Modeling fluid saturated porous media under frost attack

GAMM-Mitt. 33, No. 1, 40 – 56 (2010) / DOI 10.1002/gamm.201010004

Modeling fluid saturated porous media under frostattack

Tim Ricken1,∗ and Joachim Bluhm2∗∗1 Faculty of Engineering, Department of Civil Engineering, Computational Mechanics, Uni-

versity of Duisburg-Essen, 45117 Essen, Germany2 Faculty of Engineering, Department of Civil Engineering, Institute of Mechanics, Univer-

sity of Duisburg-Essen, 45117 Essen, Germany

Received 02 February 2010

Published online 27 April 2010

Key words Phase Transition, Multiphase Material, Constitutive Modeling, Simulation

MSC (2000) 74N10

Freezing and thawing are important processes in civil engineering. On the one hand frostdamage of porous building materials like road pavements and concrete in regions with peri-odical freezing is well known. On the other hand, artificial freezing techniques are widelyused, e.g. for tunneling in non-cohesive soils and other underground constructions as well asfor the protection of excavation and compartmentalization of contaminated tracts. Ice forma-tion in porous media results from a coupled heat and mass transport and is accompanied bythe ice expansion. The volume increase in space and time is assigned to the moving freezingfront inside the porous solid. In this paper, a macroscopic ternary model is presented withinthe framework of the Theory of Porous Media (TPM) in view of the description of phasetransition. For the mass exchange between ice and water an evolution equation based on thelocal balance of the heat flux vector is used. Examples illustrate the application of the modelfor saturated porous solids under thermal loading.

c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The behavior of fluid saturated porous media under cycling thermal loading, e.g. drying

of porous solids, the freezing of soils and concrete or geothermal investigations, is strongly

influenced by the fluid-ice phase transition and plays an important role in order to predict

fatigue and lifetime estimations as well as frost resistance in material science.

Frost damage results from two different mechanisms (internal frost attack and surface scal-

ing) FAGERLUND [1] expected one basic common cause for both, namely the expansion of

ice associated with the achievement of a critical degree of saturation in the pores. Due to the

anomaly of water an increasing volume dilatation of 9% (referred to the volume of liquid)

takes place during the phase change from liquid to ice, i.e., an equivalent water volume must

fitt in the pores. If there is no void available, internal pressure acts on the surfaces of the pores

∗ Corresponding author E-mail: [email protected], Phone: +49 (0)201 183 2681, Fax:

+49 (0)201 183 2680∗∗ [email protected]

c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

GAMM-Mitt. 33, No. 1 (2010) 41

and frost damage occurs, the so-called hydraulic pressure, compare POWERS [2], which ex-

ceeds the tensile strength of the concrete. Although this simple model gives a good advice to

the frost damage behavior, more detailed stress leading damage models have been developed

including a variety of mechanisms, see e.g. FAGERLUND [1], SETZER [3] and PALECKI [4].

It is known from experiments that isothermal capillary saturated hydrated cement with-

stands several freeze-thaw cycles almost without damage. During the freezing and thawing

cycles the degree of saturation increase, i.e., an additional water uptake of the saturated ce-

ment stone specimen is observed, see e.g. AUBERG & SETZER [5] and SETZER et al. [6],

which is called frost suction. The saturation due to the frost suction is two to three times

higher than the saturation of isothermal capillary suction.

With respect to this complexity in the presented paper a macroscopic multiphase model is

discussed. Based on the framework of the Theory of Porous Media (TPM) (Mixture Theory

combined with the Concept of Volume Fraction) we investigate quite a few main physical

phenomena of freezing and thawing processes in porous media: (i) expansion of ice during

freezing; (ii) phase transitions of water and ice and vice versa in consideration of energetic

aspects; (iii) propagation of the freezing and thawing fronts; (iv) development of temperature

in space and time during a freeze-thaw cycle.

In the multiphase approach discussed in this paper we consider an incompressible ternary

model, consisting of the constituent solid (cement stone), liquid (freezable water) and ice. In

this first approach, the model is to be understood as a basic concept in view of the complete

description of freezing and thawing processes in saturated porous media. With respect to the

simplified model, capillary effects, see RICKEN & DE BOER [7], cannot be taken into account.

The mass exchange between the phases ice and liquid is determined by the disequilibrium

of the corresponding chemical potentials. This is a result of the evaluation of the entropy in-

equality for the porous medium in connection with the dissipation mechanism of the system,

compare BLUHM et al. [8] and KRUSCHWITZ & BLUHM [9]. An ansatz for the control-

ling of the phase transition and the mass exchange, respectively, is discussed by BLUHM &

RICKEN [10]. As shown in BLUHM et al. [11], the fusion enthalpy can only be correctly de-

scribed by controlling the heat flux through the surface, especially during the thawing process.

Therefore, a new ansatz regarding the evaluation equation for the mass supply between ice and

water and vice versa is presented. This thermo-mechanical consistent ansatz is based on the

reformulation of the entropy inequality and the balance of energy for the mixture in consid-

eration of the respected phase enthalpies. It is postulated that the ice-water mass exchange

in a material point is proportional to the divergence of the temperature gradient dived by the

difference of the corresponding enthalpies. The physical verification of this new ansatz will be

confirmed by a simple example of a one-dimensional freezing process, where one-component

materials in connection with a moving ice-front (interface) are considered.

After discussion of the basic equations (saturation condition, balance equations) restric-

tions for the constitutive relations and the dissipation mechanism of the simplified model will

be derived. Furthermore, a calculation concept is presented for the description of freezing and

thawing processes. This concept leads to a highly coupled set of differential equations. The

resolving weak formulations are inserted into the finite element program FEAP. The useful-

ness of the presented model will be demonstrated by a comparison of computationally and

experimentally gained data of the CIF-Test (Capillary suction, Internal damage and Freeze-

Thaw Test), see SETZER et al. [6]. The illustrated results show that the simplified model is

capable of reproducing the experimental observations.

www.gamm-mitteilungen.org c© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

42 T. Ricken and J. Bluhm: Frost attack

2 Basics

In this section, the fundamentals of the Theory of Porous Media (TPM) are briefly introduced.

Readers interested in the foundation of the TPM are referred to BOWEN [12,13], DE BOER &

EHLERS [14], EHLERS [15,16]. A historical overview of the development of the TPM as well

as the fundamentals, the modeling of incompressible, compressible and hybrid binary porous

media and possible applications of the theory are presented in the book of DE BOER [17].

2.1 Mixture theory, concept of volume fraction and kinematics

It is assumed that all constituents κ are statistically distributed over a control space and that the

system is in ideal disorder. The single constituents are “smeared” over the control space and

occupy the whole volume simultaneously. Within the TPM a real density ραR(x, t) as well

as a volume fraction nα(x, t) are assigned to each phase. The real density of a constituent

is the mass of ϕα per unit volume dvα, while the volume fraction nα = dvα/dv of ϕα

represents the proportion of the total volume occupied by the constituent. The partial density

ρα is determined by ρα = nαραR. Since solid particles and all remaining κ − 1 constituents

are assumed to occupy all the available volume, the porous solid is said to be saturated and

κ∑α=1

nα =κ∑

α=1

ρα

ραR= 1 . (1)

Equation (1) is the so-called saturation condition. The idea of superimposed and interacting

continua implies that each constituent is assigned its unique individual Lagrangean (material)

function of motion χα:

x = χα (Xα , t ) , x′α =

∂χα (Xα , t )∂t

, x′′α =

∂2χα (Xα , t )∂t2

, (2)

The vectors x′α and x′′

α are the velocity and the acceleration field. The function χα is postu-

lated to be unique and uniquely invertible at any time t. The existence of a function inverse

to (2)1 leads to the Eulerian description of motion, viz. Xα = χ−1α (x , t ). The deformation

gradient Fα is defined as:

Fα =∂χα (Xα , t )

∂Xα= Gradα χα . (3)

2.2 Balance equations and entropy inequality

The balance equations for porous media are the balance equations of the constituents taken

from the mixture theory. Excluding additional supply terms of moment of momentum, the

balance equations for a saturated porous medium are given by the local statements of the

balance equation of mass, the balance equations of momentum and moment of momentum as

well as the balance equation of energy for each individual constituent:

(ρα)′α + ρα div x′α = ρα ,

div Tα + ρα (b − x′′α ) = ρα x′

α − pα , Tα = (Tα)T ,

ρα (εα)′α − Tα · Dα − ρα rα + div qα =

= eα − pα · x′α − ρα ( εα − 1

2x′

α · x′α ) .

(4)


GAMM-Mitt. 33, No. 1 (2010) 43

In these equations, Tα and b are the partial Cauchy stress tensor of ϕα and the external

acceleration. The internal energy, the external heat supply and the influx of energy are denoted

by εα, rα and qα. The quantities ρα, pα and eα represent the local supply terms of mass,

momentum and energy of ϕα arising out of all other constituents κ− 1 that occupy the same

position as ϕα at time t. The tensor Dα is the symmetric part of the spatial velocity gradient

L = (Fα)′α F−1α . In addition, “div” is the divergence operator and the symbol (. . . )′α =

∂(. . . )/∂t+grad(. . . )·x′α (calculation specification for a scalar quantity) defines the material

time derivative with respect to the trajectory of ϕα. For the supply terms of mass, momentum

and energy the following restrictions

κ∑α=1

ρα = 0 ,κ∑

α=1pα = o ,

κ∑α=1

eα = 0 (5)

are applied. In anticipation of the modeling of the phase transition between the phases ice

and liquid, evolution equations for the corresponding mass supply terms must be formulated.

For the clarification of the physical motivation of these equations an alternative form of the

balance of energy is required. Therefore, the balance of energy is to be reformulated by using

the enthalpy instead of the energy. In view of the introduction of the enthalpy, the stress power

Tα · Dα = (Tα)D · (Dα)D − pα (Dα · I ) , pα = − 13

(Tα · I ) , (6)

is split into two parts, where the value pα is the hydrostatic pressure of the constituent ϕα.

The upper subscript (. . . )D = (. . . )−1/3 [(. . . ) ·I ] I denotes the deviatoric part of the tensor

(. . . ). With help of the balance of mass (4)1, the second term in (6) can be expressed as

pα (Dα · I ) = ρα (1ρα

pα )′α − (pα)′α +ρα

ραpα (7)

and the balance equation of energy (4)4 can be transferred into

ρα (sαc )′α − (pα)′α − (Tα)D · (Dα)D − ρα rα + div qα =

= eα − pα · x′α − ρα ( sα

c −12

x′α · x′

α ) ,(8)

where sαc = εα + pα/ρα is denoted as the classic specific enthalpy, compare VOLLER et

al. [18] and KOWALCZYK [19]. The display format (8) of the balance of energy often is called

enthalpy equation, see SPURK [20].

The entropy inequality is a helpful tool for the development of thermodynamically consis-

tent constitutive and evolution equations. The local entropy inequality for the mixture with

different temperatures of the constituents is defined as

κ∑α=1

[ ρα (ηα)′α + ρα ηα ] ≥κ∑

α=1

[1

Θαρα rα − div (

1Θα

qα ) ] (9)

where ηα and Θα are the specific entropy and the absolute temperature of the constituent ϕα.

Within the framework of the TPM the saturation condition is understood as a constraint

in view of the individual motion of the constituents. This must be considered in respect of

the evaluation of the entropy inequality. Therefore, the inequality (9) will be extended by

the material time derivation of the saturation condition following the motion of solid together

with the concept of Lagrange multipliers. Thus, the local statement



κ∑α=1

1Θα

{− ρα [ (ψα)′α + (Θα)′α ηα ] + Tα · Dα − 1Θα

qα · gradΘα +

+ eα − pα · x′α − ρα (ψα − 1

2x′

α · x′α ) } + λ ( 1 −

κ∑α=1

nα )′S ≥ 0

(10)

of the entropy inequality of a saturated porous medium is obtained where use has been made

of the balance equation of energy and the free Helmholtz energy ψα = εα −Θα ηα.

3 Simplified ternary model

In view of the description of freezing and thawing processes of liquid saturated porous solid

materials like concrete the following assumptions and simplifications, respectively, will be

made: i) the fluid flow is low during the process (x′′α = o); ii) internal energy terms are not

taken into consideration (rα = 0); iii) the local temperatures of all constituents are equal

(Θα = Θ); iv) the motions of solid and ice are identical (χS = χI); v) the mass exchange

only acts between the ice and the liquid phase (ρS = 0, ρI = −ρL); vi) solid, ice and liquid

are incompressible (ραR = const.).

3.1 Field equations

Considering the aforementioned assumptions as well as the restrictions for the supply terms,

see (5), the set of coupled field equations for the description of freezing and thawing processes

in fluid saturated porous solids is given by the balance equations of mass,

(nS)′S + nS div x′S = 0 , (nI)′S + nI div x′

S =ρI

ρIR,

(nL)′L + nL div x′L = − ρI

ρLR,

(11)

the balance equations of momentum for the mixture, for liquid and for gas,

div T + ρb = − ρI wLS , div TL + ρL b = − ρI x′L − pL , (12)

the balance equation of energy for the mixture,

ρS (εS)′S + ρI (εI)′S + ρL (εL)′L − TSI · DS − TL · DL + div q =

= − pL · wLS − ρI ( εI − εL − 12

x′S · x′

S +12

x′L · x′

L ) ,(13)

and the material time derivative of the saturation condition along the trajectory of the solid,

(nS)′S + (nI)′S + (nL)′L − gradnL ·wLS = 0 . (14)

In these equations the abbreviations

T =∑α

Tα , TSI = TS + TI , q =∑α

qα , ρ =∑α

ρα , (15)

have been used. The quantities wLS = x′L − x′

S is the velocity of liquid relative to solid. In

soil mechanics, the difference velocity wLS is called seepage velocity.


GAMM-Mitt. 33, No. 1 (2010) 45

3.2 Constitutive theory

In this model, the material time derivative of the saturation condition (14) is an equation

in excess that restricts the motion of the incompressible constituents. Therefore, the set of

unknown field quantities

U = {χS , χL , Θ , nS , nI , nL , λ } . (16)

must be extended by a scalar, namely the Lagrange multiplier λ, which is understood as

an indeterminate reaction force assigned to the saturation condition. Considering that the

acceleration of gravity b is known, the remaining quantities

C = {TSI , TL , qSIL , ψα , ηα , pL , ρI } (17)

require constitutive assumptions or evolution equations to close the system of equations. It

is postulated that these variables can depend on the following set of process variables, where

regarding the stresses viscous effects are not considered:

P = {Θ , gradΘ , CS = CI , JL , nI , nL , wLS , gradnL , x′S · x′

S , x′L · x′

L } , (18)

where viscous effects are not considered. As a result of the mass exchange between the ice

and liquid phase, the volume fractions nI and nL can not expressed by the Jacobian JI = JS =detFS =

√detCS and JL. Thus, both the volume fraction and the Jacobian are considered

in the set of process variables. Merely for an incompressible phase the volume fraction is

proportional to the volume deformation if mass exchange is excluded. In (18), the quantity

nL is not an independent process variable. Taking into account the saturation condition and

the balance equation of mass for the solid with JS = nS0S/nS, where nS

0S = const. denotes

the volume fraction of the solid reference configuration, the volume fraction of liquid can be

expressed by CS and nI. This dependence must be considered with respect to the evaluation

of the entropy inequality.

To simplify the evaluation of the entropy inequality and in view of preceding results re-

garding the constitutive relations of other multiphase models, see e.g. EHLERS [21], for the

free Helmholtz energy functions the following dependences are postulated:

ψS = ψS ( Θ , CS ) , ψI = ψI ( Θ , CS , nI ) , ψL = ψL ( Θ , JL , nL ) . (19)

With the ansatz (19) it can be shown that the entropy inequality (10) is fulfilled for the follow-

ing constitutive relations concerning the specific entropies and the Cauchy stress tensors:

ηS = − ∂ψS

∂Θ, ηI = − ∂ψI

∂Θ, ηL = − ∂ψL

∂Θ, (20)

and

TSI = − nSI λ I + TSIE , TL = − nL λ I + TL

E , (21)

where

TSIE = − nI ρI ∂ψI

∂nII + 2FS ( ρS ∂ψS

∂CS+ ρI ∂ψI

∂CS)FT

S ,

TLE = − nL ρL ∂ψL

∂nLI + ρL JL

∂ψL

∂JLI

(22)



are the so-called effective stress tensors of solid/ice and liquid. The dissipations mechanism

of (10) yields the evolution equations the heat flux vector the supply terms of mass and mo-

mentum:

q = −α�Θ gradΘ − αwLS wLS (23)

and the supply terms of mass and momentum

ρI = − βIΨIL ( ΨI − ΨL ) ,

pL = λ gradnL + pLE , pL

E = − γL�Θ gradΘ − γL

wLSwLS ,

(24)

where the material parameters are restricted by

α�Θ ≥ 0 , αwLS + γL�Θ = 0 , βI

ΨIL ≥ 0 , γLwLS

≥ 0 . (25)

The vector pLE is the so-called effective part of the interaction of momentum of the liquid

phase.

For the Helmholtz free energy functions of the solid and ice phase the following ansatz are

postulated, compare BLUHM & RICKEN [10]:

ψS =1

ρS0S

[12

λS ( log JS )2 − μS log JS +12

μS (CS · I − 3 ) −

− 3 αSΘ kS log JS ( Θ − Θ0 ) − ρS

0S cS ( Θ logΘΘ0

− Θ + Θ0 ) ] ,

ψI =1

ρI0S

[12

λI ( log JS )2 − μI log JS +12

μI (CS · I − 3 ) −

− 3 αInI kI log JS ( nI − nI

0S )−

− 3 αIΘ kI log JS ( Θ − Θ0 ) − ρI

0S cI ( Θ logΘΘ0

− Θ + Θ0 ) ] ,

ψL =1ρL

[− 3 αLΘ kL ( Θ − Θ0 ) − ρL cL ( Θ log

ΘΘ0

− Θ + Θ0 ) ] .

(26)

Therein, the material parameters μβ , λβ for β = {S , I} and kα are the Lame constants and

the bulk modulus of the corresponding constituent. The coefficient of thermal expansion, the

heat capacity and the coefficient related to the volume fraction are denoted by ααΘ, cα and

αInI . The symbols (. . . )S0S = JS (. . . )S = const. and (. . . )I0S = JS (. . . )I �= const. denote

quantities of solid and ice referring to the reference placement of solid at time t = 0.

With the constitutive assumptions (26), the following constitutive relations for the effective

stress tensors of solid/ice is gained, see (22)1:

TSIE =

1JS

[ 2 (μS + μI )KS + (λS + λI ) ( log JS ) I−

− 3 αInI kI ( nI − nI

0S − nI log JS ) I − 3 (αSΘ kS + αI

Θ kI ) ( Θ − Θ0 ) I ] ,

TLE = − 3 αL

Θ kL ( Θ − Θ0 ) I .

(27)

where KS = 1/2 (BS − I) denotes the Karni-Reiner strain tensor, BS = FS FTS is the

left Cauchy-Green deformation tensor. With the assumptions regarding the Helmholtz free

energies the balance of energy simplifies to


GAMM-Mitt. 33, No. 1 (2010) 47

Θ [ ρS (ηS)′S + ρI (ηI)′S + ρL (ηL)′L ] + div q = − pLE · wLS − ρI ( sIp − sLp ) . (28)

with the specific enthalpies assigned to phase change of the constituents ice and liquid

sIp = εI +1ρI

( nI λ + nI ρI ∂ψI

∂nI) − 1

2x′

I · x′I ,

sLp = εL +1ρL

( nL λ + nL ρL ∂ψL

∂nL) − 1

2x′

L · x′L .

(29)

For the description of phase transitions the internal energy εα has to be extended with the

value of the specific latent heat of fusion αf of the corresponding phase, i.e. εα = εα + α

f .

The quantity εα describe the classical part of internal energy without consideration of phase

transitions. In this model only mass exchange from liquid to ice and vice versa is considered.

Thus, merely the specific heat of fusion If for the ice phase has to be taken into account, i.e.,

Sf = L

f = 0 and If �= 0. Under normal pressure (p = λ = 1.013 bar, the latent heat of

fusion for ice is in the order of If = 334 [kJ/kg].

Apart from the scalar quantity pLE ·wLS, the balance equation of energy (28) for the quadru-

ple porous medium has the structure as the balance of energy of a one-component material

with an internal source represented by ρI(sIp − sLp). With respect to the description of phase

change it will be postulated that the nascent as well as the required energy for the phase trans-

formations liquid/ice and ice/liquid is proportional to the divergence of the heat flux of all

phases, i.e.,

ρI ( sIp − sLp ) = ρI [ ΨI − ΨL + Θ ( ηI − ηL ) ] = − βI div q , (30)

where 0 ≤ βI ≤ 1. Thus, the mass supply term reads

ρI = − (βI div q) / (sIp − sLp) . (31)

Considering the evolution equation (24)1 for the mass supply term of ice and liquid, respec-

tively, the material parameter βIΨIL can be expressed as

βIΨIL =

βI div q( ΨI − ΨL ) ( sIp − sLp )

. (32)

By neglecting the influence of the difference velocity regarding the heat flux, i.e., αwLS = 0and q = −α�Θ gradΘ, see (23), for α�Θ = const. the relation (31) turns over into

ρI = (βI α�Θ div gradΘ) / (sIp − sLp) . (33)

In order to get a deeper understanding of the characteristic of ansatz (33) regarding the supply

term of mass of ice, a one dimensional freezing process of water and ice is contemplated,

see Fig. 1, where the phases water and ice are considered as incompressible one-component

materials and not as a mixture, i.e., all supply terms are equal to zero and the interaction

between the two phases is captured by the moving interface.

The whole mantle surfaces of ice and water consist of ∂BI \ Σ plus Σ and ∂BW \ Σ plus

Σ, where Σ denotes the surface of the interface. Thus, the change of the densities in time of



Fig. 1 (online colour at: www.gamm-mitteilungen.org) Illustration of a one-dimensional freezing pro-

cess of water and ice

the one-component materials ice and water in consideration of the interface can be expressed

as

ddt

∫

BI

ρI dv =∫

BI

∂ρI

∂tdv +

∫

∂BI

ρI x′I · n da −

∫

Σ

ρI (x′I − v ) · nΣ da ,

ddt

∫

BW

ρW dv =∫

BW

∂ρW

∂tdv +

∫

∂BW

ρW x′W · n da +

∫

Σ

ρW (x′W − v ) · nΣ da ,

(34)

see e.g. HUTTER & JOHNK [22], where v and nΣ denote the velocity and the normal vector of

the interface. It should be mentioned that the normal vector nΣ is outward-looking regarding

ice and inward-looking regarding water. This has been considered in (34) by the algebraic

sign of the surface integrals of the interface. The change in time of the enthalpy of ice and

water can be formulated in a similar manner:

ddt

∫

BI

ρI sI dv =∫

BI

∂ρI sI

∂tdv +

∫

∂BI

ρI sI x′I · n da −

∫

Σ

ρI sI (x′I − v ) · nΣ da ,

ddt

∫

BW

ρW sWdv =∫

BW

∂ρW sW

∂tdv +

∫

∂BW

ρW sW x′W · n da +

+∫

Σ

ρW sW (x′W − v ) · nΣ da .

(35)

With the balance equations of mass and energy for the single phases, the change of mass

and energy in time of the whole system (water, ice and interface) read

ddt

∫

BI

ρI dv +ddt

∫

BW

ρW dv =∫

Σ

[ ρW (x′W − v ) − ρI (x′

I − v ) ] · nΣ da (36)

and


GAMM-Mitt. 33, No. 1 (2010) 49

ddt

∫

BI

ρI sI dv +ddt

∫

BW

ρW sWdv =∫

Σ

[ ρW sW (x′W − v ) − qW −

− ρI sI (x′I − v ) + qI ] · nΣ da −

∫

∂BI\Σ

qI · n da −∫

∂BW\Σ

qW · n da .

(37)

where the pressure rate has been neglected.

Postulating, that there are no sources of mass and energy in the interface, the surface inte-

grals regarding the interface must vanished. Provided that the integrands are continuous, the

local statements

ρW (x′W − v ) − ρI (x′

I − v ) = o ,

ρW sW (x′W − v ) − qW − ρI sI (x′

I − v ) + qI = o(38)

are complied, where the variables of the interface are not marked in particular.

With (38)1, the local velocity of water on the interface can be expressed as

x′W =

ρW − ρI

ρWv +

ρI

ρWx′

I (39)

and it follows from (38)2:

v =ρI x′

I ( sW − sI ) + qI − qW

ρI ( sW − sI ). (40)

Assuming that the particles of ice do not change their position, i.e., x′I = o, the velocity of

the interface simplifies to

v =qI − qW

ρI ( sW − sI ). (41)

Equation (41) shows that the driving force of the velocity of the interface is proportional

to the jump of the heat flux vectors divided by the difference of the corresponding specific

enthalpies.

4 Examples

In this section an application of the model is shown. Therefore, a cuboid solid specimen with

a cross section of 15 cm2 and a height of 7.5 cm is treated. The dimension of the specimen is

taken from the so called CIF-Test (Capillary suction, Internal damage and Freeze thaw test).

For a detailed description of the CIF-Test the reader is referred to SETZER et al. [6].

Assuming that all unstressed surfaces are adiabatic to the environment, it is sufficient to

analyze the middle surface of the cuboid, i.e., the ice formation can be modeled as a 2-D

problem. Furthermore, it is postulated that all pores are only filled by macroscopic water, i.e.,

the influence of vapor and air about the discussed process is not considered here. This assump-

tion is not realistic for an adiabatic saturated cement specimen but close to one with beginning

frost damage because of a high rate of freeze-thaw cycles, see KASPAREK & SETZER [23].

Due to the anomaly of water, a volume dilatation of 9% referred to the volume of liquid

takes place during the phase change from liquid to ice, i.e., an equivalent water volume has



tem

pera

ture

20°C

-20°C

0°C

0h 4h 7h 11h 12h

time

Fig. 2 Temperature load. (online colour at:

www.gamm-mitteilungen.org)

air layer as termalisolation

10 - 50 [mm]

5 [mm]

Reference Pointtest liquid spacer 5 [mm] high

lateral sealing

lid of the chest

nextcontainer

5 [mm]

specimen

cooling liquid

Fig. 3 CIF-Test, see SETZER [1] (online

colour at: www.gamm-mitteilungen.org)

to be replaced. The volume increases in space and time subject to the moving freezing front

inside the porous solid. To solve the freezing and thawing problem the linearized weak forms

of the balance equations of mass of ice, momentum and energy of the mixture and the rate of

saturation condition within the framework of a standard Galerkin procedure are implemented

in the finite element program FEAP.

In view of the simulation, a freeze-thaw cycle of the following value and boundary con-

ditions is chosen. The bottom of the specimen will be cooled in-line from 293 K to 253 K

within 7 hours and afterward will be heated to 293 K within 4 hours. The temperature load is

assumed from the so called CIF-test, see Fig. 2. During the CIF-Test, the bottom of the partly

saturated solid is also cooled down and heated. Due to laterally air layers as thermal isolation

the heat transport is one dimensional as in the simplified model.

The material parameters of the initial configuration are given in Tab. 1. The remaining

parameters concerning the interaction terms between the constituents are chosen as follows:

αSILwLS

= 0 [ N/m2 ], γL�Θ = 0 [ N/m2 K ] and γL

wLS= 101 / 104 [ Ns/m4 ], whereas the last

value regarding γLwLS

is only used for the last example. Furthermore, for the here discussed

numerical results it is postulated that the prefactor βI of the modified ansatz with respect to

the mass supply term of ice, see (33), is equal to 1 for the freezing as well as for the thawing

process. Regarding the numerical control for the time discretization in the Newmark method

the time step �t = 1.6 [min] is used.

With respect to the simulation the specimen is considered as unjacketed, i.e., during the

freeze-thaw cycle liquid water can be sucked in or squeezed out to the three unstressed sur-

faces. The bottom of the specimen is assumed as impermeable. These boundary conditions

are not conform with the CIF-Test, but in view of the numerical calculation the condition is

essential, because the gas phase is not considered in the model. Due to the here chosen value

for γLwLS

, the influence of the pressure on the deformations is in negligible order. Regarding

the specific enthalpy, see (29)1, it is obvious that the influence of the pressure part is neglected.

Thus, the specific heat of fusion If is constant, see Table 1.

Fig. 4 a) to c) show the distributions of the temperature, the volume fraction of solid and

ice and the dilations represented by the Jacobian of solid after 5 and 11.8 hours. Furthermore,

the expansion of the specimen is plotted, where the scaling factor 10 is used.

During the cooling phase in-between 293 – 273 K, a thermal dilatation is observed in

reality as well as in the simulation. At a temperature of Θ ≤ 273 K, the phase change from

liquid to ice occurs and during cooling the volume deformation of the specimen rises up to


GAMM-Mitt. 33, No. 1 (2010) 51

Table 1 Material parameters of the initial configuration

solid ice liquid

Lame constant μα[

Nm2

]12.5e+9 4.17e+9 –

Lame constant λα[

Nm2

]8.33e+9 2.78e+9 –

compression modulus kα[

Nm2

]16.67e+9 5.56e+9 20e+09

heat dilatation coefficient ααΘ (273K)

[1K

](0.9-1.2)e-6 5.1e-5 1.8e-4

specific heat capacity cα[

Jkg K

]900 (273-373 K) 2090 (273 K) 4190 (293 K)

heat conduction coefficient αα�Θ

[W

m K

]1.1 (293 K) 2.2 (273 K) 0.58 (293 K)

specific heat of fusion �αf

[kJkg

]- 334 -

real density ραR[

kgm3

]2000 920 1000

initial volume fraction nα [-] 0.5 0.0 0.5

factor Darcy permeability αwFS

[Nsm3

]1.0e+6 - -

4.3 per cent referred to solid/ice, see Fig. 4 c). Bear in mind that the volume fraction of water

at the beginning of the phase transition is 0.5. As a result the maximum volume deformation

is 4.5 per cent, the half of the maximum phase exchange volume deformation of 9 per cent,

if all water in the pores is frozen. Upon temperatures higher than the bulk freezing point, a

contraction of the matrix can be observed whereas temperatures lower than this point lead to

an higher expansion in contrast to the contraction due to the temperature.

In Fig. 5 a) to f) the temperature, the system, the volume fractions of water and ice,

the divergence of the heat flux, the mass supply and the sum of mass supply are displayed as

functions of time concerning the marked points P1, P2 and P3 of the specimen. The functions

in Fig. 5 c) show the mass exchange between water and ice during the freezing and thawing

process. While the volume fraction of water decreases, the one of ice increases. Because

of numerical control, the volume fraction of water is still greater than zero, i.e., the mass

exchange process during freezing as well as during thawing is not completed. As point P1 is

situated nearest to the bottom of the specimen, the influence of the heat of fusion at point P1 is

less than at the points P2 and P3, see Fig. 5 a) and b). Furthermore, the temperature gradient

at point P1 is higher than at the other two points, because of the fact that the heating energy

is used for the melting process at point P1 and not at P2 and P3. Thus, the mass exchange

at point P1 occurs in an essential shorter space of time as at the points P2 and P3, see Fig. 5

c) to e), i.e., the temperature at P1 stays constant for less time in comparison to the other two

points. The calculated temperature functions are similar to the experimental measurements of

the CIF-Test, cf. KASPAREK & SETZER [23].

In the following, the internal energy and the heat of fusion concerning phase transition

during the freezing and thawing process is discussed. Therefore, two different cases regarding

the specific heat capacity will be analyzed. At first, the specific heat capacity for each phase

is defined to 900 [J/kg K] and then, the realistic material parameters of the phases are used,

see Tab. 1. Fig. 4 illustrates the quantities of sum of internal energy and sum of heat of



fusion as functions of time concerning the marked point P2. In the simulation with respect

to the first example (blue lines) the sum of heat of fusion and the sum of internal energy

was calculated to 285 [kJ] and 365 [kJ] and for the other example (red lines) to 290 [kJ] and

426[kJ]. Note that the sum of heat of fusion in both examples are approximately equal, but

the sum of internal energy of the first example is higher than of the second. Furthermore, the

simulation illustrates the influence of the heat of fusion. Before the beginning of ice formation

0

0.025

0.05

0.075

0

0.05

0.1

0.15

0

0.05

0.1

0.15 X Y

Z

0.5

0.4

0.3

0.2

0.1

0

2

volume fraction ice [-]

time = 11.8[hours]

b )

0

0.025

0.05

0.075

0

0.05

0.1

0.15

0

0.05

0.1

0.15 X Y

Z

295

286

277

268

259

250

a )

temperature [K]

time = 5[hours]

1

0

0.025

0.05

0.075

0

0.05

0.1

0.15

0

0.05

0.1

0.15 X Y

Z

295

286

277

268

259

250

a )

temperature [K]

time = 11.8[hours]

2

0

0.025

0.05

0.075

0

0.05

0.1

0.15

0

0.05

0.1

0.15 X Y

Z

0.5

0.4

0.3

0.2

0.1

0

1

volume fraction ice [-]

time = 5[hours]

b )

0

0.025

0.05

0.075

0

0.05

0.1

0.15

0

0.05

0.1

0.15 X Y

Z

1.045

1.038

1.031

1.024

1.017

1.01

1

volume deformation [-]

time = 5[hours]

c )

0

0.025

0.05

0.075

0

0.05

0.1

0.15

0

0.05

0.1

0.15 X Y

Z

1.045

1.038

1.031

1.024

1.017

1.01

2

volume deformation [-]

time = 11.8[hours]

c )

Fig. 4 (online colour at: www.gamm-mitteilungen.org) a) Temperature, b) volume fraction of ice and

c) volume deformation of solid/ice after 5 and 11.8 hours.


GAMM-Mitt. 33, No. 1 (2010) 53

Fig. 5 (online colour at: www.gamm-mitteilungen.org) a) Temperature, b) system, c) volume fraction

of ice and liquid, d) divergence of heat flux, e) mass supply and f) sum of mass supply concerning the

marked point P2

all phases are cooled down. The energy needed for the cooling process (difference between

sum of internal energy and sum of heat of fusion) is smaller than the sum of heat of fusion.

During the phase transition released energy by the formation of ice leads to a change of the

temperature gradient and a slowed freezing process. In Fig. 4 a) the break in the function of

overall internal energy results from the additionally needed energy, see Fig. 4 b), at the time

of ice formation. In Fig. 6 the sum of internal energy and the sum of heat of fusion over

the specimen after 5 and 11.8 hours, with respect to the realistic material parameters which

correspond to Fig. 4 (red lines).

The next example illustrates the influence of the heat of fusion with respect to the devel-

opment of the temperature in time. Therefore, the freeze-thaw cycle is simulated with and

without consideration of the heat of fusion, see Fig. 8. The red curve describes the temper-

ature function of time for the marked point P2 without consideration of the heat of fusion.

The blue temperature function illustrates the released energy by formation of ice during the

phase transition. This energy released by formation of ice leads to a change of the temperature

gradient. Thus, the freezing process is slowed down.



time [1000s]10 20 30 40 50

0

75000

150000

225000

300000

375000

450000 sum internal energy [J]

time [1000s]10 20 30 40 50

0

75000

150000

225000

300000

375000

450000 sum heat of fusion [J]

a )1 a )2

Fig. 6 (online colour at: www.gamm-mitteilungen.org) a) Sum internal energy and b) sum heat of

fusion concerning marked point P2, blue line: 900 [J/kg K] for each phase and red line: 900/2090/4190

for solid/liquid/ice in [J/kg K]

0 112500 225000 337500 450000

sum internal energy [J]

a )1 t = 5 [hours]

0 112500 225000 337500 450000

sum internal energy [J]

2a ) t = 11.8 [hours]t = 11.8 [hours]

0 112500 225000 337500 450000

sum heat of fusion [J]

1b ) t = 5 [hours]

0 112500 225000 337500 450000

sum heat of fusion [J]

2b ) t = 11.8 [hours]

Fig. 7 (online colour at: www.gamm-mitteilungen.org) a) Sum internal energy and b) sum heat of

fusion after 5 and 11.8 hours

By feeding energy, due to the temperature load of the CIF-Test, the temperature increases

after 7 hours. When the temperature is equal to 273 K, the so called melting point, the feeding

energy is used to melt the ice, i.e., the temperature at the point P2 during the thawing process

is approximately constant, see Fig. 8, dashed line. This energy is called heat of fusion. As

soon as the ice is completely melted, the temperature increases. The simulation clearly points

out the physical effects which can be observed in experiments.

In the last example the influence of the fluid pressure is presented. The system is also

cooled down and then it is heated again by the temperature load as before defined, see Fig. 2.

It should be mentioned that for this example γLwLS

= 104[ Ns/m4 ] is used. In this case the

maximum of the Jacobian is in the order of 1.035 [-]. Fig. 9 shows the distributions of the

temperature and the fluid pressure, where the arrows describe the velocities. By cooling down


GAMM-Mitt. 33, No. 1 (2010) 55

10 20 30 40 50255

265

275

285

295 temperature [K]

time [1000s]10 20 30 40 50

255

265

275

285

295

Fig. 8 (online colour at: www.gamm-mitteilungen.org) Temperature functions concerning the marked

point P2

the matrix contracts, the fluid pressure gets higher and water squeezed out, whereas by the ice

formation the matrix expands, the fluid pressure gets negative (below atmospheric pressure)

and water gets sucked into the system, see Fig. 9 a1) and b1). Additionally it is notice that

during the thawing process the matrix also contracts, the fluid pressure increases and water

squeezed out again, see Fig. 9 a2) and b2).

250 259 268 277 286 295

temperature [K]

a )1

-0.4 -0.29 -0.18 -0.07 0.04 0.15

liquid pressure [Pa]filter velocity [m/s]

b )1

250 259 268 277 286 295

temperature [K]

a )2

-0.4 -0.29 -0.18 -0.07 0.04 0.15

liquid pressure [Pa]filter velocity [m/s]

b )2

Fig. 9 (online colour at: www.gamm-mitteilungen.org) a) Temperature and b) fluid pressure and veloci-

ties after 5 and 11.8 hours

Furthermore, it can be noticed that for the here discussed examples the change of volume

fractions is not related to the change of volume deformations. This is associated with the

direction and the norm of the filter velocity. Thereby, the main problem will be the description

of the time dependent behavior of freezing and thawing processes in saturated porous solids.



The result of the simulation is that after 11.8 hours all water in the pores is frozen. As

mentioned, this time interval must be considered as a numerically result and must be proved

by more data interpretation of CIF-Tests as well.

References

[1] G. Fagerlund, Internal frost attack – state of the art, in: Frost Resistance of concrete, edited by

R. Auberg and M. Setzer, RILEM Proceedings Vol. 34 (E & FN Spon, London, 1997), pp. 321 –

338.[2] T. Powers, Journal of the American Concrete Institute 41, 245 – 272 (1945).[3] M. Setzer, Journal of Colloid Interface Science 243, 193 – 201 (2001).[4] S. Palecki, Hochleistungsbeton unter Frost-Tau-Wechselbelastung – Schadigungs- und Transport-

mechanismen, PhD thesis, Universitat Duisburg-Essen, Cuvillier-Verlag, Gottingen, 2005.[5] R. Auberg and M. Setzer, Influences of water uptake during freezing and thawing, in: Frost Resis-

tance of concrete, edited by R. Auberg and M. Setzer, RILEM Proceedings Vol. 34 (E & FN Spon,

London, 1997), pp. 232 – 245.[6] M. Setzer, H. P., S. Palecki, R. Auberg, V. Feldrappe, and E. Siebel, Material and Structures

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elastic porous solids, in: Transport in Concrete: Nano- to Macrostructure, edited by M. Setzer

(Aedificatio Publishers, Freiburg, 2007), pp. 41 – 57.[11] J. Bluhm, T. Ricken, and M. Bloßfeld, PAMM (2008).[12] R. Bowen, International Journal of Engineering Science (Int J Eng Sci) 18, 1129 – 1148 (1980).[13] R. Bowen, International Journal of Engineering Science (Int J Eng Sci) 20, 697 – 735 (1982).[14] R. de Boer and W. Ehlers, Theorie der Mehrkomponentenkontinua mit Anwendung auf boden-

mechanische Probleme, Teil 1, Tech. Rep. Heft 40, Forschungsberichte aus dem Fachbereich

Bauwesen, Universitat-GH-Essen, 1986.[15] W. Ehlers, Porose Medien – ein kontinuumsmechanisches Modell auf der Basis der Mischungs-

theorie, Habilitation, Heft 47, Forschungsbericht aus dem Fachbereich Bauwesen, Universitat-GH

Essen, Essen, 1989.[16] W. Ehlers, Foundations of multiphasic and porous materials, in: Porous Media: Theory, Exper-

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2002), pp. 3 – 86.[17] R. de Boer, Theory of Porous Media – highlights in the historical development and current state

(Springer-Verlag, Berlin, 2000).[18] V. Voller, A. Brent, and C. Prakasch, International Journal of Heat and Mass Transfer 32(9),

1719 – 1731 (1989).[19] W. Kowalczyk, Numerische Simulation von Fest-Flussig-Phasenubergangen bei der Hochdruck-

behandlung, PhD thesis, Technische Universitat Munchen, 2004.[20] J. Spurk, Stromungslehre – Einfuhrung in die Theorie der Stromungen (Springer-Verlag, Berlin ·

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thawing, in: Frost Resistance of concrete, edited by R. Auberg, H. J. Keck, and M. SetzerNo. 24

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Documents

Modeling fluid saturated porous media under frost attack