A micromechanics model for partial freezing in porous media

International Journal of Solids and Structures xxx (2015) xxx–xxx

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International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr

A micromechanics model for partial freezing in porous media Rongwei Yang a,c,⇑, Eric Lemarchand b, Teddy Fen-Chong c,d, Aza Azouni c a

Civil Engineering Department, Tsinghua University, 100084 Beijing, PR China Université Paris-Est, UR Navier, UMR 8205 CNRS, Ecole des Ponts ParisTech, Champs-sur-Marne, F-77455 Marne-La-Vallée cedex 2, France c Université Paris-Est, Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, Marne-la-Vallée 77420, France d Université Paris-Est, MAST, FM2D, IFSTTAR, F-77447 Marne-la-Vallée, France b

a r t i c l e

i n f o

Article history: Received 12 March 2015 Received in revised form 7 August 2015 Available online xxxx Keywords: Freezing Porous media Unfrozen water film Disjoining pressure Micromechanics

a b s t r a c t Based on the local physical characterization of partial freezing in porous media, the role of the unfrozen water film, located between the in-pore ice crystal and pore wall, is paid special attention to in this study. The disjoining pressure within unfrozen water film, the membrane stress induced by surface tension effect, the thermal stress and the initial stress are fully accounted for in the proposed micromechanics model. The micromechanics model improves the physical understanding of the macroscopic mechanical behaviors of the partially frozen porous media. The micromechanics model is applied to simulate the free swelling of a undrained cement paste (denoted by CP) and an air-entrained mortar (denoted by AM). The model results are comparable with the experimental results on cement paste. The reasons for the discrepancies between the model results and experimental results on cement paste may lie in the overestimation of the ice content, which here is estimated by the pore size distribution by mercury intrusion porosimetry (MIP) and Gibbs–Thomson equation. However, the model results agree well with the experimental results of the air-entrained mortar, the ice content of which is determined by the differential scanning calorimeter (DSC). The disjoining pressure within partially frozen porous media will become more and more significant with decreasing temperature. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Internal frost damage and surface scaling damage are considered to be the two main deteriorating phenomena of concrete structure in cold region, which is more and more concerned by the civil engineering community (Sun and Scherer, 2010a). After the pioneering work of Powers and Helmuth (1953), numerous models were developed to investigate the mechanism of the deterioration as well as to simulate the mechanical behaviors of the frozen porous media (Scherer, 1999; Zuber and Marchand, 2004; Coussy, 2005; Coussy, 2006; Coussy and Monteiro, 2008; Wardeh and Perrin, 2008; Sun and Scherer, 2010a; Zeng et al., 2011b; Fen-Chong et al., 2013a,b). In these models, poromechanics models developed by Coussy and coworkers (Coussy, 2005; Coussy, 2006; Coussy and Monteiro, 2008; Fen-Chong et al., 2013a,b) have been proved to be robust and comprehensive for studying the behaviors of partially frozen porous media. Based on the thermodynamic equilibrium of in-pore ice crystal, ‘‘crystallization pressure” is introduced in the poromechanics model for partially frozen porous media by Scherer et al. (Scherer, 1999; Sun and Scherer, 2010a). In ⇑ Corresponding author at: Civil Engineering Department, Tsinghua University, 100084 Beijing, PR China. Tel./fax: +86 10 6279 7993. E-mail address: [email protected] (R. Yang).

the previous poromechanics models, the local physics information (e.g. disjoining pressure in unfrozen water film) along with heterogeneous microstructure of frozen porous medium are not fully taken into account at local scale in these models. The existence of the unfrozen water film, located between ice crystal and pore wall, has been verified by extensive works (Faraday, 1859; Fagerlund, 1973; Derjaguin et al., 1987; Churaev et al., 1994; Scherer and Valenza, 2005). Therefore, in crystallized pores, the unfrozen water film instead of ice crystal directly exerts pressure on the pore wall (see Fig. 1). Moreover, an additional pressure denoted as ‘‘disjoining pressure” emerges in this unfrozen water film, which is believed to play a crucial role in mechanical behaviors of partially frozen porous media (Derjaguin and Obuchov, 1936; Derjaguin and Churaev, 1978; Derjaguin et al., 1987; Churaev et al., 1994). This local physical information is always not accounted for in the poromechanics models, as explained in Coussy (2006). As an alternative approach for poromechanical approach, micromechanics approach has already been used to model the mechanical behavior of unsaturated porous medium (Chateau et al., 2002; Cariou, 2010). The micromechanics methodology is capable of accounting for the heterogeneous microstructure as well as local physical behavior. By means of the homogenization and appropriate estimate scheme (e.g. Mori–Tanaka scheme in this

http://dx.doi.org/10.1016/j.ijsolstr.2015.08.005 0020-7683/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Yang, R., et al. A micromechanics model for partial freezing in porous media. Int. J. Solids Struct. (2015), http://dx.doi.org/ 10.1016/j.ijsolstr.2015.08.005

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

Fig. 1. Schematic illustration of partial freezing within porous media.

study), the micromechanics model is able to characterize the macroscopic behaviors of the partially frozen porous media. Moreover, it also allows us to discuss the impact of local physical quantity (such as the disjoining pressure in this study) on the macroscopic behaviors during freezing in porous media. The aim of this study is devoted to developing a comprehensive model for partial freezing in porous media by means of the micromechanics methodology. The effect of unfrozen water film and the disjoining pressure is emphasized in the model. The paper is organized as follows: Section 2 recalls the physical characterization of the partial freezing in porous media; based on this physical characterization, the two state equations of the micromechanics model are presented in Section 3. In Section 4, the two state equations are used to discuss and interpret the macroscopic behavior of the partially frozen porous medium in isotropic case; in Section 5, the micromechanics model is also employed to model the free swelling of the partial frozen cement paste and air-entrained mortar. this context, the curlicue letters represent fourth order tensor (e.g. C; A), the boldfaced letters denote second order tensor (e.g. r; E) while underlined letters (e.g. n; n) denote vectors. The subscripts and/or superscripts of s; c; l; f ; cf and p represent solid matrix, ice crystal, liquid pore water, unfrozen water film, unfrozen water film + ice crystal composite and pore space, respectively.

unfrozen water film interface (I sf ), ice crystal/liquid pore water interface (I cl ) and ice crystal/unfrozen water film interface (I cf ). The surface tension cab along a interface I ab induces a stress vector discontinuity (½r:n) between a and b phases, (a; b 2 l; s; f ; c); where n is the unit vector normal to the interface I ab between the a phase and b phase; ½r is the stress difference between a phase and b phase. If we assume n orientates towards a phase, then ½r ¼ ra  rb ; ra and rb are the local stress tensors within a phase and b phase, respectively. It should be noted that, herein, r is concise expression of rðzÞ; z is local position vector. With the introduction of the curvature tensor of interface, defined by b ¼ gradn, the stress vector discontinuity can be derived from the momentum balance equation (Dormieux et al., 2006):

½r:n þ cab ð1T : bÞn ¼ 0

ð1Þ

where 1T (1T ¼ 1  n  n) is the second order identity tensor of the plane tangent to the interface I ab ; 1 is the second order identity tensor. For the ice crystal-liquid pore water interface (I cl ),

rc ¼ Pc 1; rl ¼ Pl 1; ½r ¼ rc  rl , Eq. (1) can thus be rearranged as the classic Young–Laplace equation:

Pc  Pl ¼ jcl ccl c

ð2Þ l

2.1. The interfaces in partially frozen porous media

where P and P are the pressures of the ice crystal and liquid pore water, jcl ¼ ð1T : bÞ is the reciprocal of curvature radius of the interface I cl . In addition to the mechanical equilibrium in the interface I cl (see Eq. (2)), the thermodynamic equilibrium of the interface I cl between ice crystal and liquid pore water should also be obeyed (Coussy, 2011). With the assumption of hemispherical interface I cl (jcl ¼ 2 cos h=rcr ), according to the thermodynamic equilibrium and mechanical equilibrium of the interface I cl , the freezing temperature can be depressed by the capillary effect according to the following modified Gibbs–Thomson equation (Coussy, 2011):

As can be seen in Fig. 1, during freezing process, there exist four interfaces: pore wall/liquid pore water interface (I sl ), pore wall/

2ccl cos h ¼ r cr

2. Local physical characterization As shown in Fig. 1, the partial freezing in porous media can be treated as unsaturated case in which the ice crystal is non-wetting phase and the liquid pore water is wetting phase. Therefore, during freezing process, the physics within interfaces (e.g. ice crystal-liquid pore water interface I cl ) and the unfrozen water film f (as shown in the subset of Fig. 1) should be paid special attention to at local scale.




 Vl  1 ðPl  P 0 Þ  Sm dT Vc

ð3Þ

Please cite this article in press as: Yang, R., et al. A micromechanics model for partial freezing in porous media. Int. J. Solids Struct. (2015), http://dx.doi.org/ 10.1016/j.ijsolstr.2015.08.005

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where r cr is the critical radius of meniscus of interface I cl ; r cr ¼ r p  h; rp is the pore radius, h is the thickness of unfrozen water film; h is the contact angle between ice crystal and unfrozen water film; V l and V c are the molar volumes of the ice crystal and liquid pore water; P0 is the atmospheric pressure; Sm (in J K1 m3 ) is the fusion entropy per unit volume of ice transferring to liquid water, Sm ¼ 1:2MPa K1 (Brun et al., 1977); dT ¼ T  T 0 is the depressed temperature, T 0 ¼ 273 K and T are the reference temperature and the current freezing temperature, respectively.   It should be noted that, since VVcl  1  0:09 is a very low value, the first term in the right hand side of Eq. (3) is usually negligible. However, in undrained case, the liquid pore water pressure in partially frozen porous media can reach hundreds of MPa (Coussy and Monteiro, 2008), the depressed effect induced by liquid pore water pressure should be accounted for. It should also be noted that the solute in pore water can also depress the freezing temperature. For simplicity, this depression effect will not be accounted for in this study. 2.2. Disjoining pressure within unfrozen water film

and Van der Waals components of disjoining pressure are negligible in the unfrozen water film (Churaev et al., 1993, 1994, 2002). Thus, the structural component of the disjoining pressure may play a dominant role in unfrozen water film due to the molecular structure modification of the unfrozen water film (Churaev et al., 1994). Though the origin of the structural component of disjoining pressure Pst ðhÞ is still disputed, its magnitude obeys the following empirical exponential equation (Pashley, 1982; Churaev et al., 2002; Israelachvili, 2011):

PðhÞ  Pst ðhÞ ¼ K sr expðh=ksr Þ þ K lr expðh=klr Þ

ð5Þ

The first term of the right hand side of Eq. (5) relates to the short range effect of the structural component while the second term of the right hand side corresponds to the long range effect of the structural component (Churaev et al., 1994). K sr ; K lr ; ksr and klr are the four corresponding fitting parameters and are of the same order for hydrophilic surface of mica, quartz, silica and glass, at room temperature (Churaev et al., 1994).

3. Model development

At nano-scale, the stability and thickness of the unfrozen water film are determined by the so-called disjoining pressure (Derjaguin and Obuchov, 1936; Derjaguin and Churaev, 1978). From mechanical equilibrium point of view, the disjoining pressure PðhÞ is defined as the difference between the pressure within the water film P f and the pressure of the bulk water Pl adjacent to the unfrozen water film (Derjaguin and Churaev, 1978; Derjaguin et al., 1987):

PðhÞ ¼ P f  P l

3

ð4Þ

Disjoining pressure in the unfrozen water film has been proved to be detrimental to the porous media during freezing process (Derjaguin and Churaev, 1986; Churaev et al., 1993, 1994, 2002). Generally, the disjoining pressure is considered to be the sum of three components: namely Van der Waals component, electrostatic component, and structural component (Churaev and Derjaguin, 1985; Majumdar and Mezic, 1999; Gonçalvès et al., 2010). However, Churaev et al. have found that the electrostatic

Representative volume element (RVE) of a partially frozen porous medium is made up of solid matrix (domain Xs ) and pore space (domain Xp ). The latter may be occupied by two kinds of immiscible phase: liquid pore water phase (domain Xl ), ice crystal + unfrozen water film phase spherical composite inclusion (domain Xcf ) (see Fig. 1), that is, Xp ¼ Xcf [ Xl . At equilibrium state, the critical pore radius between these two different pores can be determined by Eq. (3). Herein, in ice crystallized pores, spherical composite inclusion phase can be decomposed into ice crystal phase (domain Xc ) and the unfrozen water film phase (domain X f ), that is Xcf ¼ Xc [ X f (as illustrated in Fig. 1). In this work, for the convenience of the introduction, a matrix + inclusion morphology is used to illustrate the morphological representation of the porous medium. Therefore, a morphological representation of RVE of the partially frozen porous medium is illustrated in Fig. 2. A uniform strain E is prescribed on the boundary of the RVE @ X.

Fig. 2. Morphological representation of the partially frozen porous medium.

Please cite this article in press as: Yang, R., et al. A micromechanics model for partial freezing in porous media. Int. J. Solids Struct. (2015), http://dx.doi.org/ 10.1016/j.ijsolstr.2015.08.005

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It should be aware that the domains introduced above are under the current state, their associated initial states are denoted by subscript ‘‘0”. For instance, Xl0 and Xc0 represent the initial state of the liquid pore water domain and ice crystal domain, respectively.

e¼A:E

3.1. General formulas: localization 3.1.1. The initial state of the freezing in porous media As encountered in many engineering applications, an initial stress r0 prevails within porous media. The initial configuration of porous medium is here the reference state and the initial macroscopic strain of porous medium E0 is considered to be 0 (Dormieux et al., 2006). The physical formulas for this initial state problem can be expressed as:

8 divr0 ¼ 0 > < r0 ðzÞ ¼ CðzÞ : e0  jdT 0 þ rp0 > : n0 ¼ 0

z 2 X0

af T

8z 2 Xs0 8z 2 Xcf0 8z 2 Xl0

ð6aÞ

rðzÞdX ¼ R0 þ Chom : E  jhom dT jXj X    /l P l 1 : Al þ /cf P f 1 : Acf  /0l Pl 1 : Al  /0cf P f 1 : Acf 
Z 
Z  dS dS  caf þ cab 1T : A 1T : A

3.1.2. The current state of freezing in porous media Correspondingly, in current state, the microscopic stress r, strain e and uniform displacement n applied on the boundary of the RVE obey the following relations:

z2X z2X

ð7Þ

z 2 @X

with:

8z 2 Xs 8z 2 Xcf 8z 2 Xl 8z 2 I ab

ð7aÞ

where P f ¼ Pl þ PðhÞ; one should be aware that cab 1T is the current membrane stress, cab is the surface tension within the interface I ab ; ða; bÞ 2 ff ; l; c; sg.

where /i (i 2 fl; cf g) is the volume fraction of phase (i 2 fl; cf g) is the initial volume fraction of phase i; Ai is the average strain concentration tensor of phase i; Ap is the average strain concentration tensor of pore space; Chom and jhom are the homogenized elastic tensor (stiffness tensor) and homogenized thermal stress coefficient tensor, respectively; I is the fourth order identity tensor; dT ¼ T  T 0 (< 0) is the variation between the current temperature T with reference temperature T 0 ; ss ¼ js : Ss is the thermal strain 1

coefficient tensor of the solid matrix; Ss ¼ ðCs Þ is the compliance tensor of solid matrix; B is the second order Biot tensor of the porR R and ðcab I ab 1T : A dS ous medium; ðcaf I af 1T : A dS XÞ XÞ (ða; b; fÞ 2 ff ; l; c; sg) are the initial and current membrane stress tensors induced by surface tensions caf and cab during freezing process; R0 is the macroscopic initial stress tensor. As can be found in the fourth term of the right hand side of Eq. (10), the membrane stress (surface tension effect) within the interfaces between pore wall and ith phase (i 2 fl; f g) are accounted for while the membrane stress of interface I cl can be assumed to be negligible (Cariou, 2010). Since the membrane stress is closely related to the pore size distribution of the porous medium, a macroscopic equivalent pressure P eq is introduced to account for the effect of in-pore pressure (P l and P f in the third term of the right hand side of Eq. (10)) and surface tension effect. The detailed derivation of Peq accounting for the pore size distribution of porous medium is given in Appendix A. Eq. (10) can thus be reorganized in a concise incremental form:

dR ¼ Chom : E  jhom dT  dPeq B Z

rmax

Z

aðzÞdX

ð11Þ

peq ðrÞgðrÞdr

ð11aÞ

r min

with:

X

ð10Þ

i; /0i

Peq ¼

The macroscopic quantity (tensor) a is defined as the average of the local quantity aðzÞ over X (Dormieux et al., 2006):

jXj

X

8 > Chom ¼ C : A ¼ Cs : ðI  /0 Ap Þ > > > < hom j ¼ j : A ¼ js : ðI  /0 Ap Þ ¼ js : Ss : Chom ¼ ss : Chom > > B ¼ /0 1 : Ap > > : R0 ¼ r0 : A

3.2. Macroscopic state equations

a¼

I af

ð10aÞ

I 0 ; ða; fÞ 2 ff ; l; c; sg.

1

X

I ab

8z 2 I 0af

af

8 s s ðC ; j ; 0Þ > > > > < ð0; 0; P f 1Þ ðC; j; rp ÞðzÞ ¼ > ð0; 0; Pl 1Þ > > > : ð0; 0; cab 1T Þ

Z

1

with:

where C and j are local stiffness tensor and thermal dilation stress coefficient tensor, Cs ; js and rp are respectively the fourth order stiffness tensor of the solid matrix, thermal dilation stress coefficient tensor of the solid matrix and local prestress tensor of the pore space; X0 and @ X0 are the initial RVE domain and its corresponding boundary surface; e0 ; dT 0 ; rp0 and n0 are initial local strain, initial temperature variation, initial local prestress tensor and initial boundary displacement, respectively; caf 1T is the initial membrane stress, caf is the surface tension within the interface

8 > < divr ¼ 0 r ¼ C : e þ rp  jdT > : n¼Ez

R¼

ð6Þ

z 2 @ X0

ð9Þ

Based on the energy consistent rule and treated in Levin’s theorem (Dormieux et al., 2006), the macroscopic stress tensor R can be derived by averaging the local stress tensor in Eqs. (6) and (7) in terms of Eq. (8):

z 2 X0

with

8 s ðC ; js ; r0 Þ > > > > > < ð0; 0; P f 1Þ 0 ðC; j; rp0 ÞðzÞ ¼ l > > ð0; 0; P > 0 1Þ > > : ð0; 0; c 1 Þ

3.2.1. First state equation Under the uniform boundary condition E, the fourth order strain concentration tensor A, function of the microstructure morphology of partially frozen porous medium, is introduced to relate the macroscopic strain tensor E and local strain tensor e (Dormieux et al., 2006):

ð8Þ

p ðrÞ ¼ eq

8  < Pl  2csl

r min 6 r 6 rcr

r

:

Pf 

2csf r

¼ Pl þ PðhÞ 

2csf r

ð11bÞ

r cr 6 r 6 r max

where dR ¼ R  R0 .

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As in the work of Cariou (2010), the equivalent pressure peq ðrÞ in Eq. (11b) consists of fluid pressure of x phase P x (x 2 ff ; lg) and its surface tension effect 2csx =r, which is depicted in Fig. (3). 3.2.2. Second state equation During the progressive freezing process in the connected pore network, the pore volume will vary, which can be captured by the second poroelastic state equation. With the assumption of linear elasticity and small strains of frozen porous media, the variation of the porosity is expressed by the lagrangian porosity as (Dormieux et al., 2006):

jXj  jX0 j /  /0 ¼ ¼ /0 trep jX0 j

ð12Þ

where / is the current pore volume fraction (porosity), /0 is the initial pore volume fraction. / is the current porosity, ep is the average strain of the pore space within the porous medium. The RVE may be considered to be subjected to four loading parameters: namely uniform macroscopic strain tensor E on the boundary of RVE, thermal strain loading sdT, where s is the local thermal strain tensor, the initial stress r0 and the microscopic prestress tensor rp ¼ peq ðrÞ1. On the premise of the linear elasticity, the overall variation of porosity can be decomposed into contributions of four subproblems by means of Levin’s theorem (Dormieux et al., 2006):

/  /0 ¼ /0 trep1 þ /0 trep2 þ /0 trep3 þ /0 trep4

ð13Þ

where /0 trep1 is induced by loading 1 (E – 0; dT ¼ 0; r0 ¼ 0; peq ðrÞ ¼ 0); /0 trep2 is induced by loading 2 (dT – 0; E ¼ 0; r0 ¼ 0; peq ðrÞ ¼ 0); /0 trep3 is induced by loading 3 (r0 – 0;peq ðrÞ ¼ 0; E ¼ 0; dT ¼ 0); /0 trep4 is induced by loading 4 (peq ðrÞ – 0; E ¼ 0; dT ¼ 0; r0 ¼ 0). By means of Mori–Tanaka estimate, the contributions of these four loadings to the overall variation of the porosity can be determined as:

8 /0 trep1 ¼ /0 1 : ep1 ¼ /0 1 : Ap : E ¼ B : E > > > > s p hom > p  ð1  /0 Þjs dT > < /0 tre2 ¼ /0 1 : e2 ¼ 1 : S : ½j s ¼ ð/0 1  BÞ : s dT ¼ s/ dT > > > /0 trep ¼ /0 1 : ep ¼ 0 > 3 3 > > : eq /0 trep4 ¼ PN

N

/  /0 ¼

B:E |ffl{zffl}

s/ dT |fflfflffl{zfflfflffl}

þ

Boundary strain effect Thermal dilation effect

Peq N |{z} The equivalent pressure effect

ð16Þ 4. Applications and discussion The surface tensions csl and csf appearing in Eq. (11b) are not independent. They are related to Young equation in the form (Dormieux et al., 2006):

(

csl ¼ ccl cos hcl þ ccs csf ¼ ccf cos hcf þ ccs

ð17Þ

where hcl is the contact angle between the interface I cl and the interface I sl ; hcf is the contact angle between the interface I cf and the interface I sf . With the assumption of hcl ¼ hcf ¼ 180 , according to Eq. (17), we have:

csl  csf ¼ ccf  ccl

ð18Þ

4.1. Discussion on the first state equation 4.1.1. Physical interpretation of equivalent pressure When freezing process occurs in fully liquid saturated porous medium from T cr to T cr (T cr < T cr < 0  C), the corresponding critical radius of the crystallized pore r cr (resp. r cr ) at T cr (resp. T cr ) is determined by Eq. (3), r cr < rcr (be aware that the supercooling effect is neglected here). According to Eqs. (11a) and (11b), we have:

8    R  R 2c > < Peq ¼ rrcr Pl  2cr sl gðrÞdr þ rrcrmax P f  rsf gðrÞdr min h  i R rmax 2csl R > : gðrÞdr þ rrcrmax PðhÞ þ 2cr sl  2crsf gðrÞdr ¼ Pl  rmin r

ð19Þ

ð14Þ

If ccf  ccl can be assumed reasonably, with Eq. (18), Eq. (19) simplifies to:

Correspondingly, the poroelastic constants (tensors) estimated with Mori–Tanaka scheme read:

8 1 hom > > ¼ ð1  /0 ÞCs : ½ð1  /0 ÞI þ /0 ðI  SÞ1  >C > > < B ¼ /0 1 : Ap ¼ /0 1 : ½I  ð1  /0 ÞS1 > s/ ¼ ð/0 1  BÞ : ss > > > > : 1 ¼ / 1 : ð1  BÞ : ðI  SÞ1 : P

The second poroelastic state equation (Eq. (13)) can thus be rewritten as:

Z Peq 

Pl |{z}



liquid pore water pressure

Z ð15Þ

0

where N1 is the Biot modulus, s/ is the homogenized thermal dilation coefficient of the partially frozen porous medium, S ¼ P : Cs is the Eshelby tensor, P is the fourth order Hill tensor.

þ

rmax

2csl gðrÞdr r |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r min

surface tension effect rmax

PðhÞgðrÞdr r cr |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð20Þ

disjoining pressure

From Eq. (20), it is readily found that P eq within partially frozen porous media is explicitly composed of three parts: namely liquid pore water pressure, membrane stress induced by surface tension effect and disjoining pressure.

Fig. 3. Components of peq ðrÞ during freezing, x 2 ff ; lg, modified from Cariou (2010).

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Taking saturated porous media at 0  C as initial reference state,

observations in air-entrained cement paste (Sun and Scherer, 2010a; Zeng, 2011a). However, jSr cf PðhÞj can be

we have liquid pore water pressure being Pl0 ; rcr ¼ r max , and PðhÞ ¼ 0; at temperature T (T < 0  C), the freezing occurs (in the absence of supercooling effect) and the variation of the macroscopic equivalent pressure dP eq reads:

dP eq 

Pl  Pl |fflfflffl{zfflfflffl}0

Srcf PðhÞ |fflfflfflfflffl{zfflfflfflfflffl}

þ

Liquid pore water pressure

higher than jPl j at extremely low temperature (e.g. T < 25  C in Sun and Scherer (2010a)). In this case, E2 > 0, this maybe qualitatively explain the upwarp tendency of the deformation curve of the air-entrained mortar when temperature decreases below 25  C in Sun and Scherer (2010a). (3) For undrained porous media subject to bulk supercooling effect, there is no ice crystal when the temperature lowers

ð21Þ

disjoining pressure effect

where Srcf ¼ ð/c þ /f Þ=/ is the saturation degree of ice crystal + unfrozen water film composites (see Appendix A). It can be found from Eq. (21) that the variation of the macroscopic equivalent pressure can be decomposed into two contributions: (1) the variation of liquid pore water pressure throughout the overall pore network induced by the volume change of the water to ice crystal during freezing; (2) the variation of disjoining pressure in crystallized pore space which is induced by the physico-chemical process, namely the disjoining pressure increases with decreasing temperature. Therefore we can summarize that, if ccf  ccl , the effect of the

down. As the ice crystals are unavailable, P l is totally induced by the density variation and the value of P l is thus l

low: E2 ¼ 3KbPhom is negligible.Therefore, E  E1 at the stage of supercooling, which is verified by the experimental results of Zeng (2011a). (4) Eq. (23) is also used to qualitatively interpret the mechanism of unexpected swelling of the early age cement paste saturated with benzene observed by Beaudoin and MacInnis

membrane stress on dP eq can be neglected. Liquid pore water pressure as well as disjoining pressure are left as the two main components of dP eq .

(1974). Considering Pl ¼ 0 (Fen-Chong et al., 2013b), therefore the sign of E is determined by the competition between E1 and

bSrcf PðhÞ

bSr cf PðhÞ

3K hom

3K hom

. When j

j > jE1 j, the early age cement

paste, saturated with benzene, swells even if benzene contracts when solidifying.

4.1.2. Discussion on the first state equation For the sake of the discussion, several assumptions are made: the porous medium is initially liquid saturated and its initial state is taken as natural state (Pl0 ¼ 0; E0 ¼ 0; R0 ¼ 0; R ¼ 0 and thus dR ¼ 0); the microscopic and macroscopic elasticity tensors are assumed to

be

isotropic,

namely i

Ci ¼ 3K i J þ 2li K; b ¼ b1,

where

J¼  1; K ¼ I  J; K and l are the bulk modulus and shear modulus of ith phase (i 2 fhom; sg). 1 1 3

i

According to Eq. (11), the macroscopic linear strain E can be determined as:

E¼

1 ss dP eq trðEÞ ¼ dT þ b hom 3 3 3K

where

ð22Þ

4.2. Determining the liquid pore water pressure from second state equation We derive two state equations Eqs. (11) and (16), in which two macroscopic state quantities R and /  /0 are expressed as function of macroscopic strain E and P eq and dT. An alternative second state equation which expresses variation of the mass instead of pore volume fraction is proposed to derive the liquid pore water pressure P l :

Pl  M 1




ss (> 0) is the volumetric dilation coefficient of the solid

matrix, K hom is the bulk modulus of the porous medium, b is the Biot coefficient, dT ¼ T  T 0 < 0. Inserting Eq. (21) into Eq. (22) yields:

E¼

ss 3

dT þ

b½Pl þ Srcf PðhÞ

ð23Þ

3K hom

For simplicity, we define E1 ¼ ss =3dT < 0 and E2 ¼

tion (which determines the sign of Pl ). Eq. (23) is employed to understand the mechanisms of the following four cases qualitatively. (1) For undrained porous media, E2 > 0, if jE1 j > jE2 j, the total strain E < 0, which corresponds to shrinkage at macro scale; while when jE1 j < jE2 j; E > 0, which corresponds to swelling at macro scale. l

(2) For air-entrained porous media, P < 0. If E2 < 0, which means Pl prevails over Sr cf PðhÞ, the air-entrained porous media shrink overall; this is in line with the experimental

s/ dT þ /l sl dT þ /c sc dT






q0c þ /f s f dT q0l



ð24Þ

with:

(

,

therefore, E ¼ E1 þ E2 . The sign of the linear strain E during the progressive freezing process depends on the values of thermal strain E1 and linear strain induced by macroscopic equivalent pressure E2 . For geomaterials with hydrophilic surface, PðhÞ > 0 (Churaev and Derjaguin, 1985; Derjaguin et al., 1987; Israelachvili, 2011). E2 can be positive or negative depending on the boundary condi-

q0l 0 c 0 l

 B:Eþ

/f Sr cf q Sm dT q0c  1 þ /c c  PðhÞ f þ 0 N K ql q K

/c

b½P l þSr cf PðhÞ 3K hom


m  m0

M

1

¼

)1
2 /f 1 /c q0c þ c þ fþ 0 N K l K ql K /l

ð24aÞ

The detailed derivation of P l is given in Appendix B. It can be found from Eq. (24) that the liquid pore water pressure Pl within partially frozen porous media is composed of six contributions: (1) mass change m  m0 during freezing process which is dependent on the boundary condition (e.g. drained or undrained); (2) external boundary macroscopic strain B : E; (3) thermal dilation of each phase, i.e.,

0

s/ dT þ /l sl dT þ /c sc dT qqc0 ; (4) volume change l

between ice and liquid during freezing, i.e., /c entropy of melting, i.e., /c SmKdT c

q0c , q0l




q0c q0l

  1 ; (5)

which is the origin of micro-

cryo-suction process (Coussy and Monteiro, 2008); (6) disjoining   / Sr pressure,  K ff þ Ncf PðhÞ, which decrease the liquid pore water pressure for hydrophilic pore wall.

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

4.3. Relation between micromechanics and poromechanics methodology

5. Modeling the free swelling of cement-based materials

A schematic of the disjoining pressure effect and the interfaces is illustrated in the inset of Fig. 1. The equilibria of ice crystalunfrozen water film interface and unfrozen water film-solid interface read:

8 f 2ccf > P  Pc ¼  R1 h > < P f ¼ Pl þ PðhÞ > > : c P  Pl ¼ 2rc2cl

ð25Þ

where r 2 is the curvature radius of the ice-liquid pore water interface, R1 is the curvature radius of the ice-unfrozen water film interface. The magnitude of the disjoining pressure can be estimated by Eq. (25):

PðhÞ ¼

2ccf 2ccl  r2 R1  h

ð26Þ

If ccf  ccl , Eq. (26) exactly coincides with the crystallization pressure shown in Eq. (6) of Scherer and Valenza (2005). Thus, the magnitude of the disjoining pressure PðhÞ is indeed in line with the crystallization pressure denoted by P a in Scherer (1999); Scherer and Valenza, 2005. From physico-chemical point of view, crystallization pressure originates from disjoining pressure (Scherer and Valenza, 2005). One of the differences between the micromechanics model and poromechanics model lies in the definition of the equivalent pressure. In poromechanics models for partially frozen porous media (Coussy, 2005; Coussy and Monteiro, 2008; Fen-Chong et al., 2013b), there are two main internal pressures: liquid pore water pressure Pl and ice crystal pressure Pc . The validity of the poromechanics model for partially frozen porous media is discussed at local scale, following (Coussy, 2006). As depicted in Fig. 1, use of P c instead of PðhÞ lies in the assumption: R1  h r 2 . From Eq. (26), it gives PðhÞ  2rc2cl . When this expression for disjoining pres-

The micromechanics framework has already been used to study the poroelastic properties/behaviors of the cement-based materials (Bernard et al., 2003; Dormieux et al., 2004; Ulm et al., 2004; Pichler and Hellmich, 2011). Generally, cement-based materials are multi-scale porous media and exhibit complex microstructure. Owing to the significant role of the water distribution in partial freezing problems, the water distribution morphology (pore structure) other than the solid morphology should be paid special attention to. The partially frozen cement-based materials may be considered as simply consisting of solid matrix, liquid pore water and ice crystal + unfrozen water film spherical composite inclusion. The latter two phases are embedded in the solid matrix. Therefore, the cement-based materials exhibit notable matrix + inclusion morphologies and their macroscopic properties can be estimated with Mori–Tanaka scheme (Bernard et al., 2003; Pichler and Hellmich, 2011; Zhou and Meschke, 2014b). Hence, the simplified morphologies of cement-based materials can also be represented by Fig. (2). The micromechanics model developed in the previous section is used to simulate the free swelling of a cement paste (CP) in Refs. Zeng (2011a) and Zeng et al. (2014a)) (water/cement ratio = 0.50, curing time = 360 days) and an air-entrained mortar (AM) in Sun and Scherer (2010a) (water/cement ratio = 0.55, curing time = 720 days). The CP is sealed with a thin layer of regin epoxy to ensure the undrained boundary condition, namely m  m0 ¼ 0 is satisfied in Eq. (24). The AM is entrained 6% vol. air voids which offer sufficient room for the expelling water during freezing, therefore AM is drained. The further experimental details are given in Refs. Sun and Scherer (2010a), Zeng (2011a) and Zeng et al. (2014a). Several assumptions and boundary conditions for the CP and AM are summarized as: The sample is filled with liquid pore water before freezing while the air voids are filled with gas. T 0 ¼ 0  C is taken as initial tem

sure is inserted into Eq. (25), we have:

ð27Þ



The pressure of ice crystal P is thus the pressure exerted on the solid matrix, where rs is the stress of solid matrix at the boundary of pore space (see Fig. 1). Therefore, ice crystal pressure Pc instead of disjoining pressure PðhÞ is taken into account in the poromechanical model (Coussy, 2005; Coussy, 2006; Coussy and



Pc  PðhÞ þ P l ¼ P f ¼ rs c

Monteiro, 2008; Coussy, 2011). It should be aware that, Pc and Pl can be linked by Eq. (3) in the poromechanics model. Another difference lies in the different approach for determining macroscopic poroelastic properties (e.g. bulk modulus, Biot coefficient), which are used as input parameters in the poromechanics/micromechanics models. In the poromechanics model, the macroscopic poroelastic properties are often determined by experiment (Sun and Scherer, 2010a; Fen-Chong et al., 2013b) or estimated by the empirical formulas (Zuber and Marchand, 2004; Sun and Scherer, 2010a); while the macroscopic poroelastic properties are analytically determined by the estimate schemes (e.g. Mori–Tanaka scheme) in micromechanics models (Bernard et al., 2003; Ulm et al., 2004; Pichler and Hellmich, 2011). In the present study, the Mori–Tanaka scheme is adopted to estimate the macroscopic poroelastic properties of the porous media under partial freezing.



perature and Pl0 ¼ 0; dT ¼ T (in °C). The temperature is uniform through the overall sample, the variation of the temperature is sufficient slow so that the sample is always in equilibrium state. The sample is absent of initial stress and is under free swelling during freezing, it gives: R ¼ 0. Each phase of the cement paste sample is isotropic at local and macroscopic scales. The physical properties of unfrozen water film are assumed to be identical to those of liquid pore water, q0l  q0f ; K l  K f ;

sl  s f and ccf  ccl . The parameters used in the simulation are given in Table (1). 5.1. Poromechanical properties and formulations Based on aforementioned conditions and assumptions, the macroscopic equivalent pressure is P eq ¼ P l þ Sr cf PðhÞ according to Eq. (21). In terms of Eq. (23), the linear strain of porous medium E can be determined as:

E ¼ EPeq þ Eth

ð28Þ

with

EPeq ¼ EPl þ EP

ð28aÞ

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

Pl  Sm ðT  T 0 Þ

Table 1 Parameters used in the simulation (Sun and Scherer, 2010a; Zeng, 2011a). Parameters

CP

AM

unit

/0 Ks2

0.331 31.8 19.1 1.79

0.367 – – 1.79

– GPa GPa GPa

7.81

7.81

GPa

0:0409 þ 3:9 104 T 0.917

0:0409 þ 3:9 104 T 0.917

0.999

0.999

J  M2 g:cm3 g:cm3

s2

l

Kl 3 Kc 3

ccl 4 q0c 3 q0l 3 ss sc 3 sl 5

K sr K lr 5 ksr 5 klr 5

11:49 3

10:9 3

106 =  C

155

155

106 =  C

(68.7 + 24.732 T)

(68.7 + 24.732 T)

300 2 0.3 2

300 2 0.3 2

106 =  C MPa MPa nm nm

1 /0 ¼ 0:26, measured by the MIP, is adopted in the simulation of Refs.(Zeng, 2011a; Zeng et al., 2014a), the difference of two porosities is shown the text in Section 5.2. 2 After (Ulm et al., 2004). 3 After (Coussy and Monteiro, 2008). 4 After (Sun and Scherer, 2010b). 5 After (Churaev and Derjaguin, 1985).

where

8 l > EPl ¼ 3KbPhom > > <

bð/ þ/ ÞPðhÞ

c f EP ¼ 3/ hom > 0K > > s : Eth ¼ s3T

ð28bÞ

where EPl ; EP ; Eth and EPeq are the linear strains induced by liquid pore water pressure, disjoining pressure, thermal stress, and macroscopic equivalent pressure, respectively. The poroelastic properties of CP shown in Eq. (28) can be estimated by the scalar form of Eq. (15):

8 hom s > K ¼ ð1/0 ÞK/0 > > ð1/ Þþ 0 > 1n > > > /0 > > b ¼ 1ð1/ Þn < 0

jhom ¼ K hom ss ! shom ¼ jhom =K hom ¼ ss > > > 3/0 ð1/0 Þ > 1 > > > N ¼ 4ls þ3/0 K s > > : n ¼ 3K s

ð29Þ

ð30Þ

The disjoining pressure PðhÞ can be determined by Eq. (5). There are still lack of experimental data for disjoining pressure within unfrozen water film until now. Therefore, the structural component of water film at room temperature (isothermal condition) is adopted to characterize the disjoining pressure of the unfrozen water film (Churaev and Derjaguin, 1985). The corresponding parameters for the disjoining pressure are given in Table (1). An empirical equation proposed by Fagerlund (1973) is employed to determine the thickness of unfrozen water film:

h ¼ 1:97ðTÞ1=3 ðnmÞ

ð31Þ

5.2. Volume fractions of each phase The pore size distribution of the CP is determined by means of Mercury intrusion porosimetry (MIP) and the pore size distribution function gðrÞ is fitted by Zeng (2011a) (see Fig. 4). Under free swelling, the volume strain of the CP sample subject to freezing process is infinitesimal (hundreds of micros per meter (Zeng, 2011a)), the current porosity is assumed to be the same as the initial porosity (Morin and Hellmich, 2014). The current volume fractions of each phase are thus assumed to be estimated by the pore size distribution function of cement paste (measured by MIP before freezing process) and Gibbs–Thomson equation (see Eq. (3)). The pores determined by MIP is assumed to be capillary pores of CP. The freezing of liquid pore water is expected to occur in the capillary pores other than in gel pores. Moreover, h  D can be assumed during freezing, where D is the diameter of the pore space. Therefore, the volume fractions of each phase can be evaluated as:

8 R  Dmax 6h > > < /f  /0 Dcr D gðDÞdr RD ÞgðDÞdr /c  /0 Dcrmax ð1  6h D > > : /l  /0  /f  /c

ð32Þ

where Dmax ¼ 4 105 nm, Dmin ¼ 3 nm; Dcr ¼ 2r cr ; rcr is determined by Eq. (3); /0 ¼ 0:26 is the porosity determined by the MIP tech-

3K s þ4ls

where K s and ls are the bulk modulus and the shear modulus of the cement solid matrix (with gel porosity). /0 ¼ 0:33 is the statistical average value of varied porosities measured at different drying techniques (e.g. freezing dry, acetone replacement, methanol replacement and oven-drying at 105  C) (Zeng, 2011a). ss is the volumetric thermal dilation coefficient of the cement solid matrix. For the AM, K s and ls are not given in the literature. Fortunately, K hom ¼ 8:9 GPa and b ¼ 0:5 are estimated in Sun and Scherer (2010a) and thus they are used directly in the simulation. With the assumption of isotropy at local and macro scale as well as pore space being spherical, the liquid pore water pressure (Pl ) within undrained CP can be determined by Eq. (24) and the detailed volume fractions of each phase introduced in next section. Because of the 6% vol. entrained air voids in AM, the ice crystal pressure within frozen porous media can be taken as 0 at equilibrium (Coussy and Monteiro, 2008; Sun and Scherer, 2010a). Owing to the micro-cryo-suction effect, the liquid pore water nearby is sucked into the air voids, which creates a negative liquid pore water pressure (Coussy and Monteiro, 2008; Sun and Scherer, 2010a). The latter within AM can be determined by Coussy and Monteiro (2008); Sun and Scherer, 2010a:

Fig. 4. Pore size distribution function gðrÞ fitted by MIP experimental results, /0 ¼ 0:26, after Zeng (2011a).

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

nique (Zeng, 2011a; Zeng et al., 2014a). The difference between /0 ¼ 0:33 and /0 ¼ 0:26 mainly lies in the gel pore volume fraction. The latter is not accounted for in the porosity determined by MIP, while it is a major component of /0 ¼ 0:33. Therefore, /0 ¼ 0:33 is adopted to estimate the poroelastic properties while /0 ¼ 0:26 is used to estimate the volume fractions of ice and unfrozen water films (see Eq. (32)). It should be aware that owing to the supercooling effect (lack of the nucleus), the instantaneous solidification of the supercooling water initiates only at 8 °C in Zeng’s CP sample (Zeng, 2011a; Zeng et al., 2014a). This effect is also accounted for in our simulation. Herein, there is no ice crystals and thus P l ¼ 0 when T > 8  C. That is to say, when T > 8  C, the negative liquid pore water pressure induced by the variation of liquid pore water density (Zeng, 2011a) is neglected in this study. When temperature T6 8 °C, T is taken as a variable. Combining Eqs. (3), (24) and (32), the saturation degrees of each phase and liquid pore water pressure Pl can be determined numerically. For the AM, all we have in the Sun and Scherer (2010b) is the cumulative curve of ice content with temperature, the volume fraction of unfrozen water film is not given in the literature. Indeed, the volume fraction of the unfrozen water film /f can be estimated by the cumulative curve of ice content:

/f ¼

X 3hðiÞ i

r c ðiÞ

ð/c ði þ 1Þ  /c ðiÞÞ

ð33Þ

where r c ðiÞ is the critical radius at temperature TðiÞ in cumulative curve of ice content (see Eq. (3)), i is the number of data point in cumulative curve of ice content between 0 and T; /c ðiÞ is the corresponding ice content at TðiÞ; hðiÞ is the thickness of unfrozen water film at TðiÞ. 5.3. Results and discussion Using the elastic properties of solid matrix and the porosity of CP in Table (1), the homogenized poroelastic properties of CP can be

predicted

by

Eq.

(29).

That

is:

K hom ¼ 15:1 GPa, 6 

b ¼ 0:525; N ¼ 162:2 GPa, s ¼ 11:49 3 10 = C. The former three poroelastic constants differ little from those in Ulm et al. (2004), in which the CP has the same water/ratio and /0 ¼ 0:34. Furthermore, in isotropic case, the homogenized shear modulus of the porous medium can also be estimated by the scalar form hom

(a) CP

of Eq. (15), that is to say,

s

s

ð1/0 Þð9K þ8l Þ lhom ¼ 9K s ð1þ2=3/ ¼ 9:7 GPa. s 0 Þþ8l ð1þ3=2/0 Þ

The homogenized Young’s modulus estimated by our model is hom

l ¼ 24:0 GPa, Ehom ¼ 3þl9hom =K hom

which

is

comparable

with

E ¼ 21:7 0:1 GPa of the CP with water/cement ratio (/0 ¼ 0:34) determined by Resonance Frequency (Ulm et al., 2004). The evolution of volume fractions of each phase with temperature is illustrated in (Fig. 5). As expected, below the bulk supercooling temperature (8 °C), the nucleation of supercooling water emerges instantaneously in CP, see Fig. (5)(a). However, there is no bulk supercooling effect in AM (see Fig. (5)(b)) because metaldehyde (efficient nucleation agent) was applied on the surface of AM to encourage the ice nucleation (Sun and Scherer, 2010a). As shown in Fig. (5), for both of CP and AM, the volume fraction of liquid pore water /l decreases with decreasing temperature, while both of /c and /f increase with decreasing temperature. More interestingly, /f can reach as much ca. 15% of porosity at T ¼ 35  C for both of CP and AM. Therefore, during freezing process, the unfrozen water film is expected to be more and more crucial with decreasing temperature. The variation of the disjoining pressure PðhÞ, liquid pore water pressure Pl and macroscopic equivalent pressure P eq with the temperature is plotted in (Fig. 6). It can be seen from Fig. (6)(a) that owing to bulk supercooling effect, Pl ; PðhÞ and P eq are null above 8°C. When the temperature is below 8 °C, the ice will form in the CP instantaneously. Hence, a huge liquid pore water pressure and disjoining pressure will emerge within the CP. As depicted in Fig. (6)(a), PðhÞ; P l and the corresponding Peq increase with the decreasing temperature; P l and PðhÞ can reach as much as 115 MPa and 41 MPa when the temperature is down to 35 °C, which indicates that Pl is more significant than that of disjoining pressure in undrained CP during freezing process. It should be borne in mind that, the CP is undrained while AM is drained. As discussed previously, the different boundary conditions induce distinct pressure development during freezing. As shown in Fig. (6) (b), the pressure evolution is quite different in AM from that in CP: the negative liquid pore water pressure results from the cryo-suction by ice crystal in air voids, the liquid pore water pressure decreases with decreasing temperature; PðhÞ increases with decreasing temperature and counterbalances the negative Pl . Comparing the effect of disjoining pressure in undrained CP and drained AM, it can be readily found that PðhÞ plays a more dominant role in drained AM than that in undrained CP.

(b) AM

Fig. 5. Evolution of the volume fractions of each phase with freezing temperature, the ice content curve in AM is after Sun and Scherer (2010a).

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

(a) CP

(b) AM

Fig. 6. Evolution of the components of P eq with freezing temperature.

The evolution of the linear strains induced by the components of P eq with freezing temperature is illustrated Fig. 7. As expected, for undrained CP below 8 °C, owing to the increase of the liquid pore water pressure and the disjoining pressure with the decreasing temperature (see Fig. (6)(a)), EPl ; EP and EPeq increase with decreasing temperature. Therefore, the undrained CP swells. On the contrary, as plotted in Fig. 7(b), negative Pl gives rise to negative strain EPl while the PðhÞ contributes to positive strain EP . The combination of EPl and EP results in the negative EPeq (see Eq. (equ: total strain of free swelling in isotropic case)(a)): the drained AM contracts. Likewise, the variation of EPeq ; Eth and E with temperature and the comparison with experimental results (Zeng et al., 2014a; Sun and Scherer, 2010a) are plotted in Fig. (8). As illustrated in Fig. (8)(a), for temperatures ranging from 0 °C to 8 °C, Eth agrees well with experimental results, since there is no ice formation owing to bulk supercooling effect and only the thermal shrinkage is acting. It can also been found from Fig. (8)(a) that after the onset of the freezing (T < 8 °C), the simulation results of the total strain (red solid curve) are comparable with the experimental results (blue points) though there are still some discrepancies between the two results. Thus, according to Eq. (28b), the discrepancies

between the simulation result (red solid curve) and experimental results (blue points) are likely to lie in the overestimation of macroscopic equivalent pressure Peq . The overestimation of the latter probably results from the overestimation of ice content. The latter is derived by the pore size distribution from MIP and modified Gibbs–Thomson equation. According to Sun and Scherer (2010b), not only the porosity (/0 ¼ 0:26) but also the pore size is enlarged by MIP Sun and Scherer (2010b). Hence, estimating the ice content by means of pore size distribution derived from MIP will overestimate the ice content. Accordingly, the total strains evaluated by the pore size distribution from MIP will be overestimated. Generally, it is more feasible to measure the volume fraction of ice crystal by means of thermoporometry (TPM) (with differential scanning calorimeter (DSC)) (Brun et al., 1977; Sun and Scherer, 2010b), which is in accordance with the freezing process in porous media. This is exactly what is used in our simulation for the partially frozen AM. As depicted in Fig. (8)(b), the agreement between the model result (red solid curve) and experimental results (blue points) Sun and Scherer (2010a) is better than that in Fig. (8)(a), considering the uncertainties of measurement of properties and volume fraction of cement-based materials (Sun and Scherer, 2010a).

(a) CP Fig. 7. Evolution of the linear strains induced by components of P

(b) AM eq

with freezing temperature.

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

(a) CP

11

(b) AM

Fig. 8. Comparison between model results (red solid curves) and experimental results, the experimental results in (a) and (b) are after Zeng et al. (2014a) and Sun and Scherer (2010a), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The introduction of the unfrozen water film and corresponding disjoining pressure is the major distinction between micromechanics and simplified poromechanics model. In the latter, there are the presence of only ice crystal and liquid pore water. The two models for the partial freezing in CP and AM will be compared to elucidate the effect of unfrozen water film. In micro and macro isotropic case, the linear strain of the partially frozen porous media (without unfrozen water film) under free swelling is estimated by Coussy and Monteiro (2008):

E ¼ ss T=3 þ

bð/c =/0 Pc þ /l =/0 Pl Þ

ð34Þ

3K hom

Pl and Pc in undrained frozen porous media and drained porous media can be determined as Coussy and Monteiro (2008):

( CP :

l Pl  V cVV c

c

l

1 1=K c þð/0 =/c 1Þ=K l

P ¼ P  Sm dT

( AM :

Pl ¼ Sm dT Pc ¼ 0

ð35Þ

The impact of the unfrozen water film and the corresponding disjoining pressure on the linear strains of CP and AM are depicted in Fig. (9). As can be found in Fig. (9), the effect of unfrozen water film on the linear strains of CP and AM is minor since the disjoining

(a) CP

pressure is negligible when temperature is higher than 10 °C. However, the influence of the unfrozen water film will be pronounced at low temperature (lower than 10 °C) in both CP and AM during freezing process. 6. Concluding remarks The poromechanical effect of the unfrozen water film and the corresponding disjoining pressure are stressed in this study. Based on the local characterization of the partially frozen porous media, a micromechanics model fully accounting for the initial stress, thermal stress, membrane stress induced by surface tension and disjoining pressure, is developed in this study. Conclusions can be drawn as follows: Based on the thermodynamic and mechanical equilibrium between ice crystal and unfrozen water film, it is found that R1  h r2 is the implicit assumption for the validity of poromechanics model (Coussy, 2006), where r2 is the curvature radius of the ice-liquid pore water interface, R1 is the curvature radius of the ice-unfrozen water film interface, h is the thickness of unfrozen water film.

(b) AM

Fig. 9. The influence of the unfrozen water film and the corresponding disjoining pressure on the linear strains.

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R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

When surface tension ccf  ccl is assumed, the variation of macroscopic equivalent pressure P eq only consists of two components: disjoining pressure and liquid pore water pressure. The macroscopic swelling or shrinkage of the partially frozen porous media depends on the competing effects between thermal strain and the strain induced by the disjoining pressure and liquid pore water pressure. The micromechanics model is employed to simulate the free swelling of partially frozen cement paste (CP) with undrained boundary condition and air-entrained mortar (AM) with drained boundary condition. The disjoining pressure plays a more pronounced role in drained AM than that in undrained CP. The simulated linear strains of the frozen CP under free swelling are comparable with the experimental results. The discrepancies between the two are likely to arise from the overestimation of the ice content. The latter is overestimated since both porosity and pore size of the cementitious materials are enlarged during the MIP experiment. Owing to the estimation of the ice content by means of DSC experiment, the simulated linear strain of the frozen AM reasonably agrees well with the experimental results. The poromechanical effect of the unfrozen water film and the corresponding disjoining pressure will be more and more pronounced when temperature decreases.

Z

cab

I ab

1T : A

dS

X

¼

cab Z jXj

I ab

Z 1T dS : Ap ¼ cab

r1

r2


 2gðrÞ drB r

ðA:4Þ

where the interface I ab occupies the pore space with radius ranging from r 1 to r2 . In terms of Eq. (A.4), the homogenized membrane stresses of R rcr 2gðrÞ Rr ð r ÞdrB and csf rcrmax ð2grðrÞÞdrB, where interfaces I sl and I sf are csl rmin

csl and csf are the surface tensions of solid/liquid pore water interface and solid/unfrozen water film interface. Inserting Eqs. (A.2)–(A.4) into Eq. (10) yields Eq. (11). Appendix B. P l in partially frozen porous medium with undrained boundary condition In the RVE of freezing in porous media, when crystallization occurs in one pore, some of the water is expelled out of the crystallized pore through the channel pores or unfrozen water films. Thus, the mass of water as well as liquid pore water pressure in one pore is imbalanced at local scale. However, at equilibrium state, the deviation of the local in-pore phase pressure field around its average in the RVE is negligible (Dormieux et al., 2006): according to Eq. (8), P i ¼ P i ðzÞ  Pi ðzÞ, where i 2 fl; c; f g. A similar assumption can be made about the density of the each phase in pores:

qi ¼ qi ðzÞ  qi ðzÞ ð8z 2 Xi Þ, where i 2 fl; c; f g. These are the theoAcknowledgment The authors gratefully acknowledge the original experimental data and constructive suggestion provided by professor George W. Scherer from Princeton University. Appendix A. Macroscopic equivalent pressure accounting for pore size distribution Here, it is reasonable to assume that there are no morphological differences between Xl and Xcf (Chateau et al., 2002; Dormieux et al., 2006). Hence, the average strain concentrations of the liquid pore water and ice crystal + unfrozen water film spherical composite inclusion are expected to be equal to that of pore space:

Al  Acf  Ap

ðA:1Þ

Substituting Eq. (A.1) into the third term of the right hand side of Eq. (10) yields:

/l Pl 1 : Al þ /cf P f 1 : Acf  ðSr l Pl þ Sr cf P f Þ/1 : Ap ¼ BðSr l Pl þ Sr cf P f Þ

ðA:2Þ

where Srl ¼ /l =/0 is the saturation degree of liquid pore water, Srcf ¼ ð/c þ /f Þ=/0 is the saturation degree of the ice crystal + unfrozen water film composites. In order to determine the membrane stresses within freezing porous medium, a normalized pore size distribution function gðrÞ is introduced firstly. /0 gðrÞdr represents the volume fraction of the pores within the radius range ½r; r þ dr. Out of the range of the ½rmin ; r max ; gðrÞ ¼ 0, where rmin and r max are the minimum and maximum radii of the porous medium. According to the definition of the pore size distribution function, we have:

Z

r max

r min

Z

gðrÞdr ¼ 1; Srl ¼

r cr

rmin

Z

gðrÞdr; Srcf ¼

r max

gðrÞdr

ðA:3Þ

r cr

In terms of Eq. (A.1), the homogenized membrane stress within interface I ab (a; b 2 fs; l; f g) can be rewritten as Chateau et al. (2002), Dormieux et al. (2006) and Cariou (2010):

retical premise of the determination of macroscopic liquid pore water pressure in partially frozen porous media. In terms of Eq. (8), the mass content per unit volume m is defined as the total water mass (different components) contained   in pore space Xp divided by the initial volume jX0 j of the RVE (Dormieux et al., 2006):

m¼

Z

1 jX0 j

Xp

qðzÞdV z

ðB:1Þ

The porous medium is initially saturated with liquid pore water. The initial water mass per unit volume m0 ¼ /0 q0l . In the partially frozen porous medium, the water mass content change per initial total volume of the RVE can be expressed in terms of the Lagrangian porosity (Dormieux et al., 2006):

m  m0  /c qc þ /f qf þ /l ql  /0 q0l

ðB:2Þ

where /i is the current volume fraction of the ith phase (i 2 fc; f ; lg). In order to ensure the linearity of the macroscopic state equation in terms of E and m  m0 , it is necessary to assume that the variation of the density of water (including ice) around a reference value are small:

dqi

q0i

 1; i 2 fc; f ; lg (Dormieux et al., 2006). There-

fore, a linear form of fluid state equation linking the density and the pressure and temperature can be employed (Dormieux et al., 2006):

dqi

q0i

¼

Pi Ki

 si dT

i 2 l; f ; c

ðB:3Þ

where K i is the bulk modulus of the ith phase, si is the volumetric thermal dilation coefficient of the ith phase, i 2 fl; f ; cg. The density of the unfrozen water film qf is assumed to be equal to that of liquid pore water ql , namely qf ¼ ql . Inserting Eq. (B.3) into Eq. (B.2) yields the alternative second state equation:

  8  l  Pc

q0c q0c P l c > 0 < mm  /  / þ /  s dT þ / þ  1 c  s dT 0 0 0 l 0 l c K ql ql ql K l  > P þPðhÞ : f  s dT þ/f Kf

ðB:4Þ

Please cite this article in press as: Yang, R., et al. A micromechanics model for partial freezing in porous media. Int. J. Solids Struct. (2015), http://dx.doi.org/ 10.1016/j.ijsolstr.2015.08.005

R. Yang et al. / International Journal of Solids and Structures xxx (2015) xxx–xxx

Here, / ¼ /c þ /f þ /l is the current pore volume fraction. The internal pressure of ice crystal Pc is related to liquid pore water pressure P l by Eq. (3). Therefore, inserting Eqs. (16) and (21) into Eq. (B.4) yields the expression of liquid pore water pressure P l . References Beaudoin, J.J., MacInnis, C., 1974. The mechanism of frost damage in hardened cement paste. Cem. Concr. Res. 4 (2), 139–147. Bernard, O., Ulm, F.-J., Lemarchand, E., 2003. A multiscale micromechanicshydration model for the early-age elastic properties of cement-based materials. Cem. Concr. Res. 33 (9), 1293–1309. Brun, M., Lallemand, A., Quinson, J.-F., Eyraud, C., 1977. A new method for the simultaneous determination of the size and shape of pores: the thermoporometry. Thermochim. Acta 21 (1), 59–88. Cariou, S., 2010. Couplage hydro-mécanique et transfert dans l’argilite de Meuse/ Haute-Marne: approches expérimentale et multi-échelle (Ph.D. thesis), Ecole des Ponts ParisTech. Chateau, X., Moucheront, P., Pitois, O., 2002. Micromechanics of unsaturated granular media. J. Eng. Mech. 128 (8), 856–863. Churaev, N., Bardasov, S., Sobolev, V., 1993. On the non-freezing water interlayers between ice and a silica surface. Colloids Surf., A 79 (1), 11–24. Churaev, N., Bardasov, S., Sobolev, V., 1994. Disjoining pressure of thin nonfreezing water interlayers between ice and silica surface. Langmuir 10 (11), 4203–4208. Churaev, N., Derjaguin, B., 1985. Inclusion of structural forces in the theory of stability of colloids and films. J. Colloid Interface Sci. 103 (2), 542–553. Churaev, N., Sobolev, V., Starov, V., 2002. Disjoining pressure of thin nonfreezing interlayers. J. Colloid Interface Sci. 247 (1), 80–83. Coussy, O., 2005. Poromechanics of freezing materials. J. Mech. Phys. Solids 53 (8), 1689–1718. Coussy, O., 2006. Deformation and stress from in-pore drying-induced crystallization of salt. J. Mech. Phys. Solids 54 (8), 1517–1547. Coussy, O., 2011. Mechanics and Physics of Porous Solids. John Wiley & Sons. Coussy, O., Monteiro, P.J., 2008. Poroelastic model for concrete exposed to freezing temperatures. Cem. Concr. Res. 38 (1), 40–48. Derjaguin, B., Churaev, N., 1978. On the question of determining the concept of disjoining pressure and its role in the equilibrium and flow of thin films. J. Colloid Interface Sci. 66 (3), 389–398. Derjaguin, B., Churaev, N., 1986. Flow of nonfreezing water interlayers and frost heaving. Cold Reg. Sci. Technol. 12 (1), 57–66. Derjaguin, B., Churaev, N., Muller, V., 1987. Surface Forces. Consultants Bureau, New York. Derjaguin, B., Obuchov, E., 1936. Colloid j., moscow 1, 385, 1935. Acta Phys.-chim. URSS 5 (1). Dormieux, L., Kondo, D., Ulm, F.-J., 2006. Microporomechanics. John Wiley & Sons. Dormieux, L., Lemarchand, E., Kondo, D., Fairbairn, E., 2004. Elements of poromicromechanics applied to concrete. Mater. Struct. 37 (1), 31–42. Fagerlund, G., 1973. Determination of pore-size distribution from freezing-point depression. Matér. Constr. 6 (3), 215–225.

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Please cite this article in press as: Yang, R., et al. A micromechanics model for partial freezing in porous media. Int. J. Solids Struct. (2015), http://dx.doi.org/ 10.1016/j.ijsolstr.2015.08.005