stabilize after 30–40 drying-wetting cycles. All these investigations revealed actually the importance of the drying-wetting actions on the corrosion process, which also helps to identify the difference between laboratory results and in-filed observations. It is attempted here to synthesize the acquired knowledge through a schematic model of corrosion in cracks, cf. Fig. 9. Conceptually, cracks in concrete cover will provide additional path for transport of external

3.4. Further discussion The quantitative results on the crack-altered transport properties can help to gain some insight into the long-term durability of concrete containing single cracks or multi-cracks. First, the geometry of cracks is of utmost importance and the crack opening is not always active in the alteration of transport properties. For single cracks, the flow depends strongly, in addition to the surface roughness, on the crack opening. For the mass exchange and transfer, the crack opening plays an indirect role on the kinetics through its impact on the flow rate in cracks. Accordingly, for single cracks, the opening is always an important factor to consider. For multi-cracks, the EMT and percolation results show that the impact of the opening is not significant for crack networks before percolation, and the role of crack opening becomes significant only as the cracks are highly connected. Furthermore, the experimental results show that the crack opening contributes differently to different transport processes, liquid permeation or electrical conduction, after the crack percolation.

Fig. 8. Correlation between crack geometry and altered transport properties in 2D context: liquid permeability (left) and electrical conductivity (right). 8

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Fig. 9. Enhanced corrosion model for steel bars embedded in cracked concrete cover, inspired by works from Tuutti [50] and others [42,43].

aggressive agents, such as CO2 and Cl−. Actually, the corrosion cannot be seen individually around the crack and the exposed steel surface underneath, but involves relevant zones associated with the crack. In Fig. 9 are marked three different zones: the cracking zone (A), the debond zone (B) and the intact zone (C). As the aggressive agents penetrate to the bottom of cracking zone, instantaneous corrosion can be induced but afterwards whether the corrosion will develop further depends much on the local environment: if the corrosion products are stable further corrosion can be stopped by the scarcity of reactants such as oxygen; if the corrosion products are removed constantly from the crack bottom the further corrosion can develop all along. The former is more probable in natural exposure while the laboratory accelerated schemes, with severe drying-wetting cycles, favors the latter. Meanwhile, the zone B contains damage as the cracking occurs and the steel surface is debonded from the concrete, all these factors facilitate the mass transport compared to the zone C. Accordingly, the possible kinetics of the corrosion process is the instantaneous corrosion at A bottom (Phase I), the crack is sealed by corrosion products, and the transport process in damaged zone B dominates the corrosion kinetics (Phase II), and corrosion restarting once aggressive agents penetrate through zone B (Phase III). The duration or the existence of Phase II depends much on the local environments, especially the mass transport under drying and wetting in cracks. This synergic image summarizes the different points of view in literature [42,43,50].

around 0.2 mm for severe exposure and 0.3 mm for moderate exposure. These values are chosen mainly from the empirical experiences from engineering practice. From the available experimental results, it is also rational to relate the crack width limitation to the quality and thickness of concrete cover, as JSCE code [55] and suggestions from Otieno et al. [49]. Another relevant issue relates to the long-term evolution of the crack width: during relatively long service life the drying shrinkage and the creep can all contribute to the increase of crack width. Thus, the prescription of crack width limitation should allow for this increase in long-term duration. 4.3. Crack control for durability consideration: critical analysis Both engineers and researchers would agree to impose control on loading cracks for the sake of the long-term durability. However, the methods diverge: the codes such as ACI-318 [52] and AASTHO [53], instead of prescribing limit on crack width, rely on the good practice of reinforcement distribution and steel stress control while other codes, such as EuroCode 2 [51], JSCE [55] and GB/T 50476 [56], provide direct limit on the crack width or crack width/cover thickness ratio. The former is well supported by the long-term tests on RC beams [42–46], showing that initial mechanical (service) cracks do not constitute a risk for subsequent general corrosion on the structures. Another reason for not assigning a crack width is that the surface crack width under (flexural) loading is related to the concrete cover thickness and the surface width can usually distort the real opening on the steel surface [57]. The results in [47–49] would support a more comprehensive specification on crack control for durability, considering other factors such as the concrete cover quality (resistivity), concrete cover thickness. Nevertheless, the notable difference between the laboratory conditions (using drying-wetting to activate the corrosion process) and natural exposure conditions (more moderate) should be noted. Complete knowledge on this issue is far from accomplished while the qualitative message is clear: under moderate exposure conditions

4.2. Specification on crack limitation Nowadays, the crack opening width is considered by design codes as the primary parameter to limit against the negative impact of cracking on the long-term durability of RC structures. Not all the specifications impose explicit limits on the crack width, and these specifications are capitulated in Table 3. From these values one can see that the conventional limit on the crack width, regardless of the concrete cover quality and thickness, is

Table 3 Specification on the limit of service cracks with respect to long-term durability of RC structures. Code/standard

Exposure condition

Specification on crack control

EuroCode 2 [51]

Indoor Outdoor, severe exposure Interior/exterior Not specified Dry air, Humidity, soil, moist air Deicing salts, Seawater/drying-wetting, Water retaining structure Normal Corrosive Severely corrosive Indoor, outdoor Deicing salts Seawater Chemicals

0.4 mm (X0,XC1) 0.3 mm (XC2-4, XS, XD) Maximum reinforcement spacing in terms of steel stress and concrete cover thickness Stress level in reinforcement steel 0.41 mm (dry) 0.30 mm (humid) 0.18 mm (deicing chemicals) 0.15 mm (seawater, drying-wetting) 0.10 mm (water retaining) 0.005c (normal), c for cover thickness 0.004c (corrosive) 0.0035c (severely corrosive) 0.2–0.4 mm (indoor, outdoor) 0.15–0.2 mm (deicing salts) 0.1–0.2 mm (seawater) 0.1–0.2 mm (underground water, chemicals)

ACI-318 [52] AASTHO [53] ACI-224 [54]

JCSE [55]

GB/T 50476 [56]

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the reinforced concrete can tolerate cracks with larger surface width (0.4–0.5 mm) but stricter limit should be adopted for severe exposures (0.2 mm even below). More generally, the crack control for durability is by no means limited only to the loading cracks, which are normally discrete and concentrated in certain structural zones. The cracks, originating from early-age shrinkages, such as thermal, plastic or autogenous ones, can affect the bulk material in concrete structure, and thus the consequence is more important for long-term durability. The return of experience from engineering practice confirms that the cracking due to these inherent causes has wider impact on the corrosion and other durability processes [7].

Declaration of Competing Interest

5. Conclusions and perspectives

References

Cracking introduces fracture surfaces into cement-based materials. The mass exchange and transfer between the fracture surface and the fluid phases in cracks is fundamental to understand the role of crack on the long-term durability of materials. The liquid flow, the surface adsorption and dissolution are reviewed through quantitative results. The general observations are: the surface roughness can alter greatly the flow rate, and under low pressure gradient the surface friction alone can induce a transient flow behavior without any secondary reaction; the liquid flow can notably decrease the surface ion adsorption capacity and adsorption rate; the flowing water can greatly increase the surface leaching rate. On this topic, what is not clear is the mass transport under severe actions like drying-wetting. The deepened insight on this topic would greatly help the interpretation of the local corrosion on the crack bottom and subsequent process, cf. Phase II in Fig. 9. Multi-cracks behave more than the addition of single cracks due to the interaction among the cracks. In concrete materials, the inherent cracks, from different shrinkages and internal chemical degradations, belong to this case. These cracks, contrary to the individual loading cracks, affect all the structure, thus should be treated as multi-cracks. The geometry of multi-cracks can be represented by average-based or percolation-based quantities, and both can describe the altered transport properties in their respective ranges. Conceptually, as the multicracks are evenly distributed and the connectivity is low, as in the case of the early-age plastic shrinkage, the crack density alone is enough to account for the change of properties. However, as the mutli-cracks attain a high degree of connectivity, as in case of some internal expansion reactions, both the density and connectivity are needed to describe the change in properties, and moreover the crack opening will play an important role. Much work remains to be done to apply the obtained knowledge to engineering practice for multi-cracks. The foremost task would be to establish the link between the geometry parameters of multi-cracks and the macroscopic deformation of materials, appealing for fracture mechanics applied to the multi-scale structure of cementbased materials. The crack should be controlled, if not inevitable, to maintain the long-term durability of concretes. The engineering community relies mainly on the good practice of concrete technology to limit the inherent cracking of concrete from early-age to hardening phases. The principle of reinforced concrete needs the concrete and steel to deform together on the tensile part of elements and loading cracks are inevitable. Within certain opening values (0.3–0.5 mm) the cracks are believed not to constitute a risk for durability except for severe exposure conditions such as drying-wetting actions. Some codes specify the steel stress level to control the loading cracks in concrete while others prescribe explicit limit on the crack width or crack width/cover thickness ratio. The explicit limit on crack width is problematic, and the rational specification should at least consider further the concrete cover thickness and quality and the crack width. To this purpose, the long-term observation in natural environments is of utmost importance.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement The research is supported by the National Key Research and Development Program of China Project (No. 2017YFB0309904), NSFC Grant No. 51778332, and Zhejiang Transportation Science and Technology Project No. 2018035.

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