Acta Polytechnica


DOI:10.14311/AP.2020.60.0012
Acta Polytechnica 60(1):12–24, 2020 © Czech Technical University in Prague, 2020

available online at https://ojs.cvut.cz/ojs/index.php/ap

HOMOGENIZATION OF TRANSPORT PROCESSES AND
HYDRATION PHENOMENA IN FRESH CONCRETE

Michal Beneša, Radek Štefanb,∗

a Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mathematics, Thákurova 7,
166 29 Prague 6, Czech Republic

b Czech Technical University in Prague, Faculty of Civil Engineering, Department of Concrete and Masonry
Structures, Thákurova 7, 166 29 Prague 6, Czech Republic

∗ corresponding author: radek.stefan@fsv.cvut.cz

Abstract.
The problem of hydration and transport processes in fresh concrete is strongly coupled and

non-linear, and therefore, very difficult for a numerical modelling. Physically accurate results can be
obtained using fine-scale simulations, which are, however, extremely time consuming. Therefore, there
is an interest in developing new physically accurate and computationally effective models. In this paper,
a new fully coupled two-scale (meso-macro) homogenization framework for modelling of simultaneous
heat transfer, moisture flows, and hydration phenomena in fresh concrete is proposed. A modified
mesoscale model is first introduced. In this model, concrete is assumed as a composite material with two
periodically distributed mesoscale components, cement paste and aggregates. A homogenized model is
then derived by an upscaling method from the mesoscale model. The coefficients for the homogenized
model are obtained from the solution of a periodic cell problem. For solving the periodic cell problem,
two approaches are used – a standard finite element method and a simplified closed-form approximation
taken from literature. The homogenization framework is then implemented in MATLAB environment
and finally employed for illustrative numerical experiments, which verify that the homogenized model
provides physically accurate results comparable with the results obtained by the mesoscale model.
Moreover, it is verified that, using the homogenization framework with a closed-form approach to the
periodic cell problem, significant computational cost savings can be achieved.

Keywords: Concrete, hydration, transport processes, homogenization, numerical experiments.

1. Introduction
Modelling of transport processes and hydration phe-
nomena in early age concrete is a challenging engineer-
ing problem, since hydration of cement is an exother-
mic reaction, which may result in high temperatures
and related deformations, stresses, and cracking in
concrete structures, e.g. [1–7] and references therein.
Fresh concrete is a heterogeneous porous material

highly saturated with liquid water, e.g. [4]. In the
present paper, concrete is assumed as a composite
material with two periodically distributed mesoscale
components, cement paste and aggregates, represented
by multiphase hygroscopic and heat conducting rigid
porous media partially saturated with liquid water,
e.g. [4, 8].

Coupled transport processes in porous materials are
associated with balance equations for mass of moisture
and heat energy of the whole medium, e.g. [9]. For
the mesoscopic description of transport processes in
cement paste and aggregates, we use the balance equa-
tions based on the averaging theory, see [4] and ref-
erences therein, particularly [10–12], for more details.
The governing equations at the mesoscale level are
completed by an appropriate set of constitutive equa-

tions, material data, and physically relevant boundary
and initial conditions.

It is well known and closely described in literature
that the problem of hydration and transport processes
in concrete is strongly coupled and non-linear, e.g.
[4, 7] and references therein. This, together with the
complexity of the mesoscopic structure of the concrete
composite, makes the detailed numerical simulations
of the problem extremely time consuming. Therefore,
there is an interest in developing new physically accu-
rate and computationally effective models based on
a sophisticated numerical coupling at different scales,
see e.g. [13–16], see also our previous work [17, 18].

The homogenization method is one of the most
advanced techniques in upscaling the response of the
microstructure of heterogeneous materials, e.g. [19–
28]. In this method, the solution of a fine scale problem
is used to examine the local material behaviour at
the macroscale. Combining the heat transfer together
with mass flows through fresh concrete motivates us
to construct a fully coupled multi-field and a two-scale
(meso-macro) homogenization framework.

12

https://doi.org/10.14311/AP.2020.60.0012
https://ojs.cvut.cz/ojs/index.php/ap


vol. 60 no. 1/2020 Homogenization of transport processes and hydration phenomena. . .

2. The mesoscale model
In the mesoscale model, concrete is assumed as
a composite material with two periodically distributed
mesoscale components, cement paste and aggregates,
represented by multiphase hygroscopic and heat con-
ducting rigid porous media partially saturated with
liquid water, e.g. [4, 8]. The detailed description of
the assumptions adopted for developing the model is
given in our previous work [17].
Let Ω be a polygonal domain with boundary ∂Ω.

Consider a concrete composite consisting of two flow
regions (aggregates and cement paste) periodically
distributed in a domain Ω with period εY, where
Y = (0, 1)2 is a periodicity cell split into two com-
plementary parts Ya and Yc. Throughout the paper,
subscripts a and c refer to aggregates and cement
paste, respectively. By χa and χc, we denote the
corresponding characteristic functions of Ya and Yc,
respectively, extended Y–periodically to R2. From
the geometrical point of view, ε is the characteristic
length representing the small scale variability of the
concrete composite. In this paper, we present the
homogenization result for the problem (indexed by ε)

∂t [%w (χεcφc(ξ
ε) + χεaφa)S(p

ε)]

−∇·
[
%wkR(S(pε))

µ(ϑε)
(χεckc(ξ

ε) + χεaka)∇p
ε

]
= α1χεcf(p

ε,ϑε,ξε), (1)

∂t [cw%w (χεcφc(ξ
ε) + χεaφa)S(p

ε)ϑε]
+ ∂t [(χεc%sccsc(1 −φc(ξ

ε)) + χεa%sacsa(1 −φa)) ϑ
ε]

−∇· [(χεcλc(p
ε,ϑε,ξε) + χεaλa(p

ε,ϑε))∇ϑε]

−∇·
[
cw%wϑ

εkR(S(p
ε))

µ(ϑε)
(χεckc(ξ

ε) + χεaka)∇p
ε

]
= α2χεcf(p

ε,ϑε,ξε), (2)

coupled with an integral condition

ξε(x,t) =
∫ t

0
f(pε(x,s),ϑε(x,s),ξε(x,s)) ds. (3)

The unknowns in the model are the temperature ϑε,
water pressure pε, and the memory function ξε, the
so-called degree of hydration. From the physical point
of view, equations (1) and (2), respectively, repre-
sent the mass balance of moisture (liquid water) and
the heat equation for the porous system, see e.g. [9].
Equation (3) represents an additional integral condi-
tion. Such type of equations arises in the theory of
heat conduction when inner heat sources are of special
types, in particular, in so-called problems of hydration
heat. In this case, the intensity of inner sources of
heat also depends on the amount of heat already de-
veloped, see [29]. In the mesoscale model, we assume
different hydraulic and thermal characteristics in ag-
gregate and cement paste, respectively. In particular,
φ� (� ≈ a,c) is the porosity, and S represents the
degree of saturation with liquid water. Further, k� is

the intrinsic permeability, kR represents the relative
hydraulic conductivity, λ� is the thermal conductivity
function, and µ is the temperature dependent viscos-
ity of the fluid. Material constants are as follows:
%w is the density of liquid water and cw represents
the isobaric heat capacity of water. Moreover, %s�
and cs�, respectively, are the mass density and the
isobaric heat capacity of the solid. Finally, f is the
hydration degree rate and parameters α1 and α2 on
the right hand sides of (1) and (2) express the final
values of mass of hydrated water and the amount of
released heat energy during hydration, respectively.
To complete the model, it is further necessary to

prescribe the initial and boundary conditions. The
initial conditions specify the initial fields of the water
pressure and temperature:

pε(0) = p0, ϑε(0) = ϑ0 in Ω. (4)

The so-called Newton (Robin type) boundary condi-
tions are prescribed on the boundary ∂Ω where the
water flux through the boundary of the domain is
proportional to the difference between the pressure at
the boundary and the outer pressure p∞. Similarly,
the heat flux at the boundary of the system is propor-
tional to the difference between the temperature at
the boundary and the external temperature ϑ∞.

3. The homogenized problem
The system (1)–(3) has been theoretically investigated
in [17, 29]. Instead of solving (1)–(3) on a fine mesh
resolving the small scale variability of ε, the basic idea
of the upscaling methods is to replace the heteroge-
neous medium by an “equivalent” homogeneous one.
It has been shown in [17] that there is a sequence {εj},
limj→+∞εj = 0+, such that [pεj ,ϑεj ,ξεj ] converges
in a suitable topology to the solution of the following
upscaled equations.

Water conservation equation:

∂t [%w (χ∗cφc(ξ) + χ
∗
aφa)S(p)]

−∇· [A∗(p,ϑ,ξ)∇p] = α1χ∗cf(p,ϑ,ξ) (5)

Energy conservation equation:

∂t [cw%w (χ∗cφc(ξ) + χ
∗
aφa)S(p)ϑ]

∂t [([χ∗c%sccsc(1 −φc(ξ)) + χ
∗
a%sacsa(1 −φa)]) ϑ]

−∇· [Λ∗(p,ϑ,ξ)∇ϑ + cwϑA∗(p,ϑ,ξ)∇p]
= α2χ∗cf(p,ϑ,ξ). (6)

The system is coupled with the integral condition

ξ(x,t) =
∫ t

0
f(p(x,s),ϑ(x,s),ξ(x,s))ds. (7)

In (5) and (6), the homogenized coefficient functions
are given by

χ∗c =
∫
Y
χc(y)dy, χ∗a =

∫
Y
χa(y)dy,

13



Michal Beneš, Radek Štefan Acta Polytechnica

Λ∗ij(ω,η,ζ) =
∫
Y

(χc(y)λc(ω,η,ζ) + χa(y)λa(ω,η))︸ ︷︷ ︸
λ(y,ω,η,ζ)

×

×
(
δij + δik

∂vj
∂yk

)
dy (8)

and

A∗ij(ω,η,ζ) =
∫
Y

%wkR(S(ω))
µ(η)

[χc(y)kc(ζ) + χa(y)ka]︸ ︷︷ ︸
a(y,ω,η,ζ)

×

×
(
δij + δik

∂wj
∂yk

)
dy, (9)

where wi and vi ∈ W 1,2per(Y), i = 1, 2, are periodic solu-
tions (in a weak sense) of the following cell problems,
respectively,

−∇y · (a(y,ω,η,ζ)(ei + ∇ywi)) = 0 in Y (10)

and

−∇y · (λ(y,ω,η,ζ)(ei + ∇yvi)) = 0 in Y. (11)

Here, a summation convention is used, i.e. summation
is performed over repeated indices. Further, δij (or
δik) denotes the Kronecker delta and {e1,e2} is the
canonical basis of R2. Note that the problems (10)
and (11), respectively, define wi and vi, i = 1, 2, up
to an additive constants. Usually, we choose wi and
vi such that∫

Y
wi dy = 0 and

∫
Y
vi dy = 0. (12)

It is easily seen that the homogenized coefficients (8)
and (9) do not depend on the choice of the additive
constant.

4. Numerical solution
4.1. Macroscopic FE formulation
Let 0 = t0 < t1 < · · · < tN = T be an equidistant
partitioning of time interval [0,T] with the discrete
time step ∆t, {tn}

N
n=0, ∆t = T/N. For any function

ζ, we will use the approximation ζn ≈ ζ(tn) and de-
note the backward time difference as δ∆t [ζn+1], where
δ∆t [ζn+1] :=

ζn+1−ζn
∆t ≈

dζ(tn+1)
dt

. By Th, we denote
an admissible quadrilateral partition of Ω with a mesh
size h with standard properties from the finite element
theory (see e.g. [30]). Let Wh be the standard con-
forming linear finite element space over Th. Let ph0 =
p0, ϑh0 = ϑ0, and ξh0 = 0 in Ω. For n = 0, . . . ,N − 1,
we seek [phn+1,ϑhn+1,ξhn+1] ∈ Wh × Wh × C(Ω) the

approximate solution of [p,ϑ,ξ] at time tn, such that

%wχ
∗
c

∫
Ω
δ∆t

[
φc(ξhn+1)S(p

h
n+1)

]
ϕh dx

+ %wχ∗aφa
∫

Ω
δ∆t

[
S(phn+1)

]
ϕh dx

+
∫

Ω
A∗(phn,ϑ

h
n,ξ

h
n)∇p

h
n+1 ·∇ϕ

h dx

+
∫
∂Ω
βc(x)phn+1ϕ

h dS

= α1χ∗c
∫

Ω
f(phn,ϑ

h
n,ξ

h
n)ϕ

h dx

+
∫
∂Ω
βc(x)(p∞)n+1ϕhdS (13)

holds for all ϕh ∈ Wh, further

cw%wχ
∗
c

∫
Ω
δ∆t

[
φc(ξhn+1)S(p

h
n+1)ϑ

h
n+1
]
ψh dx

+ cw%wχ∗aφa
∫

Ω
δ∆t

[
S(phn+1)ϑ

h
n+1
]
ψh dx

+
∫

Ω
δ∆t

[
χ∗c%sccsc(1 −φc(ξ

h
n+1))ϑ

h
n+1
]
ψh dx

+
∫

Ω
δ∆t

[
χ∗a%sacsa(1 −φa)ϑ

h
n+1
]
ψh dx

+
∫

Ω
Λ∗(phn,ϑ

h
n,ξ

h
n)∇ϑ

h
n+1 ·∇ψ

h dx

+
∫

Ω
cwϑ

h
nA
∗(phn,ϑ

h
n,ξ

h
n)∇ϑ

h
n+1 ·∇ψ

h dx

+
∫
∂Ω
αc(x)ϑhn+1ψ

h dS + cw
∫
∂Ω
βc(x)ϑhnp

h
n+1ψ

h dS

= α2χ∗c
∫

Ω
f(phn,ϑ

h
n,ξ

h
n)ψ

h dx

+
∫
∂Ω
αc(x)(ϑ∞)n+1ψh dS

+ cw
∫
∂Ω
βc(x)ϑhn(p∞)n+1ψ

hdS (14)

holds for all ψh ∈ Wh and

ξhn+1 = ξ
h
n + ∆tf(p

h
n,ϑ

h
n,ξ

h
n). (15)

The unknown vector field xn+1 =(
pn+1,ϑn+1,ξn+1

)T
is introduced, and the Galerkin

procedure is applied. This leads to a system of
non-linear algebraic equations

b(xn+1) −b(xn) + ∆tKnxn+1

+ ∆tfn+1(xn+1) = 0, (16)

where pn+1, ϑn+1, and ξn+1 store the unknown nodal
values of water pressure, temperature, and hydration
degree at a time tn+1, respectively. The non-linear
system (16) is solved iteratively using the Newton’s
method.

14



vol. 60 no. 1/2020 Homogenization of transport processes and hydration phenomena. . .

4.2. Periodic cell problem
The problems (10) and (11) need to be solved in each
discrete time step. For fixed ω, η, and ζ ∈ R, both
problems, (10) and (11), can be written in a general
form as a periodic cell problem (i = 1, 2)

−∇y · (A(y)(ei + ∇yui)) = 0 in Y, (17)

where A is a given step function,

A(y) :=
{
Ac ∈ R+ for y ∈Yc,
Aa ∈ R+ for y ∈Ya,

(18)

and ui is the unknown function satisfying the periodic
boundary conditions on ∂Y.

4.2.1. The classical FE method
The variational formulation of (17) reads as follows:
we seek ui ∈ W 1,2per(Y) such that (i = 1, 2)∫

Y
A(y)∇ui ·∇ϕ dy = −

∫
Y
A(y)ei ·∇ϕ dy (19)

for all ϕ ∈ W 1,2per(Y). A finite element approxima-
tion is obtained by restricting the weak formulation
(19) to a finite dimensional subspace W 1,2per,h(Y) of
W 1,2per(Y). Then, we seek the approximate solution
uhi ∈ W

1,2
per,h(Y) satisfying the equation∫

Y
A(y)∇uhi ·∇ϕ

h dy = −
∫
Y
A(y)ei ·∇ϕh dy (20)

for all ϕh ∈ W 1,2per,h(Y).

4.2.2. Computation of the homogenized
coefficients by an analytical
approximation

It is worth pointing out that the cell problems (10)
and (11) need to be solved at each integration point
and in each discrete time step n = 0, 1, . . . ,N of the
problem (13)–(15). This means that a very large
number of cell problems need to be solved in order to
compute homogenized coefficients A∗ and Λ∗ in the
whole macroscopic domain Ω.

In [20], the authors proposed an analytical approx-
imation of the solution to the periodic cell problem
(17) with a step function A defined by (18). It can be
easily shown that the functions

ũ1(y) =
∫ y1

0

ds

A(s,y2)

(∫ 1
0

ds

A(s,y2)

)−1
−y1 (21)

and

ũ2(y) =
∫ y2

0

ds

A(y1,s)

(∫ 1
0

ds

A(y1,s)

)−1
−y2 (22)

satisfy the equation (17) almost everywhere in Y, see
[20, Theorem 2.1]. The proposed solution can be seen
as an approximation to the solution of (19) in L2(Y).
The computational cost savings are evident because

of the closed-form approximation of the solution to
the periodic cell problem.

It is worth noting that, for the aggregate of a rectan-
gular shape symmetrically placed within the periodic
cell, which is the case considered in the following
numerical experiments, the aforementioned approxi-
mation leads to simple formulae for the homogenized
coefficients, see also [20–23]. For Λ∗ (and for A∗
analogously), we can write

Λ∗11(pk,ϑk,ξk) = 2d2λc(pk,ϑk,ξk)

+
1 − 2d2

2d1
λc(pk,ϑk,ξk)

+
1 − 2d1
λa(pk,ϑk)

, (23)

Λ∗22(pk,ϑk,ξk) = 2d1λc(pk,ϑk,ξk)

+
1 − 2d1

2d2
λc(pk,ϑk,ξk)

+
1 − 2d2
λa(pk,ϑk)

, (24)

and
Λ∗12 = Λ

∗
21 = 0, (25)

where pk, ϑk, and ξk, respectively, are the water pres-
sure, temperature, and hydration degree at the k-th
integration point of the macroscopic (homogenized)
model, and d1 and d2 are the geometrical parameters
of the periodic cell, see Figure 1.

Figure 1. Cell geometry.

The solution of the cell problem in Y is crucial for
obtaining the homogenized coefficients in the global
domain Ω. Therefore, a comparison of solutions of the
homogenized problem (13)–(15), with homogenized
coefficients obtained by using the analytical formulae
(23)–(25) and the one computed numerically according
to (20), is one of the major goals of the present work,
see the next section.

5. Numerical experiments
The mesoscale model and the homogenization frame-
work have both been implemented in a MATLAB [31]
code, which is employed for the following numerical
experiments. For the spatial discretization, the finite
element method is utilized by using four-node bilinear
elements with 3×3 integration points. The temporal
discretization is performed by a semi-implicit differ-
ence scheme, see Section 4.

15



Michal Beneš, Radek Štefan Acta Polytechnica

The aim of the illustrative examples is to demon-
strate some of the properties of the mesoscale model
and the homogenization framework by analysing the
effects of changing (i) the characteristic length ε, (ii)
the aggregate shape, (iii) the FE mesh size, and (iv)
the approach to solving the local problem.
For the numerical experiments presented in this

paper, the cell geometry shown in Figure 1 is assumed.
The rectangular aggregate shape was chosen for the
reason that it can be easily analysed by the analytical
approach and it can be simply discretized by four-
node bilinear finite elements. Similar examples have
also been investigated by other researches focusing on
the homogenization problem, e.g. [19–23].
For real-world problems, the cell geometry can be

modified in order to represent the material more accu-
rately. Concrete is a cementitious composite formed
by the aggregates of different shapes and sizes. This
could be implemented in the present models by the ap-
propriate setting of the periodic cell geometry. Within
the cell, multiple domains representing the aggregates
can be modelled, each of them of a different shape
and size, and the local problem can be solved by the
finite element method. Due to the limited scope of
the paper, such problem is not presented and will be
analysed in our future work.
All the material parameters and properties of con-

crete components adopted for the calculations as well
as the parameters of initial and boundary conditions
are summarized in Section 7.

5.1. Convergence analysis – layered
structure

The first example is focused on an analysis of hy-
dration and transport processes in a structure (wall)
formed by parallel layers of a fresh cement paste and
aggregates. The objective of this hypothetical ex-
ample is to show the convergence of the mesoscale
solution towards the homogenized problem solution,
as described for similar problems, e.g., in [19–23].

This type of example has been chosen as in this case
(a layered structure), the homogenized coefficients can
be obtained by an exact solution, see [23, p. 2262], cf.
e.g. [26].

The geometrical parameters of the analysed problem
are depicted in Figure 2.
The calculations are performed on the mesoscale

level for three different values of ε: 0.1, 0.05, 0.01. In
all the cases, we use the same finite element mesh of
500 elements in total with ∆x = 1 mm, see Figure 2.
The homogenized problem is solved twice. Once with
the same mesh as for the mesoscale level (∆x = 1 mm),
once with a coarse mesh with ∆x = 25 mm. The time
step is set to ∆t = 30 s in all cases.

The obtained distributions of water pressure across
the analysed structure are displayed in Figure 3. The
temperature profiles are depicted in Figure 4. The dis-
tributions of hydration degree are shown in Figure 5.

Figure 2. Scheme of the analysed problem, finite
element (FE) mesh, and boundary conditions (BC).

For the detailed convergence analysis, we focus on
the resulting distributions of temperature and water
pressure. In Tables 1 and 2, the differences between
the temperatures and water pressures determined by
the mesoscale (ϑε, pε) and homogenized (ϑ, p) models
for selected times are illustrated by the respective L2
error norms, see [19–23].

ε
‖ϑε −ϑ‖2 (K m)

t = 8 h t = 16 h t = 24 h

0.1 8.9 × 10−3 5.8 × 10−3 2.4 × 10−3
0.05 2.2 × 10−3 1.5 × 10−3 7.6 × 10−4
0.01 8.3 × 10−5 7.9 × 10−5 1.0 × 10−4

Table 1. Error norms for the temperature determined
by the mesoscale and homogenized models for selected
times.

ε
‖pε −p‖2 (Pa m)

t = 8 h t = 16 h t = 24 h

0.1 2560.0 1520.0 819.8
0.05 734.9 358.4 197.9
0.01 29.9 14.5 9.3

Table 2. Error norms for the water pressure deter-
mined by the mesoscale and homogenized models for
selected times.

From Figures 4–3 and Tables 1 and 2, it is clear that
with the decreasing value of ε, the mesoscale solution
converges towards the homogenized problem solution.
Moreover, it can be seen that for the homogenized
problem, a coarse finite element mesh provides almost
the same results as a fine mesh.

5.2. Illustrative example – concrete
column cross-section

In the second example, the hydration and transport
processes in a concrete column of a square cross-
section are simulated. The objective of the example
is to illustrate the usage of the numerical procedures

16



vol. 60 no. 1/2020 Homogenization of transport processes and hydration phenomena. . .

0 100 200 300 400 500

x1 (mm)

-2.7

-2.6

-2.5

-2.4

-2.3
W

a
te

r
p
re

ss
u
re

(P
a
)

#106 t = 8 h

p", " = 0:1

p", " = 0:05

p", " = 0:01

p, -ne mesh

p, coarse mesh

0 100 200 300 400 500

x1 (mm)

-3.85

-3.8

-3.75

-3.7

-3.65

-3.6

-3.55

-3.5

W
a
te

r
p
re

ss
u
re

(P
a
)

#106 t = 16 h

p", " = 0:1

p", " = 0:05

p", " = 0:01

p, -ne mesh

p, coarse mesh

0 100 200 300 400 500

x1 (mm)

-4.35

-4.3

-4.25

-4.2

-4.15

-4.1

W
a
te

r
p
re

ss
u
re

(P
a
)

#106 t = 24 h

p", " = 0:1

p", " = 0:05

p", " = 0:01

p, -ne mesh

p, coarse mesh

Figure 3. Distribution of water pressure across the
analysed structure.

for the determination of the homogenized material
parameters, as described in Section 4.2. The scheme
of the analysed problem is depicted in Figure 6.

As can be seen from Figure 6, we assume two vari-
ants of the aggregate shape: square (denoted as Vari-
ant S) and rectangle (denoted as Variant R).
On the mesoscale level, the simulations are per-

formed for ε = 0.025, i.e. with 8 × 8 cells for the
analysed section. In both cases (Variants S and R),

0 100 200 300 400 500

x1 (mm)

29

29.5

30

30.5

31

31.5

32

T
e
m

p
e
ra

tu
re

(/
C

)

t = 8 h

#", " = 0:1

#", " = 0:05

#", " = 0:01

#, -ne mesh

#, coarse mesh

0 100 200 300 400 500

x1 (mm)

42

44

46

48

50

52

T
e
m

p
e
ra

tu
re

(/
C

)

t = 16 h

#", " = 0:1

#", " = 0:05

#", " = 0:01

#, -ne mesh

#, coarse mesh

0 100 200 300 400 500

x1 (mm)

46

48

50

52

54

56

58

60

T
e
m

p
e
ra

tu
re

(/
C

)

t = 24 h

#", " = 0:1

#", " = 0:05

#", " = 0:01

#, -ne mesh

#, coarse mesh

Figure 4. Distribution of temperature across the
analysed structure.

we use the same finite element mesh of square ele-
ments of the size of 2.5 mm (6400 elements in total),
see Figure 7.

For the homogenized model, a finite element mesh
of 10 × 10 square elements is used on the macroscale
level (element size of 20 mm). The homogenized ma-
terial parameters are determined using two different
approaches (see Section 4.2): (i) using the finite ele-
ment solution of the periodic cell problem with the

17



Michal Beneš, Radek Štefan Acta Polytechnica

0 100 200 300 400 500

x1 (mm)

0

0.05

0.1

0.15

0.2

H
y
d
ra

ti
o
n

d
e
g
re

e
(-

)

t = 8 h

9", " = 0:1
9, -ne mesh
9, coarse mesh

0 100 200 300 400 500

x1 (mm)

0

0.05

0.1

0.15

0.2

H
y
d
ra

ti
o
n

d
e
g
re

e
(-

)

t = 8 h

9", " = 0:05
9, -ne mesh
9, coarse mesh

0 100 200 300 400 500

x1 (mm)

0

0.05

0.1

0.15

0.2

H
y
d
ra

ti
o
n

d
e
g
re

e
(-

)

t = 8 h

9", " = 0:01
9, -ne mesh
9, coarse mesh

Figure 5. Distribution of hydration degree across the
analysed structure.

mesh of 10×10 square elements for one quarter of the
cell (element size of 50 mm; due to symmetry, only
one quarter of the cell is modelled), and (ii) using the
simplified approach proposed by Sviercoski et al. [20].
In the following figures, the results obtained by the
homogenized model using these two approaches are
indexed as (FE) and (Approx.), respectively.
The time step is set to ∆t = 60 s in all the cases.

Figure 6. Scheme of the analysed problem, cell ge-
ometries, boundary conditions (BC).

Figure 7. Finite element mesh used for the mesoscale
simulations: Variant S (left), Variant R (right).

The obtained profiles of water pressure, tempera-
ture, and hydration degree along the diagonal of the
analysed section (xD, see Figure 6, see also e.g. [19])
are displayed in Figure 8.

A more detailed comparison of the temperature dis-
tribution is presented using the isolines maps (contour
plots) in Figure 9, cf. e.g. [21–23].
For illustration, selected results (hydration degree

and saturation) are shown using the filled contour
maps in Figures 10 and 11, cf. e.g. [23].
As can be seen, the presented homogenized model

provides a sufficiently accurate approximation to the
mesoscale problem of hydration and transport pro-
cesses in early age concrete. This holds true mainly
for the temperature, which is, from the engineering
point of view, the most important variable.
The analysed example also indicates that, in com-

parison with the classical FE-approach, the approx-
imate solution of the periodic cell problem (deter-
mination of the homogenized material coefficients)
proposed by Sviercoski et al. [20] gives almost the
same results while achieving significant computational
cost savings.

18



vol. 60 no. 1/2020 Homogenization of transport processes and hydration phenomena. . .

The computational cost savings are evident because
of the closed-form approximation of the solution to
the periodic cell problem, which needs to solved at
each integration point (at the macroscale) and in each
discrete time step. By the analytical solution, the
homogenized coefficients are obtained directly by a
closed-form formulae – see formulae (23)–(25), while
for the finite element approach, the local problem
needs to be solved numerically.

6. Conclusions
In the paper, we have proposed a fully coupled two-
scale (meso-macro) homogenization framework (theo-
retically investigated in [17, 29]) for the modelling of
simultaneous heat transfer, moisture flows and hydra-
tion phenomena in fresh concrete.
A modified mesoscale model has been introduced

by assuming concrete as a composite material with
two periodically distributed mesoscale components,
cement paste and aggregates, partially saturated with
liquid water and gas.
For the mesoscale problem, an upscaling method

has been utilized in order to obtain the homogenized
model. The homogenized coefficients have been de-
termined based on the solution of the periodic cell
problem.
Two different approaches have been adopted for

solving the periodic cell problem: (i) the classical finite
element method, (ii) a simplified approach based on
an analytical approximation proposed by Sviercoski
et al. [20].

The resulting algorithm, based on the finite element
discretization in space and a semi-implicit finite dif-
ference discretization in time, has been implemented
in MATLAB environment and employed for several
numerical experiments.
The results of the presented examples have con-

firmed the following.
• With a decreasing value of the characteristic length
ε, representing the small scale variability of the con-
crete composite, the mesoscale solution converges
towards the homogenized problem solution.

• For the homogenized problem, a coarse finite ele-
ment mesh provides almost the same results as the
fine mesh.

• The presented homogenized model provides a suf-
ficiently accurate approximation to the mesoscale
problem of hydration and transport processes in
early age concrete.

• For the homogenized model, the approximate so-
lution of the periodic cell problem (determination
of the homogenized material coefficients) proposed
by Sviercoski et al. [20] gives almost the same re-
sults as the classical FE approach while achieving
significant computational cost savings.

In our future work, we will focus on:

• numerical experiments investigating the effect of
the cell geometry (aggregates of different shapes
and sizes);

• utilization of the homogenization approach for the
analysis of transport processes in concrete exposed
to high temperatures.

7. Appendix – Model parameters
Based on a literature review, the model parameters
employed for the numerical experiments presented in
this paper have been adopted as follows (see also our
previous work [18]).
The values of density and heat capacity of con-

crete components are taken as: %w = 1000 kg m−3,
%sa = 2830 kg m−3, %sc = 3220 kg m−3, cw =
4180 J kg−1 K−1, csa = 840 J kg−1 K−1, csc =
750 J kg−1 K−1 [16, Table 1], [32, Example 13.1].

The porosity of cement paste, φc(ξ), is determined
by [4, (29)], with φc,∞ = 0.2, and Aφ = 0.35 [4, Fig.
1]. The porosity of aggregate, φa, is set to a constant
value φa = 0.05, see e.g. [33].

The intrinsic permeability of cement paste, kc(ξ), is
given by [4, (32)], with kc,∞ = 10−18 m2 [4, Table I],
[32, p. 308] and Ak = 8 [4, p. 312]. The intrinsic
permeability of aggregate, ka, is set to a constant
value ka = 10−20 m2, see e.g. [33].

The thermal conductivity of cement paste and ag-
gregate, λc(p,ξ) and λa(p), respectively, is determined
by [4, (37)] with the constant thermal conductivity of
a dry material, namely λc,dry = 1.0 W m−1 K−1 and
λa,dry = 2.4 W m−1 K−1 [16, p. 136, Table 1].
The degree of the saturation with liquid water, S,

is taken from [34, (20)] with the material parameters
specified in [34, Table 5] for ordinary concrete (a =
18.6237 MPa, b = 2.2748). Here, we follow a simplified
assumption that the capillary pressure, pc, and the
liquid water pressure, p, are in the relation pc =
patm −p with patm being the atmospheric pressure [4,
pp. 305–306], [13, Chapter 4.2.2].
The relative hydraulic conductivity, kR(S), and

the dynamic viscosity, µ(ϑ), respectively, are adopted
from [34, (21)], [35, (38)], and [35, (AI.8)].
The hydration degree rate f is determined as (e.g.

[4, (23)], [16, (7)], [32, (13.31)–(13.35)])

f = A(ξ)βϑ(ϑ)βRH(RH(p,ϑ)), (26)

where the formulae and most of the material pa-
rameters for A(ξ), βϑ(ϑ), and βRH(RH(p,ϑ), are
adopted from [16, 32] with: B1 = 23.4/(24×3600) s−1,
B2 = 7×10−4, η = 6.7, Qe/R = 4600 K, and αe = 7.5.
The notation is taken from [32]. The relative humid-
ity, RH, is calculated from [4, (12)], by assuming
pc = patm − p (see above). The ultimate hydration
degree, ξ∞, is set to ξ∞ = 0.8 (see [2, (13)], [16,
(9),(10)]).

The constants α1 and α2 in the source terms in
equations (1),(2), or (5),(6), respectively, are assumed

19



Michal Beneš, Radek Štefan Acta Polytechnica

0 50 100 150 200

xD (mm)

-2.82

-2.815

-2.81

-2.805

-2.8

-2.795

-2.79

-2.785

W
a
te
r
p
re
ss
u
re
(P
a
)

#106 t =8 h, Variant S

#
p
2

p"

p(FE)
p(Approx:)

0 50 100 150 200

xD (mm)

-3.07

-3.065

-3.06

-3.055

-3.05

-3.045

-3.04

W
a
te
r
p
re
ss
u
re
(P
a
)

#106 t =8 h, Variant R

#
p
2

p"

p(FE)
p(Approx:)

0 50 100 150 200

xD (mm)

36

37

38

39

40

41

42

T
e
m
p
e
ra
tu
re

(/
C
)

t = 8 h, Variant S

#
p
2

#"

#(FE)
#(Approx:)

0 50 100 150 200

xD (mm)

50

52

54

56

58

60

62

64

T
e
m

p
e
ra

tu
re

(/
C

)

t = 8 h, Variant R

#
p

2

#"

#(FE)
#(Approx:)

0 50 100 150 200

xD (mm)

0

0.05

0.1

0.15

0.2

0.25

H
y
d
ra

ti
o
n

d
e
g
re

e
(-

)

t = 8 h, Variant S

#
p

2

9"

9(FE)
9(Approx:)

0 50 100 150 200

xD (mm)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

H
y
d
ra
ti
o
n
d
e
g
re
e
(-
)

t = 8 h, Variant R

#
p
2

9"

9(FE)
9(Approx:)

Figure 8. Water pressure, temperature, and hydration degree profiles along the diagonal of the analysed section.

as ([4, 16, 32], see also [29] and references therein)

α1 = −Qw,potmc, (27)
α2 = Qh,potmc, (28)

with Qw,pot = 0.25, Qh,pot = 510 × 103 Jkg−1, and
mc = (1 −φc(0))%sc.
The initial conditions are set to p0 = −2.155 ×

106 Pa (S0 = 0.99) and ϑ0 = 293.15 K.
The parameters of the boundary conditions are

taken as: βc = 10−30 s m−1, p∞ = −6.9 × 107 Pa

(RH∞ = 0.6), αc = 4 W m−2 K−1, and ϑ∞ =
293.15 K, for the boundaries denoted as BC1 in Sec-
tion 5. The boundaries denoted as BC2 in Section 5
are assumed to be perfectly insulated, i.e. βc = 0 and
αc = 0.

List of symbols
cw Isobaric heat capacity of water [J kg−1 K−1]
csa Isobaric heat capacity of aggregates [J kg−1 K−1]
csc Isobaric heat capacity of cement paste [J kg−1 K−1]

20



vol. 60 no. 1/2020 Homogenization of transport processes and hydration phenomena. . .

Temperature (/C), t = 8 h, Variant S

38

38.5

38.5

3
9

39

39

3
9
.5

3
9
.5

39.5

39.5

4
0

40

40

40

4
0
.5

40.5

40.5

41

4
1

41

41.5

41.5

0 50 100 150 200

x1 (mm)

0

50

100

150

200
x
2
(m

m
)

#"

#(FE)
#(Approx:)

Temperature (/C), t = 8 h, Variant R
5354

55
55

5
6

56

56
5
7

5
7

57

57
5
8

58

58

58

59

59

59

60

6
0

60

6
1

61

62

62

0 50 100 150 200

x1 (mm)

0

50

100

150

200

x
2
(m

m
)

#"

#(FE)
#(Approx:)

Temperature (/C), t = 24 h, Variant S

5456
5
8

58

58

6
0

60
60

60

6
2

62

62

64

64

66

0 50 100 150 200

x1 (mm)

0

50

100

150

200

x
2
(m

m
)

#"

#(FE)
#(Approx:)

Temperature (/C), t = 24 h, Variant R

75
80

80

8
5

85

85

85

9
0

90

90

95

0 50 100 150 200

x1 (mm)

0

50

100

150

200

x
2
(m

m
)

#"

#(FE)
#(Approx:)

Figure 9. Distribution of temperature in the analysed cross-section.

ka Intrinsic permeability of aggregates [m2]
kc Intrinsic permeability of cement paste [m2]
kc,∞ Intrinsic permeability of the matured concrete [m2]
kR Relative permeability of liquid phase [–]
mc Mass of cement [kg m−3]
p Liquid pressure [Pa]
p0 Initial value of liquid pressure [Pa]
p∞ External liquid pressure [Pa]
patm Atmospheric pressure [Pa]
pc Capillary pressure [Pa]
Qh,pot Potential hydration heat [J kg−1]
Qw,pot Chemically bound water [kg kg−1]
RH Relative humidity [–]
RH∞ External relative humidity [–]
S Degree of saturation with liquid water [–]
S0 Initial degree of saturation with liquid water [–]

Greek symbols
αc Convective heat transfer coefficient [W m−2 K−1]
βc Convective mass transfer coefficient [m s−1]
δij Kronecker delta [–]
ε Scale parameter (characteristic length) [–]
ϑ Temperature [K]
ϑ0 Initial value of temperature [K]
ϑ∞ External temperature [K]
λc Thermal conductivity of cement paste [W m−1 K−1]
λc,dry Thermal conductivity of dry cement paste

[W m−1 K−1]
λa Thermal conductivity of aggregates [W m−1 K−1]
λa,dry Thermal conductivity of dry aggregates

[W m−1 K−1]
µ Water viscosity [Pa s]
ξ Degree of hydration [–]
ξ∞ Final degree of hydration [–]

21



Michal Beneš, Radek Štefan Acta Polytechnica

Figure 10. Distribution of hydration degree in the
analysed cross-section.

%w Liquid density [kg m−3]
%sa Density of aggregates [kg m−3]
%sc Density of cement paste [kg m−3]
φc Porosity of cement paste [–]
φc,∞ Porosity at final stage of hydration [–]

Figure 11. Distribution of saturation degree in the
analysed cross-section.

φa Porosity of aggregates [–]

χa Characteristic function of aggregates [–]

χc Characteristic function of cement paste [–]

22



vol. 60 no. 1/2020 Homogenization of transport processes and hydration phenomena. . .

Acknowledgements
The first author has been financially supported by
the European Regional Development Fund (project No.
CZ.02.1.01/0.0/0.0/16−019/0000778) within activities of
the Center of Advanced Applied Sciences (CAAS).

The second author has been supported by the Czech Sci-
ence Foundation, project no. GA17-23067S. The support
is gratefully acknowledged.

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[35] C. T. Davie, C. J. Pearce, N. Bićanić. Coupled Heat
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24

http://dx.doi.org/10.1002/nme.3014
http://dx.doi.org/10.1016/j.commatsci.2011.10.017
http://dx.doi.org/10.1002/zamm.201700158
http://dx.doi.org/10.1007/978-94-024-1138-6
http://dx.doi.org/10.1016/j.conbuildmat.2008.04.009
http://dx.doi.org/10.1016/S0008-8846(99)00102-7
http://dx.doi.org/10.1080/10407780500503854

	Acta Polytechnica 60(1):12–24, 2020
	1 Introduction
	2 The mesoscale model
	3 The homogenized problem
	4 Numerical solution
	4.1 Macroscopic FE formulation
	4.2 Periodic cell problem
	4.2.1 The classical FE method
	4.2.2 Computation of the homogenized coefficients by an analytical approximation


	5 Numerical experiments
	5.1 Convergence analysis – layered structure
	5.2 Illustrative example – concrete column cross-section

	6 Conclusions
	7 Appendix – Model parameters
	List of symbols
	Acknowledgements
	References