Unsaturated poroelasticity for crystallization in pores

Computers and Geotechnics 34 (2007) 279–290 www.elsevier.com/locate/compgeo

Unsaturated poroelasticity for crystallization in pores Olivier Coussy a

a,*

, Paulo Monteiro

b

Navier Institute, Ecole des Ponts, 6-8 Avenue Blaise Pascal – Cite´ Descartes – F 77455 Marne-la-Valle´e Cedex 2, France Structural Engineering, Mechanics, and Materials, 725 Davis Hall, Department of Civil and Environmental Engineering, University of California at Berkeley, CA 94720, United States

b

Available online 2 April 2007

Abstract This paper shows how poroelasticity provides a quantitative explanation of the role of the pore size distribution in the deformation of porous materials subjected to the frost action. Unsaturated poroelasticity is the natural extension of saturated poroelasticity, once recognized that any crystallization process within a porous material results from two distinct physical processes: (i) an invasion process by the forming crystal, which is driven by surface energy effects; (ii) a deformation process under the crystallization pressure, whose intensity is governed by the poroelastic properties of the porous solid and by the current liquid saturation. The analysis reveals that the origin of the expansion is twofold: (i) the pressure build up of the unfrozen part of the porous network, that originates from the action of the excess of liquid expelled from the freezing sites; (ii) the cryo-suction process that constantly drives the liquid back towards the frozen sites as the cooling further increases. This explains the long known efficiency of air-voids against the frost action. These voids act both as expansion reservoirs and as cryo-pumps. The analysis finally provides an assessment for the recommended spacing factor between adjacent air-voids.  2007 Elsevier Ltd. All rights reserved. Keywords: Freezing; Unsaturated poroelasticity; Cryo-suction; Void; Pore size distribution

1. Introduction The mechanical behaviour of building materials subjected to in-pore crystallization relates to various sustainability issues. For instance the crystallization of sea-salts is recognized as being an important weathering phenomenon in dry environments close to the sea [21]. It often leads to serious deterioration in porous sedimentary rocks used for building in coastal areas [7,24]. Besides, the delayed formation of ettringite crystals, which may be encountered in cementitious materials after hardening [46], can seriously damage concrete structures. Furthermore, the durability of water-infiltrated materials subjected to frost action is a major concern in cold climates [31,36]. Ice formation in concrete is the cause of billions of euros in damage under*

Corresponding author. E-mail addresses: [email protected] (O. Coussy), monteiro @ce.berkeley.edu (P. Monteiro). 0266-352X/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2007.02.007

gone by concrete structures, even in temperate regions: in 1980 the supervision of French civil engineering works concluded that most damage experienced by buildings, bridges, etc. was originated by frost action [30]. A better understanding of the mechanics of confined crystallization within deformable porous materials could improve the resistance of building materials in environmental conditions and, thereby, reducing the maintenance and repair costs. The material deterioration originating from confined crystallization is caused by the pressure that the crystal growth generates on the internal solid walls of the porous solid [42,43]. The crystal pressure induces a tensile stress within the surrounding solid matrix, whose overall strength may be ultimately overpassed. All of us have definitively experienced the failure of a sealed water-filled bottle subjected to the frost action! Unfortunately, the general analysis is not as straightforward as the one applying to this familiar case, where the pressure build up and the failure result only from the undrained 9% expansion of liquid

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water solidifying within a single large pore. Actually, cement pastes can slightly expand if the saturating liquid water is replaced by benzene, which, unlike water, contracts when solidifying [2]. By contrast, water saturated porous materials can even exhibit a contraction if air-voids are present [37]. These observations bring to light that the mechanics of confined crystallization does involve the material microstructural properties, and particularly the pore size distribution. Furthermore, the deformation and the failure analysis cannot be restricted to that of a single pore so that a global approach in the context of the mechanics of porous solids is needed. Poroelasticity is that branch of mechanics which deals with elastic porous solids whose internal walls are subjected to the pressure of the fluid saturating the porous volume. Through the years, since the pioneering work of Biot [3], applications of poroelasticity have mainly lain in rock mechanics, geophysics [41,5,19,48], and biomechanics [18]. More recently, poromechanics has revealed to be an useful tool to approach the mechanics and the physics of cement-based materials [38]. Based on continuum thermodynamics [11], poromechanics extends poroelasticity to non-elastic porous solids [12] and to unsaturated conditions [13], and is apt to capture various thermo-chemohydro-mechanical couplings (see [14] for a comprehensive presentation). In the spirit of previous works [49,15], the aim of this paper is to show how poromechanics can be a relevant framework to deal with the mechanics of confined crystallization. This will be examplified for the water crystallization within porous materials subjected to cooling under 0 C. In the first part we extend standard concepts of saturated poroelasticity to unsaturated conditions in order to include surface energy effects in the mechanics of porous solids. In the second part, we apply unsaturated poroelasticity to the analysis of the deformation of waterinfiltrated porous solids subjected to freezing. We finally compare the deformation of mortar samples cast with different porosities and exposed to low temperatures as predicted by unsaturated poroelasticity with the measurements made in [9] of their observed average expansion during thaw-freezing cycles. A particular attention is given to the role of the pore size distribution and to the presence of air-voids in order to explain the experimental observations reported above. 2. Unsaturated poroelasticity The thermodynamics of heterogeneous substances goes back to the works of Gibbs [26] and Duhem [20], which have furnished the long celebrated Gibbs–Duhem equation extensively used in the thermodynamics of mixtures. The thermodynamics accounting for the role of the surface energy equally also goes back to the famous works of Gibbs, while the thermodynamics of irreversible process was in great part initiated by the pioneering works of Duhem. Affiliated with the same energy approach, the thermodynamics of deformable saturated porous materials has been initiated by Biot

[4], allowing to extend the Clausius–Duhem inequality to such a case [14]. This section combines these general approaches to provide the thermodynamics of unsaturated elasticity of porous solids, or more briefly unsaturated poroelasticity, in the context of continuum mechanics. 2.1. State equations of unsaturated poroelasticity As a preamble, let us first stress that, in all the forthcoming developments, no shear stresses will be assumed to significantly develop within the crystals of ice. This hypothesis is usual in the physics of in-pore crystallization [22,42] and is actually acceptable owing to the water inpore confinement. This confinement allows supercooled liquid water to surround everywhere the crystals of ice. Accordingly, since the pressure of the supercooled liquid water is homogeneous within the porous material infinitesimal element, the liquid–solid thermodynamic equilibrium condition (35) requires the crystal pressure to be homogeneous too. Owing to this crystal pressure homogeneity, the mechanical equilibrium of the crystals of ice is eventually achieved with no shear stress present within the solid water. As a result, liquid–solid transformation of water will constantly adjust the volume of water currently frozen to match the deformation of the solid internal walls delimiting it, in order to achieve an in-pore homogenous volumetric deformation of ice, and consequently to avoid any shear deformation. With this prerequisite in mind, let us then extract an infinitesimal representative element from a porous continuum. In the reference configuration the volume of the element is dX0, and its porous volume is /0 · dX0, where /0 stands for the (initial) porosity. At current time t the porous volume is / · dX0, / standing for the current porosity. Since porosity / refers the current porous volume to the reference volume dX0, the difference /  /0 captures the overall change of the porous volume with regard to the reference configuration. As any strain related to a (porous) solid, it is a Lagrangian variable and, accordingly, / can be coined as the Lagrangian porosity. At time t the porous volume is filled by two constituents: a wetting liquid referred to by index J = L; a non-wetting constituent referred to by index J = C, which anticipates the forthcoming applications where the non-wetting constituent will be eventually identified to the solid crystal. Let then nJ · dX0 be the number of moles of phase J currently present in the porous element so that nJ represents the (Lagrangian) macroscopic or apparent molar density of phase J. The volume currently occupied by phase J is /J · dX0 where /J is the current (Lagrangian) partial porosity related to phase J (see Fig. 1). We have / ¼ /L þ /C :

ð1Þ

Accordingly, if T is the temperature, and if pJ ; sJ and lJ denote respectively the pressure, the current molar entropy, and the current molar chemical potential of constitutent J, the Gibbs–Duhem equality applied separately to constituent J can be written in the form

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

dΩ= (1 + εkk) × dΩ0

Energy balance (5) implies that the skeleton free energy Fsk is a function of eij, /L, /C and T only. As a result the unsaturated poroelastic state equations are derived in the form

dΩ0 =

=

φ0 SC × dΩ0 φ0 SL × dΩ0

rij ¼

φL × d Ω 0

φC × d Ω0

oF sk ; oeij

oF sk ; o/L

pL ¼

pC ¼

oF sk ; o/C

(φ0 SL + ϕL ) × d Ω0

(φ0 SC + ϕC ) × d Ω0

Rsk ¼ 

oF sk : oT ð6Þ

In this paper the sign convention of continuum mechanics is adopted for the stress and the strain components rather than the convention generally used in soil mechanics: tensile stresses and dilation strains are counted positively.

solid matrix solid matrix

reference configuration

281

current configuration

Fig. 1. Schematic definition of: (a) Lagrangian saturation SJ capturing the invading process; (b) Lagrangian change uJ in partial porosity /J capturing the deformation process.

/J dpJ  nJ sJ dT  nJ dlJ ¼ 0:

ð2Þ

Let now rij be the overall stress components to which the infinitesimal porous element is currently subjected; and let eij be the current overall infinitesimal strain components capturing the deformation of the infinitesimal volume dX0, with regard to a stress-free reference configuration where the constituents are at atmospheric pressure. As a first approach the assumption of infinitesimal strains is still acceptable since the order of magnitude of the strain undergone by mortars subjected to freezing does not exceed few 103 [9]. Finally, let R and F be respectively the entropy and the overall Helmholtz free energy of the open element per unit of its initial volume dX0. When no dissipation occurs (poroelastic material), the first and the second laws of thermodynamics applied to the open system dX0 [14] combine to provide the Clausius–Duhem equality in the form rij deij þ lL dnL þ lC dnC  R dT  dF ¼ 0:

ð3Þ

In energy balance (3) rij deij is the infinitesimal strain work related to the deformation of the infinitesimal element dX0, while lJ dnJ is the infinitesimal free energy associated with the change in the molar content of constituent J. The sum of these energy supplies is equal to the infinitesimal free energy dF that the material eventually stores since no mechanical dissipation is involved. Let then Rsk and Fsk be the entropy and the Helmholtz free energy per unit of initial volume of the ‘skeleton’. The ‘skeleton’ is here defined as the material obtained by removing the bulk phases J = L, C from the material porous element, but still including the interfaces which have their own free energy. As a direct consequence of the additive character of free energy, the Helmholtz free energy of the skeleton can be written in the form Rsk ¼ R  nL sL  nC sC ; F sk ¼ F  ðnL lL  /L pL þ nC lC  /C pC Þ:

ð4Þ

Using (2) and (4) into (3), we get the free energy balance related to the poroelastic skeleton only rij deij þ pL d/L þ pC d/C  Rsk dT  dF sk ¼ 0:

ð5Þ

2.2. Saturation and state equations The ‘skeleton’ has been defined by removing the bulk phases only. Thereby the current ‘skeleton’ still includes the interfaces between the solid matrix, the wetting and non-wetting constituents. Actually, the change in partial porosities /J results from two distinct processes. The first process is a drainage process of the wetting consitutent L, by the non-wetting constituent C. This drainage process is associated with a change in the inner interface between the constituents L and C, and the solid matrix. In contrast, the second process concerns the skeleton deformation only, resulting from the action of pressures pL and pC on the internal walls of the porous network. As a direct consequence, the current partial porosities /J can be split into two independent additive contributions according to /J ¼ /0 S J þ uJ ;

S L þ S C ¼ 1:

ð7Þ

The term /0SJ accounts for the drainage process. It is such that /0SJ dX0 represents the porous volume currently occupied by the constituent J, yet prior to any deformation. In other words /0SJ dX0 represents the porous volume whose internal solid walls delimiting it in the reference undeformed configuration will be in contact with the constituent J in the current deformed configuration (see Fig. 1). Actually SJ can be coined as the Lagrangian saturation degree of phase J [15], since it relates to the initial overall porous volume /0 dX0, in opposition to the Eulerian definition which would apply to the current porous volume / dX0. The term uJ relates to the skeleton deformation only. It is such that uJ dX0 represents the infinitesimal change undergone by the currently J-filled volume /0SJ dX0 under the effect of pressure pJ. As a result and as it can be checked by substituting (7) into (1), the current (Lagrangian) overall porosity, which eventually concerns the skeleton deformation only, is consistently given by / ¼ /0 þ uL þ uC :

ð8Þ

The energy balance (5) and, consequently, the ‘skeleton’ free energy Fsk(eij, /L, /C, T), does not account separately for the energy required for the skeleton deformation, and for the energy required in the change in the inner interfaces during the drainage process. To the aim at addressing separately the two energy contributions, the free energy can be now expressed with the help of both the variables

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O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

associated with the skeleton deformation, namely eij, uL, uC, and the variable associated with the invasion process, namely SL = 1  SC. We write F sk ðeij ; /L ; /C ; T Þ ¼ Wsk ðeij ; uL ; uC ; S L ; T Þ:

ð9Þ

Substitution of (7) and (9) into (5) leads to rij deij þ pL duL þ pC duC  /0 ðpC  pL ÞdS L  Rsk dT  dWsk ¼ 0: ð10Þ

In energy balance (10) the first three terms identify to the infinitesimal work rate associated with the deformation of the solid skeleton only, irrespective of the redistribution of the interfaces during the invasion process. In contrast, the fourth term represents the infinitesimal work done against the interfacial forces during the drainage process. This work is required to make the inner C/L front propagate within the porous network, and consequently to make the interface between the inner constituents and the solid matrix change according to the infinitesimal volume /0 dSC · dX0 = /0dSL · dX0 newly invaded by the wetting constituent C. From energy balance (10) we now derive the state equations in the equivalent but more convenient form oWsk ; ouL oWsk /0 ðpC  pL Þ ¼  ; oS L oWsk Rsk ¼  : oT rij ¼

oWsk ; oeij

pL ¼

pC ¼

oWsk ; ouC

ð11aÞ ð11bÞ ð11cÞ

The constitutive state Eq. (11) apply whatever the constituents filling the porous volume actually are (gas, liquid or crystal). They capitalize the general information provided by the first and the second laws of thermodynamics. 2.3. Interface energy, saturation and pore size distribution The mechanical equilibrium of the interface between two distinct constituents involves interface energy effects. At the macroscopic scale the equilibrium of these interfaces is expressed by the means of state Eq. (11b). Actually, owing to the additive character of energy, the Helmholtz free energy Fsk of the skeleton can be split into two parts: (i) the (elastic) free energy W of the porous solid considered alone, that does not include the interfaces; (ii) the interface energy U (per unit of initial porous volume /0 dX0) of the interfaces between the constituents forming the porous material. We may assume that the interface energy does not significantly vary as the porous solid deforms and the saturations SJ are held constant. Accordingly, U is assumed not to depend upon the state variables eij, uL, uC associated with the deformation of the porous solid. In addition the dependence of the interface energy on the temperature is assumed to be negligible. We finally write Wsk ¼ p0 / þ W ðeij ; uL ; uC ; S L ; T Þ þ /0 U ðS L Þ

Substitution of (12) into state Eq. (11b) provides /0 ðpC  pL Þ ¼ 

oW dU  /0 : oS L dS L

In linear thermo-poroelasticity the energy W of the porous solid is a quadratic form of arguments eij, uL and of the temperature variation dT with respect to the initial temperature. The coefficients of the quadratic form are related to the current thermo-poroelastic properties and, thereby, depend on the current saturation SL. Accordingly, oW/oSJ is obtained by replacing in W the coefficients of the quadratic form by their derivative with respect to SJ. Under the assumption of infinitesimal transformations, oW/oSL is therefore a second order term, since it only involves quadratic terms such as eijekl, eijuJ, eijdT, uJuK, uJ, dT. As a result, neglecting oW/oSJ in (13), we get pC  p L ¼ 

dU : dS L

ð14Þ

According to (14) the infinitesimal interface energy dU per unit of porous volume /0 dX0 required to drain the infinitesimal fraction /0 dSC = /0 dSL of the initial porous volume dX0 is therefore identified to (pC  pL)dSL. This macroscopic interface energy balance can be illustrated at the microscopic level in the following way. Let h be the contact angle between the solid matrix and the wetting constituent, and let cJK be the J/K interface energy (with index S for the solid matrix). At the microscopic level the mechanical equilibrium of the L/C interface of mean curvature radius . is then governed by Laplace’s law pC  p L ¼

2cCL ; .

ð15Þ

while the equilibrium of the junction line between the three constituents (point A in Fig. 2) is governed by Young’s law cCS ¼ cLS þ cLC cos h:

ð16Þ

The pore radius of the cylindrical pore of Fig. 2 is r = . cos h, whose substitution in (15) provides 2cCL cos h : r Combining (17) and (16) we get pC  p L ¼

pC  p L ¼ 

2pr  dL  ðcCS  cLS Þ : pr2  dL

ð12Þ

where p0 stands for the reference atmospheric pressure.

ð13Þ

Fig. 2. Interface energy effects at the microscopic level.

ð17Þ

ð18Þ

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

The infinitesimal drainage sketched out in Fig. 2 will make the L/C interface cover the infinitesimal distance dL. The related contribution to the macroscopic drainage dSL is proportional to the denominator of the right hand side of (18), that is pr2 · dL, while the numerator, that is 2pr · dL · (cCS  cLS) is proportional to the overall interface energy change dU. Thereby, (18) is recognized as the microscopic counterpart of the macroscopic energy balance (14). In turn, according to state Eq. (14) the mechanical equilibrium of the L/C interface is captured at the macroscopic level in the form of a relation linking the saturation SL of the wetting constituent and the pressure difference pC  pL, so that we write S L ¼ sCL ðpC  pL Þ

ð19Þ

The next step is to determine the still unknown function sCL. The comparison of (17) and (19) shows the existence of a relation between the current wetting fluid saturation SL, and the entry radius r of pores still filled by the wetting fluid during the drainage process. In other words the pores having a smaller entry radius than r are still filled. We write S L ¼ SðrÞ

ð20Þ

where S(r) is the function   2cCL cos h SðrÞ ¼ sCL : r

ð21Þ

The function S(r) characterizes the pore size distribution, irrespective of the non-wetting and wetting fluids actually used for its experimental determination. In fact, the function 1  S(r) represents the cumulative pore volume fraction occupied by pores having a pore entry radius greater than r. In practice the function 1  S(r) is often determined from mercury porosimetry, where the invading non-wetting fluid is liquid mercury while the wetting fluid is mercury vapour. Fig. 3 shows such a determination after [9], for mor-

283

tar samples containing silica fume, small spheres (0.1 lm in diameter) of amorphous silica. Conversely, by means of (21), the function related to any association of a non-wetting fluid and a wetting fluid can be assessed from the function S(r) once and for all determined from mercury porosimetry. For instance, when the non-wetting fluid is a crystal, and the wetting fluid is an aqueous solution, we write   2cCL SL ¼ S : ð22Þ pC  pL 2.4. Linear thermo-poroelasticity Substitution of (12) into state Eqs. (11a) and (11c) provides rij ¼

oW oW oW oW : ; pL  p0 ¼ ; pC  p 0 ¼ ; S sk ¼  oeij ouL ouC oT ð23Þ

In (12) we assumed that U did not depend on uL and uC. If it was not so, state Eq. (11a) would have introduced a term depending on the interface energy U in the left hand side of Eq. (23) related to pJ  p0. This term would affect the pressure xJ finally applied on the internal walls of the solid matrix in contact with constituent J [14,15]. Actually, due to the energy cJS of the J/S interface the pressure xJ finally transmitted to the solid matrix by the constituent J is not equal to pJ. For instance, for the cylindrical pore represented in Fig. 2b, Laplace’s law gives c xJ ¼ pJ  JS : ð24Þ r The interface energy per unit of volume of cylindrical pore is UJ = 2pr · cJS/pr2 so that (24) can be rewritten in the form 1 xJ ¼ pJ  U J : 2

ð25Þ

Cumulative pore volume fraction 1−S (-)

1

The factor multiplying UJ is strongly dependent on the pore geometry. For a spherical pore the factor is 2/3. However, this rough estimation shows that the term to subtract to pJ in order to obtain the pressure actually transmitted to the solid matrix has the same order of magnitude as U. Accordingly, the fact that U does not significantly depend upon uC and uL can be experimentally checked by showing that U, as provided by the integration of (14) in the form Z r Z SL dSðqÞ ð26Þ U  U0 ¼  ðpC  pL ÞdS ¼ 2cCL q 1 1

0.9 silica fume 30%

0.8

silica fume 10% silica fume 0%

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

100

101

103

102

104

105

2 r (nm) Fig. 3. Pore size distribution for mortar samples with different silica fume contents (from [9]).

is actually negligible when compared to the magnitude of pressures pJ. This is assumed in what follows. Introducing the Legendre–Fenchel transform W * of W with respect to uJ, W  ¼ W  ðpL  p0 ÞuL  ðpC  p0 ÞuC ; state Eq. (23) are partially inverted in the form

ð27Þ

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O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

oW  ; oðpL  p0 Þ oW  oW  uC ¼  ; Rsk ¼  ; oðpC  p0 Þ oT

rij ¼

oW  ; oeij

uL ¼ 

ð28Þ

which are the generalized state equations of unsaturated thermo-poroelasticity. As already mentioned above, in linear poroelasticity W * is a quadratic form of its arguments, namely eij, pJ  p0 and dT. From now on let us restrict to isotropic linear poroelasticity. Letting then  = ekk be the volumetric dilation, and from now on omitting p0 for the sake of simplicity, W * is expressed in the form 1 W  ¼ ðK  2G=3Þ2 þ Geij eji  3aK  dT 2  X  pp  bJ pJ    3aJ pJ  dT þ J K ð29Þ 2N JK J;K¼L;C where the term depending only on dT has been ignored. Expression (29) and the first state equation in (28) provide rij ¼ ðK  2G=3Þdij þ 2Geij  bC pC dij  bL pL dij  3aKdT dij ; ð30Þ where properties K, G and a are respectively the bulk modulus, the shear modulus, and the thermal volumetric dilation coefficient of the true porous solid, that is the ones related to the empty porous material, with no interfaces and no pore pressure (pL = pC = 0); properties bJ are the generalized Biot coefficients [3]. Further, the second and the third equations of (28) gives p p uL ¼ bL  þ L þ C  3aL dT ; N LL N LC p p uC ¼ bC  þ L þ C  3aC dT ; ð31Þ N LC N CC where NJK are the generalized Biot coupling moduli, with NCL = NLC owing to Maxwell’s symmetry relations, while aJ is the coefficient related to the thermal dilation of the pore volume occupied by phase J. We will not explore the last state equation in (28) since the temperature will be imposed in the forthcoming applications. Current macroscopic poroelastic properties involved in constitutive Eqs. (30) and (31) depend upon the elastic properties of the matrix forming the solid part of the skeleton, and upon the current saturation degree SL = 1  SC, that delimits the current parts of the porous volume occupied by constituents L and C respectively. These poroelastic properties have to satisfy the standard meso–macro relations ensuring the compatibility of the macroscopic poroelastic constitutive Eqs. (30) and (31), with the underlying elastic properties of the solid matrix, that is the solid part of the porous solid. Letting ks and as be respectively the bulk modulus and the linear thermal dilation coefficient of the solid matrix, these relations are [14] K 1 1 bJ  / 0 S J ; þ ¼ ; ks N JJ N LC ks aJ ¼ as ðbJ  /0 S J Þ:

bC þ bL ¼ b ¼ 1  a ¼ as ;

ð32Þ

In view of the prediction of the mechanical behaviour of mortars exposed to freezing, let us now give some data related to these materials. For mortar samples a recent direct experimental estimation of K and ks has provided the values K = 17,700 MPa and ks = 42,400 MPa [44]. Accordingly, the first relation in (32) furnishes the estimation b = 0.58. This value agrees fairly well with the order of magnitude of a previous independent estimation b = 0.48  0.54, combining nanoindentation experiments and upscaling procedures [47]. We will adopt this value in the forthcoming applications, together with as = 18 · 106 K1 [33], and /0 = 0.34. With no further knowledge about the microscopic morphology of the porous space, the poroelastic properties bC, bL, NLL, NCC, NLC cannot be expressed as separate functions of the elastic properties of the solid matrix and of the saturation SL. Specific assumptions are needed. For instance, if the morphologies of the porous spaces respectively occupied by the constituents L and C are close enough to each other, they deform the same [8] and we may write     uL uC ¼ : ð33Þ /0 S L pJ ¼dT ¼0 /0 S C pJ ¼dT ¼0 Substitution of (33) in (31) leads then to the simple relation bJ ¼ bS J :

ð34Þ

3. Poromechanics of confined crystallization 3.1. The SL–DT curve The thermodynamic equilibrium between ice crystals and liquid water in contact requires their chemical potentials to be equal [26]. Restricting to the terms of higher order of magnitude, this condition implies (see for instance [42,32,15]) pC  pL ¼ Rm ðT m  T Þ;

ð35Þ

where Tm and Rm stand respectively for the melting temperature and the melting entropy (Tm . 273 K, Rm = 1.2 MPa K1 at 273 K [35]). According to relation (19), where the non-wetting constituent is identified to the crystal, the liquid saturation SL is a function of the pressure difference pC  pL. Substitution of (35) in (19) gives S L ¼ sCL ðRm DT Þ;

ð36Þ

where DT now denotes the cooling Tm  T. From now on Tm will be the reference temperature in (30) and (31) so that DT = dT. The relation (36) reveals that the liquid saturation SL, as a function of the current cooling DT, is an equilibrium state function. As such, the SL–DT curve can be determined from a direct experiment, exploiting for instance the huge contrast existing between the dielectric properties of ice and liquid water [25,23]. However, we will now examine how the SL–DT curve can be derived from the knowledge

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

of the pore size distribution. To this aim let first note that (17) and (35) combine to the celebrated Gibbs–Thomson law r¼

2cCL cos h ; Rm DT

ð37Þ

which furnishes the access radius r of the pore whose freezing is associated with the current cooling DT. As a result, the state function (36) can be explicitly determined as soon as the pore size distribution S(r) of the porous material is known. Actually, (20) and Gibbs–Thomson law (37) combine to provide the SL–DT curve in the form   2cCL cos h S L ¼ S L ðDT Þ ¼ S : ð38Þ Rm DT Adopting h = 0, cCL = 0.0409 J m2 [6], Rm = 1.2 MPa K1 at 273 K [35], Fig. 4 shows the determination of the SL–DT curve associated with the mortar samples with pore size distributions shown in Fig. 3. The liquid saturation of the 30% silica fume mortar drops dramatically in the range of 0 C to 2.5 C compared with the 0% and 10% silica fume mortars. Actually, in this temperature range the 30% silica fume mortar possesses a larger number of pores which can freeze. The reverse happens below 2.5 C. Based upon the SL–DT curve, a material characteristic cooling (DT)ch can be eventually introduced, that weights the cooling DT by the number of pores having the radius r associated with DT through Gibbs–Thomson law (37). This definition leads to express (DT)ch in the form ðDT Þch ¼

Z

DT ðrÞ  dSðrÞ:

ð39Þ

When using this definition, the values of the characteristic temperature associated with the mortars containing 0%, 10% and 30% silica fume of Figs. 3 and 4 are found to be respectively 6.7 C, 10.3 C and 13.5 C.

Let us consider an infinitesimal element dX0 of porous material whose porous volume is initially saturated by liquid water. This porous material is subjected to a progressive cooling under 0 C and we would like to express the resulting change of porosity. The current mass density qJ of constituent J is linked to pressure pJ (or more precisely the overpressure with regard to the atmospheric pressure p0), and to the current cooling DT through the linearized constitutive equation:   1 1 pJ ¼ 1  3aJ DT qJ q0J KJ

mW ¼ qL ð/0 S L þ uL Þ þ qC ð/0 S C þ uC Þ:

silica fume 10% silica fume 0%

0.8 0.7 0.6 0.5 0.4 0.3

Substituting poroelastic constitutive Eqs (31) and (40) in (41), we get mW ¼ q0L /0 þ q0L ðvDq þ vu Þ;

ð42Þ

where  vDq ¼

 q0C  1 /0 S C q0L

ð43Þ

and vu ¼ b þ

pL p þ C þ 3ð/0 S L aL þ /0 S C aC þ aL þ aC ÞDT ; ML MC ð44Þ

while MJ is defined by

0.2 0.1

ð41Þ

ð45Þ

The term vDq captures the pore volume change due to the change in ice saturation and to the related liquid-crystal mass density difference. The term vu relates to the water source term due to the deformation of the pore volume under the thermo-mechanical loading, whose above expression holds irrespective of the liquid-crystal phase tranformation. Using now the liquid-crystal equilibrium condition (35), we finally obtain ! X p L Rm vu ¼ b þ þ DT þ 3 /0 S J aJ þ aC þ aL DT ; M MC J¼L;C

silica fume 30%

0.9

ð40Þ

where KJ and aJ are respectively the bulk modulus and the thermal volumetric dilation coefficient of constituent J. The total mass of water mW currently contained in the porous material per unit of initial volume dX0 in both liquid and solid form is

1 1 1 / SJ ¼ þ þ 0 : M J N JJ N LC KJ

1

liquid saturation SL (-)

3.2. Cryo-change of porosity

S¼1

S¼0

285

ð46Þ 0

2

4

6

8

10

12

14

16

ΔT(K) Fig. 4. The SL–DT curve for the three mortars of Fig. 3.

18

where 1/M denotes 1 1 1 ¼ þ : M ML MC

ð47Þ

286

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

3.3. Cryodeformation and resistance to frost action

0.03 silica fume 30%

In a stress-free experiment (r = 0), the constitutive Eq. (30), relations (32) and equilibrium condition (35) combine to

vDq þ vu ¼ 0:

where

 /0 bM th ¼ 3 as þ ðS C aC þ S L aL  as Þ DT ; Ku   M bC bL  Rm ¼ Rm DT ; Ku ML MC   /0 bM q0C SC 1  0 ; Dq ¼ Ku qL

(%)

0.02

0.015

0.01

0.005

ð49Þ

Substitution of (43) and (46) in (49) provides an equation linking  and pL. Eliminating pL between the latter and (48), while using (32), we finally get  ¼ th þ Rm þ Dq ;

silica fume 10% silica fume 0%

Σm

bp þ ðbC Rm  3as KÞDT ¼ L : ð48Þ K In undrained conditions, where the liquid is prevented from escaping from the sample throughout the freezing process, the total water mass mW remains constantly equal to the initial water mass q0L /0 . From (42) it results

0.025

ð50Þ



ð51aÞ ð51bÞ ð51cÞ

where Ku denotes K + b2M and can be identified as the ‘undrained’ bulk modulus of a material whose porous volume would be saturated by a unique fictitious fluid having 1/(SL/KL + SC/KC) as bulk modulus. According to (50), the origin of the volumetric cryodeformation is triple:  The contribution th accounts for the combined thermal deformation of all the constituents. In what follows we will adopt the values 3aL = 286.3 · 106 K1 at 263 K for supercooled water [45], 3aC = 155 · 106 K1 at 263 K for ice crystal [29].  Nullifying with the entropy of melting Rm, the contribution Rm is the deformation induced by the micro-cryosuction process [15], that constantly adjusts the amount of frozen water in the frozen sites in order for the pressure difference pC  pL to meet at any time the liquidcrystal equilibrium condition (35). Use of (34) provides Rm in the form   Mb 1 1 Rm ¼  /0 S C S L  ð52Þ  Rm DT : Ku KL KC Since KC > KL (KL = 1.79 · 103 MPa at 263 K for supercooled water [45], and KC = 7.81 · 103 MPa at 263 K for ice crystal [28,29]), Rm is positive and contributes as a swelling to the overall volumetric strain . As a consequence, even for liquids which, unlike water, contract when solidifying, that is such as q0L < q0C , a freezing porous material can still exhibit a slight swelling, as it has been observed for cement pastes saturated by ben-

0

0

5

10

15

20

25

30

35

40

45

50

ΔT (K) Fig. 5. Contribution Rm to the overall volumetric cryodeformation  plotted against cooling DT.

zene [2]. In Fig. 5 we plotted the swelling Rm against the cooling for the three mortars of Figs. 3 and 4.  The contribution Dq is the expected cryo-swelling induced by the difference of density (q0C < q0L ) between liquid water and ice crystals. Its magnitude depends on the bulk moduli of liquid water and ice crystals, and on the skeleton poroelastic properties. These properties govern the internal pore pressures, and eventually the amount of liquid water that can actually freeze in the pores whose access radius is suited to the current temperature according to Gibbs–Thomson law (37). As a result the remainder of the liquid part, initially saturating the freezing sites at the onset of freezing, is expelled towards the still unfrozen part of the porous network, resulting in an increase of the liquid pressure pL. This is the so-called hydraulic effect. This effect has been early recognized as playing the main role in the swelling of freezing mortars [39,40]. Actually, in the undrained freezing of porous materials, the liquid pressure pL, which is responsible of the hydraulic contribution Dq, is much greater than the pressure difference pC  pL responsible of contribution Rm . This can be checked in Fig. 6, where we plotted the liquid and crystal pressures pL and pC against DT for the three mortars of the previous figures. This can be also checked by the inspection of the upper curves of Fig. 7 representing the total volumetric strain  as given by (50). Actually these curves show that the main contribution to  is Dq, since the thermal contribution th is negative, while  has an order of magnitude two times greater than the order of magnitude of Rm (compare the scales of the positive values of the vertical axis of Figs. 5 and 7). Experimentally, it is well established that the presence of silica fume in the cement paste modifies the frost resistance of the cement paste. Actually it is reported in [9] that, for mortar specimens prepared with a 0.6 water to cement and silica fume ratio, the presence of silica fume signifi-

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290 250 silica fume 30% silica fume 10%

200

silica fume 0%

MPa

150

100

pC pL

50

0 0

2

4

6

8

10

12

14

16

18

ΔT (K)

Fig. 6. Liquid pressure pL and crystal pressure pC plotted against cooling for the three mortars of Figs. 3–5.

cantly improved the frost performance. The reference sample (0% silica fume) resisted only 128 cycles of freezing and thawing (18 C to +5 C), whereas the samples containing 10% and 30% silica fume never failed even after 700 cycles. The average expansion per 100 thaw-freezing cycles experimentally observed for the samples were 0.912, 0.004, and 0.05% for the reference, 10% silica fume mortar, and 30% silica fume mortar, respectively. There are presently no quantitative explanations of the improvement of the frost resistance due to the presence of silica fume. The results obtained in Figs. 6 and 7 provide some first quantitative insights explaining this improvement. Actually, one of the main effects of the presence of silica fume is to affect the pore size distribution (see Fig. 3). The pore size distribution is the main input of the mechanical poroelastic model developed here with regard to the frost action. The pore pressures in 6, and

1.2 silica fume 30%

1

silica fume 10%

no void

silica fume 0%

0.8

287

the expansion in 7, predicted for the reference sample with no silica fume below 4 C are much higher than the ones predicted for the samples containing silica fume for the same range of temperatures. Therefore it is expected to have a more limited frost resistance, as it is experimentally observed, with the same order of magnitude for the predicted expansion and for the average experimental expansion per 100 thaw-freezing cycles. In the range of temperatures 4 C to 18 C, contrarily to the observations the sample containing 30% silica fume is predicted to exhibit a lower expansion than the 10% silica fume sample; however the reverse happens in the range of 0 C to 4 C, where the sample containing 30% exhibits a dramatic expansion increase with regard the two other samples owing to a larger number of pores which can freeze in this temperature range (see upper curves in Fig. 7). This may explain why, after 100 cycles, the average strain for the 10% silica fume mortar was smaller than for the 30% silica fume mortar. Actually, with regard to the initial pore size the pore size observed after cycling in [9] showed a more significant increase for the sample containing 30% silica fume than the other ones. However it must be kept in mind that the theoretical approach developed here has remained poroelastic. Actually it cannot capture the irreversible aspects associated with the progressive fissuration induced at each cycle. An irreversible analysis requires the development of irreversible constitutive equations for unsaturated porous solids [16]. This is left for further investigations. 3.4. Effects of air included and spacing factor In order to limit the liquid pressure build up caused by the hydraulic effect as the main contribution to the in-pore pressurization, good practice consists of including air-voids [39,40]. Actually, voids embedded in a saturated freezing porous solid act as expansion reservoirs, so that the pressurized liquid water expelled from the freezing sites can flow towards these zones at zero reference (atmospheric) pressure. At restored equilibrium the liquid pressure pL is zero everywhere so that the crystal pressure is pC = RmDT everywhere in order to meet the Gibbs–Thomson equilibrium condition (35). The resulting volumetric strain is provided by letting pL = 0 in (48). We get bC Rm 3as KDT :  K K

jpL ðRÞ¼0 ¼

0.4

The volumetric strain  results from the expansion provoked by the crystal pressurization (first term on the right hand side of (53)), in competition with the thermal contraction (second term on the right hand side of (53)). For the three mortars of Figs. 3–6, the volumetric strain is eventually a contraction as it has been observed by direct experiments [37], and as it is captured by the lower curves in Fig. 7 referred to by pL(R) = 0. Usual mortars could not resist against the pore pressures estimated in Fig. 6. Accordingly, the durability of concrete structures in cold climates

(%)

0.6

0.2

with void 0

-0.2

pL pC 0

2

4

6

8

10

12

14

16

18

ΔT(K) Fig. 7. Upper curves: volumetric strain as given by (50). Lower curves: volumetric strain in presence of an air void.

ð53Þ

288

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

does require the inclusion of voids, which are created by entrained air during the material manufacturing. However, when subjected to the frost action, mortars are not systematically destroyed in the absence of voids. Actually, cement-based materials are never fully water saturated and dispersed zones remain unsaturated. Although not as efficient as voids, these zones may play the role of possible expansion reservoirs, limiting the pore pressure build up induced by the liquid expulsion from the freezing sites. As the liquid enters the air void, it becomes no more confined so that it instantaneously freezes as it has again been observed by direct experiments [37]. Accordingly, the initial beneficial effect of a void acting as an expansion reservoir is enhanced by the cryo-suction the void further induces. Owing to the room offered by the large air void, the ice crystals so formed remain at zero pressure. The liquid within the shell, still confined but immediately in contact with the ice already crystallized on the air void surface, de-pressurizes. Actually, the confined liquid has to meet the liquid-crystal equilibrium condition (35), which requires pL = RmDT since pC = 0 in the void. The distant liquid water within the porous shell is finally driven to the void, which eventually acts as a cryo-pump. At equilibrium, the conditions pC = 0 and pL = RmDT prevail everywhere. Their substitution in (48) provides the resulting volumetric strain in the modified form jpC ðRÞ¼0 ¼ 

bL Rm 3as KDT :  K K

ð54Þ

The resulting overall strain is a volumetric contraction now provoked by the final liquid de-pressurization (first term on the right hand side of (54)), and the thermal contraction (second term on the right hand side of (54)). For the three mortars of Figs. 3–6, this contraction is captured by the lower curves in 7 referred to by pC(R) = 0. The final assessment (54) of the volumetric strain  assumes the thermodynamical equilibrium within the shell surrounding the void. In other words, the cooling rate is assumed to be infinitely slow when compared with the rate of the liquid water flowing towards the void. The characteristic time sDT, scaling the cooling rate with regard to the solidification of water within a porous material, can be assessed in the form

j D¼ gL



 b2 ð1 þ mÞ 1 1 / b  /0 þ ¼ 0þ ; 3Kð1  mÞ ML ML K L ks

ð57Þ

where m is the Poisson coefficient of the porous solid and j, its intrinsic permeability, while gL (.1.79 · 103 Pa s at 273 K [27,29]) is the liquid water viscosity. The equilibrium is achieved at any time if sw  sDT, providing the condition 

2

L  4D  ðDT Þch = DT :

ð58Þ

In practice, according to [34] the order of magnitude of in situ cooling rates does not exceed a few K h1, and we 

adopt the value DT ¼ 3 K h1 . For the three mortars analyzed above, we found that the order of magnitude of (DT)ch was 10 K. Furthermore, when adopting m = 0.3 and the poroelastic properties reported above, the hydraulic diffusivity takes the form D (m2 s1) = 2.7 · 1012j (m2). Use of these data in (58) provides the condition L2 ðm2 Þ  1:3  1017 j ðm2 Þ:

ð59Þ

The order of magnitude of the lowest values of the intrinsic permeability of high strength mortars is j . 1021 (m2) [1]. Even for these extreme permeability values, the condition (58) is eventually not restrictive since (59) provides L  1 cm, while the usual order of magnitude of the void radius is R . 50 lm [36]. A more severe condition governing the choice of the spacing factor L is that the void must be large enough in order to avoid any in-void pressure build up. The corresponding condition has to express that the void volume is capable to welcome the totality of the liquid water expelled, owing to the ice–liquid difference of density, from the freezing surrounding spherical shell with inner radius R and outer radius R + L/2. This condition provides " #  0  3 qL 4p L 4p 3 Rþ  1  /0 ½1  S L ðDT Þ   R < R3 : 3 2 3 q0C ð60Þ 12 silica fume 30% silica fume 10% silica fume 0%

10

sDT ¼ ðDT Þch = DT ;

ð55Þ

where (DT)ch is the characteristic cooling defined in (39), 

while DT is the imposed cooling rate. The characteristic time scaling the hydraulic diffusion of liquid water towards the void can be expressed in the form L2 ; ð56Þ 4D where L is the so-called spacing factor, defined in such a way that the distance between the centers of two neighbouring spherical voids of radius R is equal to 2R + L. The expression of the hydraulic diffusivity D is [14]

L /R (-)



8

6

sw ¼

4

0

2

4

6

8

10

12

ΔT (K) Fig. 8. Maximum spacing factor L with regard to the void radius R in order not to generate a final in-void pressure build up.

O. Coussy, P. Monteiro / Computers and Geotechnics 34 (2007) 279–290

In Fig. 8 we plotted L/R against DT for the three previous mortars. The values of the ratio L/R reported in Fig. 8 agrees with the order of magnitude of the experimental values [36] known to preserve the cement-based materials from bulk damage under freezing. For instance, adopting R = 50 lm, the results shown up in Fig. 8 provides the value L = 250 lm as a safe choice for the three mortars down to T = 12 C.

[2] [3] [4]

4. Concluding remarks [5]

This paper aimed at showing how poroelasticity allows a convenient approach to a first quantified understanding of the role of the pore size distribution in the deformation of porous materials subjected to the frost action. A poroplastic approach is left for a further investigation [17]. Besides the approach can apply to capture the overall mechanical effects associated with any crystallization process occuring within a porous material of given pore size distribution, and more generally with any phase transition. Actually, unsaturated poroelasticity can capture the deformation of a porous solid irrespective to the physical mechanism responsible of the pressure build up. The only law specific to the phase transition at hand is the law governing the thermodynamic equilibrium of the two phases of the same substance simultaneously present. Here, these two phases were ice and liquid water, whose thermodynamic equilibrium was governed by (35). In the drying phenomenon, where the two phases are the liquid water and its vapour, the relative humidity plays an analogous role to that played here by the temperature. Accordingly, (35) must be switched to the celebrated Kelvin’s law. The weathering phenomenon evoked in the introduction is caused by the sea salt crystallization under drying. In this more complex situation, two other phases of the same substance simultaneously present are the solute and the salt crystal. In this case, the supersaturation, that is the ratio of the current solute molar fraction x, and the one x0 related to the equilibrium at atmospheric pressure, plays an analogous role to that played by the temperature and the relative humidity. Accordingly, (35) must be switched to Correns’ law [10] RT x pC  pL ¼ ln ð61Þ tC x0 where pC and pL stand respectively for the salt crystal pressure and the solution pressure, while R is the constant of ideal gases, and tC is the crystal molar fraction. The analysis of stone weathering has been recently analysed along these lines [16].

[6] [7]

[8]

[9]

[10] [11] [12] [13]

[14] [15] [16] [17]

[18]

[19] [20] [21] [22] [23]

[24]

Acknowledgements

[25]

The authors acknowledges the support for this work of the Berkeley-France funds.

[26]

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