1-s2.0-s0029801811000059-main

7/26/2019 1-s2.0-S0029801811000059-main

http://slidepdf.com/reader/full/1-s20-s0029801811000059-main 1/10

Review

Mesoscale modeling of water penetration into concreteby capillary absorption

Licheng Wang a,n, Tamon Ueda b

a State Key Laboratory of Coastal and Offshore Engineering, No. 2 of Linggong Road, Dalian University of Technology, Dalian 116024, Chinab Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan

a r t i c l e i n f o

Article history:

Received 3 June 2010Accepted 31 December 2010

Editor-in-Chief: A.I. IncecikAvailable online 19 January 2011

Keywords:

Water penetration

Capillary absorption

Mesoscale modeling

Lattice network model

Concrete

a b s t r a c t

Water penetration into concrete by capillary absorption plays a very important role in the ingress of

contaminative substances when the structures are built in aggressive environments. In the presentpaper the lattice network model is proposed based on the unsaturated flow theory to predict the water

penetration into concrete. On the mesocale level, concrete is treated as a three-phase composite. By

means of the Voronoi diagram meshing strategy, the lattice network model of concrete with different

types of lattice elements is developed. The corresponding transport properties are assigned to the

lattice elements in the network falling in different phases. As a result, the lattice elements are idealized

as conductive ‘‘pipes’’ in which uni-directional flow can be realized between the two nodes of the

elements. Parameters in the lattice network model, such as the sorptivity and porosity of the mortar

and the ITZs are quantitatively determined. With help of the approach, the water content distribution

within a concrete sample after any elapsed time, especially the penetration depth of water frontier, can

be easily predicted. The cumulative water absorption calculated by the lattice network model is shown

to be well agreed with the experimental results.

& 2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

2. Unsaturated flow theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

2.1. Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

2.2. The sorptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

3. Lattice network model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

3.1. Mesoscale composite structure of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

3.2. Lattice network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

4. Application of the model to water absorption by mortar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

5. Water absorption of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

5.1. Sorptivity of mortar within concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

5.2. Water transport property of the ITZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

5.3. Numerical results by lattice network model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

1. Introduction

Degradation of concrete structures built in aggressive environ-

ments (e.g. coastal regions, near salted roadways) has attracted

more and more attention in recent years although concrete was

once looked as a durable maintenance-free construction material.

During the past few decades, it has been realized that the water

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/oceaneng

Ocean Engineering

0029-8018/$- see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2010.12.019

n Corresponding author.

E-mail address: [email protected] (L. Wang).

Ocean Engineering 38 (2011) 519–528

http://-/?-

http://www.elsevier.com/locate/oceaneng

http://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.oceaneng.2010.12.019

mailto:[email protected]



mailto:[email protected]


http://www.elsevier.com/locate/oceaneng

http://-/?-

7/26/2019 1-s2.0-S0029801811000059-main


ingress and aggressive agent transport are two key issues that

influence the long-term performance of concrete facilities. In each

of these processes, water is either the principal agent responsible

for the deterioration of concrete or the principal medium by

which aggressive agents (such as chloride or sulfate ions) are

transported into the concrete (Yang et al., 2006).

Except in the submerged parts of marine structures, in most

cases concrete is rarely saturated, such as the highway bridge

affected by deicing salts and the tide splashing zone of the coastand marine structures, since they are usually subjected to cyclic

wetting and drying process. Although both the diffusion and

capillary absorption are the primary transport mechanisms by

which harmful ions ingress into concrete, water penetration by

capillary suction near the unsaturated concrete surface is gen-

erally regarded as the dominant invasion process for dissolved

ions since the diffusion process alone is quite slow (Saetta et al.,

1993; Martys and Ferraris, 1997; Lunk, 1998). Therefore, it is of

utmost importance to predict the water movement into concrete

in order to realistically reflect the way of water-borne contami-

nants ingress.

Many attempts have been done to reveal the water transport

mechanisms in cementitious materials, which can be broadly

categorized as three types, i.e., experimental approach, analytical

method and numerical simulation (Wang, 2009). To monitor the

dynamic processes of water movement in porous material,

several experimental methods are available, such as the nuclear

magnetic resonance, gamma rays and neutron radiography

(Hanzic and Ilic, 2003). However, all these methods are too

complicated and the equipments used are too expensive to be

always available in all laboratories. Due to this fact, an alternative

technique to measure the volume of water absorbed, namely the

gravimetric method, is now widespread (Goual et al., 2000). As a

result of this, in most previous studies, the sorptivity is widely

used to characterize the water absorption property by capillary

suction since it is a well-defined physical parameter and could be

easily measured by means of the gravimetric technique

(Lockington et al., 1999).

The common test setup of the gravimetric method is shown inFig. 1, in which the specimen is initially dried to a controlled

saturation degree and one surface is exposed to free water source.

By this method, the cumulative mass of water absorbed (as

measured for example by the weight gain of the sample) is found

to be proportional to the square root of time with the sorptivity

defined as the coefficient of proportionality. However, the short-

coming of this method is apparent since it cannot provide the

information about the internal distribution of water within the

sample. It is also much difficult to determine the position and

movement of the advancing wetting front. However, from the

durability point of view, the profile of moisture content,

especially the penetration depth of wetting front is of great

importance to precisely predict the aggressive agent ingress

(Hanzic and Ilic, 2003; Parlange et al., 1984). The above analysis

implies that the numerical simulation approach could be a good

option if the gravimetric method is incorporated.

This paper focuses on developing a numerical modeling

method on a mesoscale description of concrete that could

simulate the water penetration process into concrete by capillary

absorption. The objective of such a model is to determine thedistribution of water content and the penetration depth of

wetting front within concrete sample. Based on composite struc-

ture of concrete on mesoscale, the lattice network model is built

up and the equation for unsaturated flow through porous media is

used as the governing equation. Finally, the available experiment

results are used to demonstrate the application of this numerical

modeling method as well as its feasibility. The computing

program adopted in this study is written by C++ language and all

the analysis is performed on a personal computer on Windows.

2. Unsaturated flow theory

2.1. Governing equation

According to the moisture content, water transport in porous

materials can be classified into two types, i.e. permeability and

sorptivity. Permeability relates to the passage of a fluid through a

saturated material under the action of a pressure differential;

sorptivity is linked to characterize an unsaturated material’s

ability to absorb and transmit water through it by capillary

suction forces (McCarter et al., 1992; Sabir et al., 1998). It has

been pointed out that the permeability is a wrong parameter for

modeling the capillary flows in cement-based materials because

most of them are rarely saturated in use (Hall, 1989). Therefore,

the unsaturated flow theory has been proved to be a realistic

method to investigate the water penetration in cement-based

materials by capillary suction (Hall, 1989, 2007). The equation for

unsaturated flow through porous media is described by the

extended Darcy’s equation

q!¼ K ðyÞr C ð1Þwhere q

!is the vector flow velocity; C is the capillary potential

with dimension L and K is the hydraulic conductivity function

with dimensions LT 1; y denotes the normalized water content,

scaled to be zero and one for the initial and saturated volumetric

water contents. Thus y can be written as

y¼ YYi

YS Yi

ð2Þ

in which Y is the volumetric water content at any state (i.e.

volume of water/bulk volume of concrete); Yi and YS are the

water contents before contacting with water and at the saturatedcondition, respectively.

Combining Eq. (1) with a mass-conservation equation and

then writing the resultant equation in terms of y by using the

substitution D(y)¼K (y)(dC/dy) (Goual et al., 2000; Hall, 1989;

Carpenter et al., 1993), the following expression can be obtained:

@y

@t ¼ r UðDðyÞr yÞ ð3Þ

In Eq. (3), D(y) is a material property, generally now called the

hydraulic diffusivity with dimensions L 2 T 1, which depends both

on the type of material and on the water content. As pointed

out by Hall (2007), calling D(y) a diffusivity merely reflects the

fact that Eq. (3) is a parabolic non-linear partial differential

equation (PDE) of the diffusion type, but it does not represent

Θi

x

Water

Sealed

Water front

Fig. 1. Schematic illustration of sorptivity measurement test.

L. Wang, T. Ueda / Ocean Engineering 38 (2011) 519–528520

7/26/2019 1-s2.0-S0029801811000059-main


mechanism about the transport physics since diffusion describes

the transport of moisture or dissolved ions as a result of a

concentration gradient.

Therefore, for an one-dimensional semi-infinite system subject

to a boundary condition y¼1 at x¼0 and an initial condition

y¼0, x40, t ¼0 (uniform initial water content within the sample

of material), Eq. (3) may be described mathematically by the

following form:

@y@t

¼ @@ x

DðyÞ @y@ x

0o xo1 ð4Þ

Using the Bolzmann transform, f ¼ x= ffiffi

t p

, Eq. (4) is reduced to

an ordinary differential equation

1

2f

dy

df

¼ d

dy DðyÞ dy

df

ð5Þ

As a result, the hydraulic diffusivity D(y) at any relative water

content y will be obtained by integrating Eq. (5) as follows:

DðyÞ ¼ 1

2

1

ðdy=dfÞy

Z y0fdy ð6Þ

Eq. (6) can be interpreted geometrically by the following

expression:

DðyÞ ¼ 1

2

1

pðyÞOðyÞ0y ð7Þ

where p(y) is the slope of curve y(f) at certain relative water

content y and O0y is the area delimited by the curve y(f), the

vertical axis and the horizontal line y¼0 and y, as depicted in

Fig. 2 (Goual et al., 2000).

Because the functional relationship between D and y is

strongly non-linear and is not always feasible to be determined,

for the purpose of predicting the water content profile by

absorption, it is commonly approximated by the exponential-law

D yð Þ ¼ D0eny ð8Þin which D0 and n are empirically fitted constants. Hall (1989)

stated that if the initial state is fully dried in laboratory (i.e.Yi¼0), n has a value between 6 and 8, varying little between

materials. The similar findings have also been reported in the

other studies (Lockington et al., 1999; Leech et al., 2003).

2.2. The sorptivity

The sorptivity S (dimensions LT 1/2), as a material property

(for a given fluid), is defined as the rate of water uptake by

any porous material when exposed to a free water source. It is a

simple parameter to be determined and is increasingly being used

as a measure of concrete resistance to exposure in aggressive

environments (Lockington et al., 1999). When the gravimetric method

is applied, the cumulative volume of absorbed water per unit area of

the inflow surface at elapsed time t , i (dimension L ), is expressed as

i ¼ St 1=2 ð9ÞBy introducing the Boltzmann variable, f¼ xt 1/2, the cumulative

water absorption at any time t can also be mathematically calculated

through integral according to the water distribution in the sample

i ¼Z YS

Yi

xdY¼ t 1=2

Z YS

Yi

fdY ð10Þ

Hence,

S ¼Z YS

Yi

fdy¼ ðYS YiÞZ 1

0fdy ð11Þ

The approximate solution of S has been given as (Lockington

et al., 1999)

S 2 ¼ ðYS YiÞ2

Z 1

0

ð1þyÞDðyÞdy ð12Þ

By substituting Eq. (8) in Eq. (12) and then integrating, it yields

S 2

¼ D0 en

ð2n

1

Þn

þ1

½ n2 ðYS YiÞ2

ð13ÞEq. (13) allows the direct estimation of the parameter D0 in

Eq. (8) assuming that the other parameters, i.e., S , n, YS and Yi

have already been determined based on assumptions or experi-

mental data. Usually, S is experimentally determined in a defined

initial state of dryness with one surface exposed to free water. It

implies that YS equals the porosity of the concerned material and

Yi equals zero. Based on this assumption, Lockington et al. (1999)

proposed an approach to determine the constants n and D0 with

the experimentally regressed value of S , in which the constant n is

suggested to take a universal value as 6.0 since giving it a value

from the lower end of the expected range will result in a slight

overestimate of the water penetration depth (Lockington and

Parlange, 2003), and then D0 will be approximately estimated by

D0 ¼ s2=123:131 ð14Þwhere s¼S /(YS Yi), is the normalized sorptivity. The compar-

ison between experimental data of Hall (1989) and the approx-

imate method proposed by Lockington et al. (1999) has illustrated

the reliability of Eq. (14). Therefore, in the subsequent analysis of

this paper, n is set as 6.0 and D0 will be calculated in terms of

Eq. (14) with the corresponding value of S .

3. Lattice network model

3.1. Mesoscale composite structure of concrete

On the mesolevel, concrete is a composite material with avariety of inhomogeneities (Zaitsev and Wittmann, 1981). There-

fore, in the viewpoint of modeling, its composite behavior may be

studied using the approach on mesoscale, which treats concrete

as a three-phase system consisting of aggregates embedded in a

matrix of hardened cement paste, and the interfacial transition

zones (ITZs) on the interface between the aggregate particles and

the surrounding cement paste (Oh and Jang, 2004; Care and

Herve, 2004; Sadouki and Van Mier, 1997; Zhou and Hao, 2008)

(see Fig. 3). It should be noted here, in Fig. 3, that the layer of ITZ

around coarse aggregates is enlarged to facilitate the under-

standing of the composite structure of concrete because the real

thickness of ITZ is usually less than 50 mm. As far as the widely

used coarse aggregates are concerned, it is reasonable to regard

them as non-sorptivity materials. This implies that the mortar with

N o r m a l i z e d w a t e r c o n t e n t

Boltzmann transform

=1

=0

Fig. 2. Schematic illustration of water content profile as a function of the

Boltzmann transform (Goual et al., 2000).

L. Wang, T. Ueda / Ocean Engineering 38 (2011) 519–528 521

7/26/2019 1-s2.0-S0029801811000059-main


fine aggregate dispersed in it will be supposed as a homogeneous

material. The ITZ, formed due to the water filled pores near the

aggregate and the wall effect, is modeled as a thin boundary layer

around the coarse aggregate. Thus, in contrast to analysis of mortar

itself, for the investigation of transport property of concrete, it is

desirable to be able to predict the diffusivity of each componentcomposing the concrete, i.e., the mortar, coarse aggregate and ITZs.

The size distribution of coarse aggregates in a geometrical

domain is generated using certain given cumulative grading

distribution curve. In this study, a standard pseudo-random

number generator is used to generate probabilities from which

the diameter of the aggregate is determined. This procedure is

repeated until the chosen volume fraction of aggregates is

obtained. After accomplishing the database of aggregate particles,

the particles are randomly placed one by one into the domain of

concrete in such a way that there is no overlapping with particles

already placed (Grassl and Pearce, 2010; Wang et al., 1999).

3.2. Lattice network model

It has been proposed that on the mesocale level, mass transport in

concrete can be described by means of lattice-type model, which is a

discrete numerical method and has the advantage of mesh-indepen-

dence and accurate descriptions of basic properties of the continuum

response (Sadouki and Van Mier, 1997; Grassl, 2009). In the current

paper, a lattice network model is established to simulate the water

absorption in concrete based on the Voronoi tessellation strategy as

shown in Fig. 4, which is obtained by randomly placing the nodes in

the structural domain (Grassl, 2009). The advantage of this meshing

technique is to avoid mesh bias on potential crack direction

concerning the mechanical analysis (Bolander and Le, 1999; Nagai

et al., 2004). In Fig. 4, the Voronoi polygons can be categorized into

two types: the polygonal cells on mortar and those on the coarse

aggregates because the segment of the circumference of any coarse

aggregate is always fixed as the common border of a pair Voronoi

polygons adjacent to the aggregate–mortar boundary. The properties

of different types of Voronoi cells depend on the characteristics of the

material upon which they are superimposed. As for the role of ITZ on

aggregate–mortar boundary, generally its corresponding property

will be assigned to the common segment of the two neighboring

cells in terms of the mechanical analysis or transport calculation. To

form a network through which the water transport takes place, each

of the Voronoi cells is connected by lattice elements with nodes atthe centers of Voronoi cells, i.e. the nuclei of the Voronoi polygons,

and the intermediate points on Voronoi cell boundaries (see Fig. 5).

This is a refined approach of truss network model proposed by

Nakamura et al. (2006), in comparison to Bolander and Berton (2004)

approach where only the Voronoi cell nuclei are linked to generate

truss elements. The advantage of this modification is that it can

easily represent the movement of mass through ITZs or the potential

cracks. As a result of this process, for an undamaged concrete sample

(no cracks) it yields two major types of lattice elements, namely

those connecting nuclei and intermediate nodes on Voronoi cell

boundaries, and the lattice elements running along the Voronoi cell

boundaries as shown in Fig. 5. Furthermore, it can be seen from Fig. 5

that the first type elements are either on aggregate or on mortar and

the second type elements are either on mortar–mortar, mortar–

aggregate (ITZ) or aggregate–aggregate boundary. As stated above,

cracks are supposed to be present on the boundary of Voronoi cells,

therefore, if a specimen is cracked, the lattice elements along cracks

will be given their cross-sectional area and diffusivity based on the

cracks width. But in the current study, the cracking effect is not

considered and implemented.

The use of an irregular lattice network for the description of

the water movement in concrete is significant since it can alter a

two-dimensional transport problem to an one-dimensional pro-

cess and also reduce the mesh-dependency of elements. The

cross-sectional area and diffusivity of a certain kind of lattice

element are subsequently determined according to its geometri-

cal shape and transporting property of the corresponding material

(aggregate, mortar or ITZ) (Wang et al., 2008).

In the lattice network model, as shown in Fig. 5, for the case of mortar or aggregate, a continuous area is represented by a

discrete lattice element. Moreover, each kind of lattice elements

has their own cross-sectional area and diffusivity. For example,

the diffusion area of D jmn is represented by the one-dimensional

lattice element jk with the length of L (see Fig. 5). Since the lattice

element jk falls in the mortar cell, its diffusivity will have the

value of mortar. In other word, the mass penetrating across the

area of D jmn will be mathematically considered along the uni-

directional lattice element jk as like through a ‘‘pipe’’. For the

lattice elements on ITZ, their cross-sectional area and diffusivity

can be determined based on the previous researches on the

properties (thickness and diffusivity) of ITZ. The similar rule can

also be applied to other lattice elements. Thus, for an one-

dimensional non-stationary potential flow problem as expressedby Eq. (4), the discrete form of the governing equation can be

formulated by using the Galerkin method as follows:

ADðy jÞL

1 1

1 1

y j

yk

( )þ 1

m AL

6

2 1

1 2

@y j

@t

@yk

@t

8<:

9=;

þ ADðy jÞ

@y@ x

x ¼ x j

@y@ x

x ¼ xk

8>>>><>>>>:

9>>>>=>>>>;

¼ 0

0

ð15Þ

in which A is the cross-section area of the lattice element,

equaling to the area of the corresponding facet of Voronoi

Fig. 4. Elements geometry of concrete sample meshed with Voronoi diagram.

Phase III: ITZ

Phase I: mortar

Phase II: aggregate

Fig. 3. The three-phase structure of concrete.


7/26/2019 1-s2.0-S0029801811000059-main


polygon; L is the length of the lattice element; y j and yk are the

potential quantities at the nodes j and k, respectively (see Fig. 5).

m is a parameter to account for the dimension conversion and is

set as 2.0 for a two-dimensional lattice element arrangement

(Nakamura et al., 2006).

The following two types of boundary conditions are alterna-

tively adopted for the general mass transport process:

y¼ f ð xÞ for x on G1 ð16Þ

q¼

D@y

@n

for x on G1

ð17

Þwhere q is the outward flux normal to the boundary (in direction n);

G1 and G2 are the boundary segments with prescribed potential

and normal flux, respectively. For case of the common water

absorption tests, the specified surface exposed to water is

assumed in contact with free water (i.e., saturated state), which

implies G1 boundary condition is appropriate with the exposed

surface of the specimen always to be fully saturated. In other

words, the relative water contents on end nodes of the lattice

elements contacting to water source are taken as one during all

the time of water absorption analysis.

After introducing the discrete time step Dt , the finite differ-

ence equation in time of Eq. (15) can be formed by a step-by-step

procedure with the Crank–Nicholson method. Given the initial

state of all lattice elements in a specimen, a system of equations isset up at each time step with the hydraulic diffusivity determined

by the current relative water content for the lattice elements, and

then solved for the new nodal potential values.

4. Application of the model to water absorption by mortar

The lattice network model described above was applied to the

moisture transport problem of mortar in order to verify its

feasibility and efficiency since in such case, mortar can be

accordingly regarded as a homogeneous material. The experi-

mental results obtained by Hall (1989) on a mortar bar (1:3:12

OPC/lime/sand by volume) are selected to be compared with the

numerically calculated results by the lattice network model. In

the experiment, the normalized water content distribution in a

long bar of cement mortar was determined at a series of elapsed

times after bringing an end face of the sample into contact with a

water source (as shown in Fig. 6). In addition, the sorptivity and

porosity of the mortar specimen was experimentally determined

as 2.57 mm/min1/2 and 0.27, respectively, which allow one to

evaluate the parameter D0 by the method described in Section 2.2.

In Fig. 6, the Voronoi meshing diagram of the specimen, whichis the first step of the construction of lattice network, is also

presented.

As stated above, the hydraulic diffusivity D expressed as a

function of y for this mortar can be easily determined asD¼0.736 exp(6y) by Eqs. (8) and (14). Here, n¼6 is taken as a

standard value for mortar since it provides a conservative pre-

diction of the depth of water penetration (Lockington et al., 1999).

The calculated profiles of normalized water content by means of

lattice network model are shown in Fig. 7 to compare with

the experimental findings. To further demonstrate the numerical

approach, Fig. 8 provides the comparison of relationships

between the normalized water content and Boltzmann variablef (f¼ x.t 1/2) after various elapsed times. It is evident from

Figs. 7 and 8 that the results predicted by lattice network model

Nodes on lattice elements

Lattice elements in Voronoi elements

Lattice elements on boundaries

k

j

L

θk

θ jITZ

j

nk m

Aggregate

m

j

nk

Fig. 5. Construction of the lattice network model.

x =0

Water

reservoir

235mm

33mm

Exposed

surface

Fig. 6. Experimental and numerical model of the mortar specimen.


7/26/2019 1-s2.0-S0029801811000059-main


match very well with the experimental data and therefore

support the possibility of applying the lattice network model to

water transport analysis in concrete.

5. Water absorption of concrete

5.1. Sorptivity of mortar within concrete

There are many factors that may influence the sorptivity of

mortar and concrete, such as the water/cement ratio (w/c ),

cement content, curing method and time, dried condition (con-

ditioning temperature), admixture type and replacement quan-

tity, etc. Although up to now, a few tests have been done to study

water sorptivity of mortar whose measured value varied mark-

edly and typically ranging from 0.15 to 2 mm/min1/2 (Hall,

1989; Sabir et al., 1998), the lack of sufficient knowledge on this

subject does not yet allow reliable and precise quantification of

their effects. As the fundamental constitute of the lattice network

model of concrete, the sorptivity of mortar needs to be firstly

quantified. However, most of the experiments were conducted on

water sorptivity of concrete and seldom measured the sorptivity

of the corresponding mortar. To solve this problem, in the current

study, an alternative method is introduced to approximately

estimate the sorptivity of mortar within a concrete based on the

test carried out for concrete. Once the sorptivity of mortar is

estimated, the hydraulic diffusivity of lattice elements of mortarin the lattice network will be consequently determined.

The effect of the addition of low-permeability or non-sorptive

particles to a homogeneous porous material on water transport

characteristics has been investigated experimentally and theore-

tically (Hall et al., 1993). Let a be the volume fraction of non-

sorptive inclusions (¼volume of inclusions/volume of concrete, in

the case of concrete, ¼volume of coarse aggregate/volume of

concrete), then the theoretical relation for a porous material is

approximately given by:

S c =S m ¼ gð1aÞ 1=2 ð18Þ

where gffi11.5a+0.588a2; S c and S m are defined as the sorptiv-

ity of a material containing non-sorptive inclusions and without

inclusions (e.g., S c is the sorptivity of concrete and S m is the

sorptivity of mortar), respectively. Thus, when the sorptivity of a

concrete, S c has been determined, the sorptivity of the companion

mortar within it can be estimated by Eq. (18).

5.2. Water transport property of the ITZ

Because of the larger pores and higher porosity of ITZ when

compared with the bulk cement paste, the existence of ITZ plays a

negative role in terms of transport property of concrete by

forming fast-conduction pathways through the material. As a

result, the ingress and movement of substances are accelerated.

From this point of view, the water moving speed in ITZ is thoughtto be much higher than that in the matrix paste. According to the

theoretical analysis and numerical calculation (Shane et al., 2000),

the ITZ is reported to be up to 500 times more permeable than the

cement paste. Furthermore, the transport speed of water in

mortar with 55 vol% of sand content is estimated to be about 50

times higher than that in the cement paste. By comparing these

two ratios, it may be deduced that the transport speed of water in

ITZ is probably 10 times higher than that in the bulk mortar. As

stated earlier in this paper, in the composite structure of concrete,

mortar is treated as a homogeneous phase and the transport

property of which could be determined by Eq. (18). Therefore, in

terms of the lattice network model of concrete, it is assumed that

the hydraulic diffusivity of lattice elements located in ITZ is 10

times higher than that of the lattice elements falling in themortar.

In addition, the widely reported thickness of ITZ is in the range

15–50 mm (Shane et al., 2000), but in most of cases it is generally

assumed as 20–30 mm (Bentz et al., 1997; Garboczi and Bentz,

1998; Care and Herve, 2004; Oh and Jang, 2004; Yang, 2003).

Therefore, in this paper, the width of the lattice elements on the

ITZ will be taken as 20 mm for concrete, but the sensitivity of

thickness of ITZ on water absorption is illustrated by an example

in the following section. It should be pointed out that although

such values of diffusivity and width of ITZ can be implemented

into the numerical model, the limited research in this field is not

sufficient for the accurate prediction. It implies that more knowl-

edge, especially experimental research is still required in order to

give reliable results.

0

0.0

0.2

0.4

0.6

0.8

1.0

Hall 1989

38 min

57 min

97 min

208 min

271 min

w a t e r c o n t e n t

x (mm)

Calculation results:

38 min 57 min 97 min 208 min 271 min

40 80 120 160 200

Fig. 7. Comparison of water content profiles between predicted and experimental

results after various elapsed times.

Fig. 8. Relationship between water content y and Boltzmann variable f after

various elapsed times.


7/26/2019 1-s2.0-S0029801811000059-main


5.3. Numerical results by lattice network model

The Voronoi meshing diagram of a concrete sample, as well as

the boundary condition, is shown in Fig. 9, which is modeled with

844 Voronoi elements and 8961 lattice elements (for sake of

clearness, the lattice elements are not depicted). As shown in

Fig. 9, the left and right surfaces of the specimen are assumed to

be exposed to free water and fully dried atmosphere, respectively,

whereas the two side surfaces are completely sealed. In thedomain of the specimen, the volume fraction of coarse aggregate

is 49%, very close to that used in the experiment by Wittmann

et al. (2006). The test data in that paper will be used to compare

and confirm our numerical results. Referring to Wittmann et al.’s

(2006) experiment results, the sorptivity of concrete (S c ) has been

determined as 0.1225 mm/h1/2 with w/c ¼0.6 by means of Eq. (9).

Therefore, by Eq. (18), the sorptivity of mortar (S m) then can be

calculated as 0.258 mm/h1/2.

If w/c is given, the capillary porosity of cement paste, fcap, can

be calculated by Hansen (1986)

fcap ¼ ðw=c Þ0:36oðw=c Þþ0:313

ð19Þ

where o is the ultimate degree of hydration, which has been

empirically expressed as the function of w/c (Oh and Jang, 2004)

o¼ 1expð3:15ðw=c ÞÞ ð20Þ

But when w/c is larger than 0.42, o is usually set as 1.0

(Hansen, 1986).

Thus, the capillary porosity of cement paste is estimated as

0.263 by Eq. (19). The porosity of mortar, fmortar , can be easily

obtained by the following equation assuming that sand does not

influence the pore structure of cement paste:

fmortar ¼ ð1 psandÞfcap ð21Þ

in which psand is the volume fraction of sand in the mortar. Since

the sand volume fraction is 51% in the test, fmortar is then

calculated as 0.129. It yields that YS ¼0.129, where YS is the

water contents at the saturated condition. Additionally, because

the specimens were dried at a high temperature for a long time,

the initial water content is supposed to be zero, meaning thatYi¼0. Then the coefficient of D0 can be estimated by Eq. (14) asD0¼(0.258/0.129)2/123.131¼0.0296 mm/h1/2. The analytical

conditions are summarized and listed in Table 1.

Fig. 9 is the medium density mesh with the average size of the

Voronoi elements of 2 mm. The other two element sizes, fine

density mesh of 1 mm and coarse density mesh of 3 mm, are also

chosen to investigate the effect of element size. From Fig. 10, it is

clear that the cumulative water absorption curves are almost

completely independent of the size of the Voronoi polygons, i.e.,

the size of the lattice elements. Additionally, the influence of the

thickness of ITZ is also performed by setting three thicknesses

separately and keeping invariable of other conditions. The results

are depicted in Fig. 11, which shows a small increase of the

absorbed water even when the thickness is tripled.The cumulative water absorption obtained with the medium

density lattice network is used to compare with the experimental

data from Wittmann et al. (2006). It can be seen from Fig. 12 that

the two cumulative water absorption curves are quite close to

each other. If concrete is treated as a homogenous material, as

represented by Eq. (9), the relationship between cumulative

water absorption and the square root of the elapsed time must

exhibit a straight line. However, the inclusion of coarse aggre-

gates, which is generally considered as the non-sorptive phase,

will reduce the water absorption velocity. But with the increase

of water penetration depth, the influence of coarse aggregates is

less pronounced. So that the cumulative absorption curve shows

H2O

H2O

x

Fig. 9. Specimen model (width height¼40 60 mm2).

Table 1

Analysis conditions for lattice network model from different literatures.

Analysis ca se Vo l. of coarse

aggregate (%)

Porosity

of mortar

S c (mm/h0.5) S m (mm/h0.5) D0 (mm2/h) Ref.

1 51 0.129 0.1225 0.258 0.0296 Wittmann et al. (2006)

2 41 0.2 0.886 1.829 0.679 Neithalath (2006)

Fig. 10. Influence of the size of lattice elements on water absorption.


7/26/2019 1-s2.0-S0029801811000059-main


a deviation from the straight line, i.e., a downward curvature at

the initial stage. This observation leads to the conclusion that the

mesoscale method concerning three-phase composite structure of

concrete can represent well this phenomenon. Moreover, the

water penetration profiles as well as the relationship between

water content y and Boltzmann variable f after various elapsedtimes are shown in Figs. 13 and 14, respectively, which exhibit

the similar tendency as that found in experiment study. The data

given in Figs. 13 and 14 are all collected from the lattice elements

on mortar cells since water contents in aggregates are zero due to

their impermeability.

Another experimental result obtained by Neithalath (2006) is

also dealt with and used in this study. The lattice network model

is applied to further confirm the above conclusion. In the experi-

ment of Neithalath (2006), the water-to-cementitious material

ratio was 0.37 since 25% Class C fly ash was used as part of

the binder by weight. The content of coarse aggregate was

1106.93 kg/m3 determined by the mixture design. The specimens

were sealed on their outer sides using electrical tape except the

top surface, which was chosen as the source of water transport

and filled with water to a height of 15 mm. The sorptivity was

determined by measuring a series of the amounts of water

uptake. The test results and conditions used for our calculation

are also given in Table 1. The comparison of the cumulative water

absorption predicted by lattice network model and measured

from test is shown in Fig. 15. It is obvious that both match well.The good agreements in Figs. 12 and 15 illustrate the possibi-

lity and efficiency of lattice network model on mesoscale in

simulating the water capillary absorption by concrete, which

provides a helpful tool for the determination of structural dur-

ability and service life prediction of concrete structures, especially

for those under a partially saturated state.

6. Conclusions

In the present work, a lattice network model on mesoscale

aimed to solve the problem of substance transport in concrete is

developed. In this model, concrete is idealized as a three-phase

composite consisting of coarse aggregates, mortar with fine

Fig. 11. Sensitivity analysis of thickness of ITZ on water absorption.

00

100

200

300

400

500

600

700

A b s o r b e d w a t e r ( g / m 2 )

Time (hr 1/2)

Wittmann et al. 2006

Numerical calculation result

1 2 3 4 5

Fig. 12. Cumulative absorbed water and its comparison with experimental data

(Wittmann et al., 2006).

00.0

0.2

0.4

0.6

0.8

1.0

w a t e r c o

n t e n t

Depth from exposed surface (mm)

1 hr

8 hrs

16 hrs

24 hrs

2 4 6 8 10 12 14 16

Fig. 13. Calculated water penetration profile in concrete for various elapsed times.

00.0

0.2

0.4

0.6

0.8

1.0

w a t e r c o n t e n t

(mm/hr 1/2)

1 hr

8 hrs

16 hrs

24 hrs

1 2 3 4

Fig. 14. Calculated relationship between water content y and Boltzmann variable f.


7/26/2019 1-s2.0-S0029801811000059-main


aggregates dispersed in it and the interfacial transition zones(ITZs) between the aggregate particles and the surrounding

mortar. Water uptake process by capillary absorption is described

by a non-linear diffusion equation according to the unsaturated

flow theory. The hydraulic diffusivity of each component of

concrete composite structure, in other words, the different type

of lattice elements, is determined separately based on the

previous findings. With help of the proposed model, the following

conclusions can be drawn:

1. Water movement in concrete by capillary absorption can be

described by the lattice network model established on mesos-

cale structure of concrete. The benefit of this method is that it

is capable of predicting the water content distribution within

the sample and the position of wetting front at any time whenthe initial and boundary conditions are given.

2. It is found that the cumulative water absorption predicted by

the lattice network model is in good agreement with the

experimental results. In addition, a deviation of the cumulative

absorption curve from the initial straight line is demonstrated

by the inclusion of non-sorptive coarse aggregates because

they can reduce the water transport velocity.

3. Because ITZ in the meso-structure of concrete is a zone of

weakness due to its high porosity, it plays an important role in

the transport properties of concrete. The lattice network model

addresses this problem by giving a higher hydraulic diffusivity

to ITZ, for example, about 10 times higher than that of the

mortar.

Furthermore, with consideration of the virtues of Voronoi

diagram on simulating the crack formation, it can be foreseen

that the lattice network model could be successfully applied to

predicting the substances transport in cracked concrete.

Acknowledgements

This study was financially supported by the Key Project of

Chinese Ministry of Education (No. 109046) and the Scientific

Research Foundation for the Returned Overseas Chinese Scholars,

State Education Ministry. The supports of Center for Concrete

Corea, Korea to the Yonsei University of Korea and the Asia–Africa

Science and Technology Strategic Cooperation Promotion Program

by the Special Coordination Funds for Promoting Science and

Technology are also acknowledged.

References

Bolander, J.E., Berton, S., 2004. Simulation of shrinkage induced cracking in cementcomposite overlays. Cement and Concrete Composites 26 (7), 861–871.

Bolander, J.E., Le, B.D., 1999. Modeling crack development in reinforced concretestructures under servce loading. Construction and Building Materials 13 (1),23–31.

Bentz, D.P., Detwiler, R.J., Garboczi, E.J., Halamickova, P., Schwartz, L.M., 1997.Muti-scale modeling of the diffusivity of mortar and concrete. In: Nilsson, L.O.,Ollivier, J.P. (Eds.), Proceedings of Chloride Penetration into Concrete. RILEMpulications, Saint-Remy-les-Chevreuse, France, pp. 85–94.

Care, S., Herve, E., 2004. Application of an n-phase model of the diffusioncoefficient of chloride in mortar. Transport in Porous Media 56 (2), 119–135.

Carpenter, T.A., Davies, E.S., Hall, C., Hall, L.D., Hoff, W.D., Wilson, M.A., 1993.Capillary water migration in rock: process and material properties examinedby NMR imaging. Materials and Structures 26 (5), 286–292.

Garboczi, E.J., Bentz, D.P., 1998. Multiscale analytical/numerical theory of thediffusivity of concrete. Advanced Cement Based Materials 8 (2), 77–88.

Goual, M.S., de Barquin, F., Benmalek, M.L., Bali, A., Queneudec, M., 2000.Estimation of the capillary transport coefficient of clayey aerated concreteusing a gravimetric technique. Cement and Concrete Research 30 (10),1559–1563.

Grassl, P., 2009. A lattice approach to model flow in cracked concrete. Cement and

Concrete Composites 31 (7), 454–460.Grassl, P., Pearce, C., 2010. Mesoscale approach to modeling concrete subjected to

thermomechanical loading. Journal of Engineering Mechanics 136 (3),322–328.

Hall, C., 1989. Water sorptivity of mortar and concretes: a review. Magazine of Concrete Research 41 (147), 51–61.

Hall, C., 2007. Anomalous diffusion in unsaturated flow: fact or fiction? Cementand Concrete Research 37 (3), 378–385.

Hall, C., Hoff, W.D., Wilson, M.A., 1993. Effect of non-sorptive inclusions oncapillary absorption by a porous material. Journal of Physics D: AppliedPhysics 26 (1), 31–34.

Hansen, T.C., 1986. Physical structure of hardened cement paste: a classicalapproach. Materials and Structures 19 (6), 423–436.

Hanzic, L., Ilic, R., 2003. Relationship between liquid sorptivity and capillarity inconcrete. Cement and Concrete Research 33 (9), 1385–1388.

Leech, C., Lockington, D., Dux, P., 2003. Unsaturated diffusivity functions forconcrete derived from NMR images. Materials and Structures 36 (6), 413–418.

Lockington, D.A., Parlange, J.-Y., 2003. Anomalous water absorption in porousmaterials. Journal of Physics D: Applied Physics 36 (6), 760–767.

Lockington, D.A., Parlange, J.-Y., Dux, P., 1999. Sorptivity and the estimation of water penetration into unsaturated concrete. Materials and Structures 32 (5),342–347.

Lunk, P., 1998. Penetration of water and salt solutions into concrete by capillarysuction. Journal of Restoration of Buildings and Monuments 4, 399–422.

Martys, N.S., Ferraris, C.F., 1997. Capillary transport in mortars and concrete.Cement and Concrete Research 27 (5), 747–760.

McCarter, W.J., Ezirim, H., Emerson, M., 1992. Absorption of water and chlorideinto concrete. Magazine of Concrete Research 44 (158), 31–37.

Nagai, K., Sato, Y., Ueda, T., 2004. Mesoscopic simulation of failure of mortar andconcrete by 2D RBSM. Journal of Advanced Concrete Technology 2 (3),359–374.

Nakamura, H., Srisoros, W., Yashiro, R., Kunieda, M., 2006. Time-dependentstructural analysis considering mass transfer to evaluate deteriorationprocess of RC structures. Journal of Advanced Concrete Technology 4 (1),147–158.

Neithalath, N., 2006. Analysis of moisture transport in mortars and concrete usingsorption-diffusion approach. ACI Materials Journal 103 (3), 209–217.

Oh, B.H., Jang, S.Y., 2004. Prediction of diffusivity of concrete based on simpleanalytic equations. Cement and Concrete Research 34 (3), 463–480.Parlange, J.-Y., Lisle, I.G., Prasad, S.N., Romkens, M.J.M., 1984. Wetting front

analysis of the nonlinear diffusion equation. Water Resources Research 20(5), 636–638.

Sabir, B.B., Wild, S., Farrell, M.O., 1998. A water sorptivity test for mortar andconcrete. Materials and Structures 31 (8), 568–574.

Sadouki, H., Van Mier, J.G.M., 1997. Meso-level analysis of moisture flow in cementcomposites using a lattice-type approach. Materials and Structures 30 (10),579–587.

Saetta, A.V., Scotta, R.V., Vitaliani, R.V., 1993. Analysis of chloride diffusion intopartially saturated concrete. ACI Materials Journal 90 (5), 441–451.

Shane, J.D., Mason, T.O., Jennings, H.M., Garboczi, E.J., Bentz, D.P., 2000. Effect of the interfacial transition zone on the conductivity of Portland cement mortar.

Journal o f the American Ceramic society 83 (5), 1137–1144.Wang, L., 2009. Analytical methods for prediction of water absorption in cement-

based material. China Ocean Engineering 23 (4), 719–728.Wang, L., Soda, M., Ueda, T., 2008. Simulation of chloride diffusivity for cracked

concrete based on RBSM and truss network model. Journal of Advanced

Concrete Technology 6 (1), 143–155.

0.0 2.00

300

600

900

1200

1500

1800

A b s o r b e d w

a t e r ( g / m 2 )

Time (hr 1/2)

Neithalath 2006

Numerical calculation results

0.4 0.8 1.2 1.6

Fig. 15. Cumulative absorbed water and its comparison with experimental data

(Neithalath, 2006).


7/26/2019 1-s2.0-S0029801811000059-main


Wang, Z.M., Kwan, A.K.H., Chan, H.C., 1999. Mesoscopic study of concrete I:generation of random aggregate structure and finite element mesh. Computersand Structures 70 (5), 533–544.

Wittmann, F.H., Zhang, P., Zhao, T., 2006. Influence of combined environmentalloads on durability of reinforced concrete structures. Journal of Restoration of Buildings and Monuments 12 (4), 349–362.

Yang, C.C., 2003. Effect of the interfacial transition zone on the transport and theelastic properties of mortar. Magazine of Concrete Research 55 (4), 305–312.

Yang, Z., Weiss, W.J., Olek, J., 2006. Water transport in concrete damaged by tensileloading and freeze-thaw cycling. Journal of Materials in Civil Engineering 18(3), 424–434.

Zaitsev, Y.B., Wittmann, F.H., 1981. Simulation of crack propagation and failure of concrete. Materials and Structures 14 (5), 357–365.

Zhou, X.Q., Hao, H., 2008. Mesoscale modelling of concrete tensile failuremechanism at high strain rates. Computers and Structures 86 (21–22),2013–2026.


1-s2.0-s0029801811000059-main

Documents