## Abstract

Expansive reactions damage porous materials through the formation of reaction products of a volume in excess of the available space left by the reactants and the natural porosity of the material. This leads to pressurizing the pore space accessible to the reaction products, which differs when the chemical reaction is through-solution or topochemical or both in nature. This paper investigates expansive reactions from a micromechanical point of view, which allows bridging the scale from the local chemo-mechanical mechanisms to the macroscopically observable stress-free expansion. In particular, the study of the effect of morphology of the pore space, in which the chemical expansion occurs locally, on the macroscopically observable expansion is the main focus of this paper. The first part revisits the through-solution and the topochemical reaction mechanism within the framework of micro–macro-homogenization theories, and the effect of the microscopic geometry of pores and microcracks in the solid matrix on the macroscopic chemical expansion is examined. The second part deals with the transition from a topochemical to a through-solution-like mechanism that occurs in a solid matrix with inclusions (cracks, pores) of different morphology.

## 1. Introduction

Expansive reactions in concrete structures are critical to reduced life-span expectations of concrete structures. These chemical reactions operate at a scale much below the macrolevel of material description, i.e. the typical size of laboratory test specimen. The mechanism by which expansive reactions damage concrete is through the formation of reaction products of a volume in excess of the available space left by the reactants and the natural porosity of the material. A typical example of deleterious expansive reactions is the alkali–silica reaction (ASR). The ASR deterioration can be attributed, on the micro-structural level, to the formation of a hydrophilic gel from reactive silica in the aggregates (S^{2+}), alkalis in the cement klinker (namely K^{+} and Na^{+}) and water in the concrete pore solution. Reactive silica is mainly provided by reactive aggregates, and alkalis by the cement klinker and other sources in the cementitious matrix. In the presence of water the gel swells, creating an increasing internal pressure in localized regions of the cementitious matrix, that induce deformation, and can initiate micro- to macro-cracking, excessive expansion, misalignment of the structure, etc. (see for instance Hobbs 1988; Poole 1992; West 1996). The typical time curve of stress-free (laboratory) expansion is *s*-shaped, as shown in figure 1, and consists of three more or less well-developed regimes, which can be described as initiation, development and rest (Courtier 1990). From a practical point of view, the most important periods are initiation and development, characterized, respectively, by the latency time *τ*_{L} and the characteristic time *τ*_{c} (Larive 1998), as indicated in figure 1.

There is an ongoing debate whether expansive reactions are topochemical or through-solution in nature. For instance, in the case of the ASR, it has been argued that the two time scales displayed in figure 1 are associated with two distinct micromechanisms (Larive 1998; Bažant & Steffens 2000; Ulm *et al*. 2000): the latency time *τ*_{L} with the dissolution of reactive silica from the aggregates; the characteristic time with the production of damaging gel probably through water inhibition in pores and microcracks of the cementitious matrix. The first mechanism is assumed topochemical in nature, occurring at the interface of the aggregate and alkaline solution, where hydroxyl ions in the alkaline pore solution attack the disordered or poorly crystallized silica network of reactive aggregates. From a mechanical point of view, this localized mechanism would pressurize in a non-uniform way the aggregate–cement paste interface, also known as Interfacial Transition Zone (or Aureole de Transition), which is the weakest link in concrete materials. In turn, the second mechanism, associated with the characteristic time *τ*_{c} seems rather through-solution in nature (Dron & Brivot 1992*a*,*b*). From a mechanical point of view, such a through-solution mechanism would imply a rather uniform pressure build-up in the connected porosity of the cementitious matrix. As for the macroscopic expansion, there seems evidence that both mechanisms are of importance, depending on the local calcium concentration in the pore solution, but also on the size of pre-existing cracks, flaws and pores, in which the reaction products can freely expand. However, little is known how these micromechanisms, whether topochemical or through-solution, relate—through the microscopic morphology of the matter constituting concrete—to the macroscopically observable expansion.

It is the purpose of this paper to investigate stress-free expansive reactions from a micromechanical point of view which allows bridging the scale from the local chemo-mechanical mechanisms to the macroscopically observable expansion. Developed around the topic of ASR-expansion, the paper is composed of two parts: the first part revisits the through-solution and the topochemical reaction mechanism from the point of view of ‘continuum micromechanics’ (*cf*. Suquet 1997), and the effect of the microscopic geometry of pores and microcracks in the solid matrix on the macroscopic chemical expansion will be examined. The second part deals with the transition from a topochemical to a through-solution mechanism that occurs in a solid matrix with inclusions (cracks, pores) of different morphology.

## 2. Micromechanics of through-solution and topochemical expansion

We consider a representative elementary volume (typical size of laboratory test specimen) composed of a solid matrix and pore space saturated by a fluid phase. The overall volume in the current configuration is noted *Ω*, and *Ω*_{0} is the initial volume. *ϕ* is the Lagrangian porosity, which represents the volume fraction occupied by the fluid phase with regard to the initial volume; *ϕ*_{0} is the initial porosity.

At a level below, i.e. the typical size of the reaction sites, we consider a reaction product, to which we refer as gel in what follows, and which progressively fills the pore space in the vicinity of the reaction site. Once the free expansion space is filled, the continuous gel formation leads to pressurizing the pore space. We will note the isotropic stress generated within the gel; and the state equation that specifies the constitutive behaviour of the gel will be a relation, which relates the gel pressure *p*^{g} to the volume mass density *ρ*^{g}(2.1)For instance, a linear state equation reads(2.2)where *K*^{g} is the gel compressibility and is the initial gel density. What we aim to derive is a relation that links the gel pressure build-up at the microscopic scale characterized by the gel state equation (2.1) to the macroscopic deformation, noted , that is recorded in a stress-free expansion experiment. To this end, in a first approach, we will consider the following assumptions:

The gel mass formed is considered to be a given function of time

*t*in the problem, quantified by the volume mass*m*^{g}(*t*) of the gel (of dimension [*m*^{g}]=L^{−3}M) in the elementary volume*Ω*, such that*m*^{g}(*t*)*Ω*_{0}is the gel mass which is contained at time*t*in*Ω*. The gel mass can be assessed either experimentally (see for instance Larive 1998), or by advanced micro-models of the reaction kinetic, in which the chemo-physical processes involved in the ASR are considered (see for instance Xi*et al*. 1999; Bažant & Steffens 2000). In turn, considering*m*^{g}(*t*) an input to the problem comes to assume that deformation and stress do not significantly affect the product formation. This seems consistent with experimental evidence highlighted by Larive (1998): macroscopic stress application does little affect the chemical product formation.The solid matrix surrounding the pore space behaves linear elastic, and

**c**^{s}denotes the elasticity tensor that characterizes this microelastic behaviour. Hence, the model will focus on the chemoelastic micro–macro-relations that link the chemical product formation at the microlevel to the macroscopic strain recorded in a stress-free laboratory expansion test. This gives a first estimate, where and how irreversible skeleton deformation induced by an internal pressure excess is to be expected.

With these assumptions, we will distinguish two cases: (*a*) the gel is uniformly distributed in the entire connected porosity of the elementary volume; (*b*) the pressure is not uniformly distributed in the matrix, but the gel localizes in cracks having one or several preferential directions within the bulk material. As we will see, the first case is representative of a through-solution mechanism, while the second one is rather associated with a topochemical mechanism at the reaction sites.

### (a) Uniform gel pressure distribution

For the gel pressure to be uniformly distributed in the material bulk, the reaction must be through-solution in nature. For instance, in the case of ASR, it has been argued that the reaction products diffuse from the reaction site at the aggregate–pore solution interface through the connected porosity. From a mechanical point of view, the uniform distribution of the reaction products in the material bulk suggests a poromechanics approach, following Biot's theory, in which the gel saturating the connected pore space exerts a pressure on the matrix. For the assumed elastic behaviour of the solid matrix, the poroelastic state equations read(2.3)(2.4)where is the macroscopic stress tensor, related by equilibrium to external surface forces; **C**^{hom} is the drained macroscopic elasticity tensor. The second-order tensor of Biot coefficient and the Biot modulus *M* characterize the overall poroelastic material behaviour, and are related to the mesoscopic material property of the solid matrix by (Chateau & Dormieux 1998)(2.5)(2.6)where and * I* denote, respectively, the second and fourth-order unit tensor. In the case of an isotropic behaviour of both the solid matrix and the porous skeleton, the previous equations take the well-known form (e.g. Coussy 1995)(2.7)where

*K*

^{hom}is the overall drained bulk modulus and

*K*

^{s}the bulk modulus of the matrix.

In a stress-free experiment, , from state equation (2.3) with equation (2.5), we obtain(2.8)and in the isotropic case(2.9)Furthermore, if we assume that the entire connected pore space is permanently saturated by the gel, porosity *ϕ* and the volume mass *m*^{g} are related by(2.10)Then, use of equations (2.8) and (2.10) in equation (2.4) gives(2.11)and in the isotropic case(2.12)In the case of an incompressible gel (i.e. ), equation (2.11) gives access to the gel pressure *p*^{g} that develops in the connected pore space surrounded by a deformable solid. Use of this pressure in equation (2.8) yields a linear relation between the gel mass increase and the macroscopic observable expansion(2.13)Equation (2.13) is precisely of the form postulated in macroscopic chemoelastic models, in which a dimensionless gel mass increase *μ*^{g}(*t*) is related by means of a linear operator, the tensor of chemoelastic expansion coefficients , to the macroscopic observable expansion (see for instance, Coussy 1995; Huang & Pietruszczak 1996; Larive 1998; Ulm *et al*. 2000). However, in addition to these macroscopic approaches, the chosen approach provides a link with the microscopic properties of the matter(2.14)In the case of an isotropic matrix behaviour, reduces to(2.15)Hence, in the case of an incompressible gel, the chemoelastic coupling, in fully-saturated conditions, is only affected by the initial porosity *ϕ*_{0} and the Biot coefficient *b*, i.e. the bulk-modulus ratio *K*^{hom}/*K*_{s}. If, for purpose of argument, the solid were also incompressible (i.e. *K*_{s}→∞, *b*=1), the volume variation of the material would result only from the dimensionless mass gel increase, i.e. . In turn, typical values of concrete are *b*=0.2–0.25 and *ϕ*_{0}=0.1, which roughly gives . These results are consistent with the results obtained from the classical Hashin–Sthrikman approach, for which(2.16)Use in equation (2.15) gives(2.17)For an incompressible solid (*ν*^{s}=0.5), it is *X*^{s}=0 and *T*=1/3. For a Poisson ratio *ν*^{s}=0.3, it is *X*^{s}=0.6, and for the typical concrete porosity *ϕ*_{0}=0.1, we obtain *T*≈1/2. Moreover, for a typical asymptotic value of ASR-expansion of (Larive 1998), the model predicts a dimensionless gel mass increase on the order of , where *ϕ*_{∞} is the asymptotic connected porosity. In addition, equation (2.12) provides an assessment of the asymptotic pore pressure of the gel confined in an elastic matrix(2.18)The order of magnitude of the pressure must be seen in relation with the assumptions of the model, i.e. the assumed elastic behaviour of the solid phase and the assumed incompressibility of the gel; but reveals that a small change in porosity suffices to crack the solid matrix.

In return, in the case of a compressible gel , substitution of the gel state equation (2.1) into equation (2.11) yields a nonlinear relation between *p*^{g} and *μ*^{g}, respectively—according to equation (2.8)—between and *μ*^{g}. For instance, for the linear state equation (2.2), equation (2.11) takes the form(2.19)whereis the dimensionless compressibility factor. Solving this equation for *p*^{g} yields(2.20)In the isotropic case, we obtain by substituting equation (2.20) into equation (2.9)(2.21)whereFigure 2 illustrates this effect of the compressibility on the gel pressure *p*^{g} in a plot of the normalized pressure *p*^{g}/*K*^{g} and *p*^{g}/*K*^{hom} versus compressibility factor *k* for a fixed value of *μ*^{g}=0.25% and typical values of concrete properties (*b*=0.25, *ϕ*_{0}=0.1). The gel can be considered as incompressible for values of *k*≥2 (i.e. roughly *K*^{g}/*K*^{hom}≳10). In turn, for a highly compressible gel, for which *k*≪1, equation (2.19) yields (see also figure 2)(2.22)In this case, use of equation (2.22) in equation (2.8) yields again a linear relation between strain and gel mass increase *μ*^{g}. In the isotropic case, this relation reads(2.23)In summary, when the gel pressure is distributed uniformly throughout the connected porosity, associated with a through-solution mechanism, the governing parameters that determine the isotropic chemoelastic behaviour are the initial porosity *ϕ*_{0}, the Biot coefficient *b*=1−*K*^{hom}/*K*_{s} and the gel compressibility *K*^{g}/*K*^{hom}.

Finally, for purpose of illustration, consider a simplified mass formation kinetics, with a constant formation rate for times 0<*t*<*t*_{c}, and a zero formation rate beyond *t*>*t*_{c}. For such a simplified reaction kinetics, we can distinguish three phases of macroscopic expansion (see figure 3):

A filling phase of the connected porosity around the reaction sites. The pressure during this filling phase is zero, and hence no macroscopic expansion is observed.

Pressurizing of the connected porosity: once the connected porosity is filled, the continuous gel formation leads to a pressure build-up in the connected pore space. In the limit cases of an incompressible or a highly compressible reaction product, equations (2.13), (2.18) and (2.23) indicate that the macroscopic expansion is proportional to the dimensionless gel mass formation

*μ*^{g}in excess of the mass required to fill the initial porosity.Beyond

*t*_{c}, there is no supplementary gel supply. The pressure remains constant, and the macroscopic expansion stops.

The three expansion phases are compatible with the typical *s*-curve of macroscopic expansion (see figure 1). The non-zero expansion during the initiation period can be associated with a filling phase, during which pressurizing starts before complete saturation. The development phase is associated with an increasing pressure build-up due to continuous gel formation. Once the reaction stops, the deformation stops; and chemical equilibrium, therefore, explains here the asymptotic nature (‘rest’) of the chemically induced mechanical expansion.

### (b) Non-uniform gel pressure distribution

A non-uniform gel pressure distribution may occur, when the reaction products expand in microcracks and (occluded) pores in the close surrounding of the reaction sites. In contrast to the uniform gel pressure distribution associated with a through-solution mechanism, a non-uniform gel pressure distribution is therefore rather characteristic of a topochemical reaction. This topochemical mechanism is the focus of this section. For purpose of analysis we will assume the gel incompressible, . Furthermore, the expansion sites will be associated with ellipsoidal cracks, characterized by the aspect ratio *ω*, and which have a rotational symmetry.

#### (i) One single crack family

We first consider a single crack family. By this we mean one type of cracks characterized by the same aspect ratio *ω*=*b*/*a* (2*a*=crack diameter in the -plane; 2*b*=crack opening in the -direction; crack volume , and the same orientation that coincides with the direction of the axis of rotational symmetry parallel to the sample axis (see figure 4). This situation idealizes a current situation encountered in laboratory tests: in the cast direction of a sample, water is trapped below the aggregates, and this space is a preferential place where ASR products expand (Larive 1998). If we note *N* the crack density (number of cracks per macroscopic volume) and *ϵ*=*Na*^{3} the crack density parameter introduced by Budiansky & O'Connell (1976), the volume fraction of cracks (volume of cracks over initial volume), or the (Lagrangian) crack porosity is given by(2.24)If we assume that the gel pressure is the same in all cracks (i.e. in crack porosity *ϕ*_{c}), it intuitively appears that the results developed in §2*a* apply, when replacing the solid elasticity tensor **c**^{s} by the elasticity properties that characterize the ‘sane’ material surrounding the cracks. Note that in general , since the sane material may be composed of a solid matrix and a connected pore space, whose characteristic size is at least one order smaller than one of the cracks, which is not considered to be filled by the reaction products. Our argument is a micromechanics one, and will confirm this intuitive reasoning by means of a micro–macro-averaging procedure.

Considering the cracks as inhomogeneities of ellipsoidal shape, the (drained) macroscopic elasticity of the overall material, characterized by **C**^{hom}, and the elasticity of the sane material, characterized by , are related by(2.25)* A* is the fourth-order concentration tensor, and denotes the volume average of

*over all cracks contained in*

**A***Ω*(domain

*Ω*

_{c}).

What we aim to derive is a relation that links the gel pressure *p*^{g}, the macroscopic strain , and the local strain in the cracks. In the line of micromechanical arguments developed by Eshelby (1957), represents the local strain in a crack saturated by a gel, which is caused by a gel pressure *p*^{g} and a macroscopic strain applied at infinity. Provided that the interaction between cracks is negligible (i.e. what is referred to as a diluted scheme), application of Eshelby's solution to this inhomogeneity problem yields(2.26)**S**_{c} is the Eshelby tensor, which depends on , the orientation and the aspect ratio of the cracks. The average 〈* A*〉

_{c}of the strain concentration tensor

*over all cracks contained in*

**A***Ω*can be estimated by (Zaoui 1997)(2.27)In addition, following transformation field analysis (Levin 1967; Dvorak & Benveniste 1992; for a comprehensive review see e.g. Dvorak & Bahei-El-Din 1997), the macroscopic stress is related to the macroscopic strain, and the uniformly distributed pore pressure in the crack porosity by(2.28)Use of equation (2.27) in equation (2.26) yields(2.29)The local volume strain in the cracks is derived from (2.26)(2.30)One relation is still missing: analogously to equation (2.10), we have to relate the gel mass increase to the local volume variation. The crack volume change is equal to . Therefore, for an incompressible gel, we obtain(2.31)Finally, for a stress-free experiment, equations (2.28), (2.30) and (2.31) yield a linear relation between the macroscopic strain and the relative gel mass increase(2.32)where is the chemoelastic coupling tensor(2.33)This confirms that the chemoelastic solution can be derived within Biot's framework of poroelasticity of a porous medium composed of the sane porous material as matrix, and the oriented crack system as pore space. The micromechanical approach through equations (2.25), (2.29) and (2.30), however, gives a precise estimate of the chemoelastic coupling tensor for the considered topochemical process in one crack family, with the only knowledge of Eshelby's tensor

**S**_{c}for ellipsoidal inclusions in an elastic matrix. The tensor of chemoelastic expansion coefficients depends on the elasticity of the sane porous material , the aspect ratio

*ω*, the orientation of the cracks, and the crack density parameter

*ϵ*=

*Na*

^{3}.

By way of illustration, consider the expansion coefficient , which relates the gel mass formation rate to the strain rate in the -direction coinciding with the direction of the axis of rotational symmetry of the cracks(2.34)Figure 5 shows versus the crack aspect ratio *ω* for different values of crack density *ϵ*=*Na*^{3}. The results were obtained by considering an isotropic behaviour of the sane porous material, for which affects through only Poisson's ratio *ν*_{0}. The results shown in figure 5 were obtained with *ν*_{0}=0.3.(2.35)As expected, the higher the crack density, the lower the chemical expansion coefficient, but not necessarily the macroscopic expansion. In fact, if the number of cracks is associated with the number of reaction sites, the effect of *ϵ* on the expansion coefficient can be counterbalanced by an increase in mass production. Conversely, if the number of reaction sites is fixed, and hence the crack density independent of these reaction sites, both the chemical expansion coefficient and the macroscopic expansion are reduced as *ϵ* increases. The same effect is found as the aspect ratio *ω* increases: the flatter the cracks, the greater the crack surfaces oriented in the -direction, which are exposed to the gel pressure, and hence the more pronounced the anisotropy of the expansion. In turn, the more circular the crack shape, the more ‘isotropic’ becomes the overall volumetric swelling, which becomes isotropic for *ω*=1 corresponding to spherical pores of crack porosity *ϕ*_{c}=4*πϵ*/3.

Finally, a comparison of the expansion caused, respectively, by the trans-solution mechanism (i.e. equations (2.13) and (2.14)) and by the topochemical mechanism (i.e. equation (2.33)) is due. An obvious difference is the chemoelastic coupling tensor. However, with the order of magnitude of *T*=*T*_{33}≈0.5 for the through-solution mechanism, we find that is on the order of 0.5–1 in the range *ϵ*∈[0.01,0.5]. Hence, despite the difference in morphology of the two mechanisms (one occurring in the connected pore space, the other in the pore space of the interface transition zone), the expansion rates are not significantly different for an identical gel mass formation rate. The main difference—beside the anisotropy induced by the crack orientation—between the trans-solution mechanism and the topochemical mechanism is the initial porosity, *ϕ*_{0} and *ϕ*_{0,c}, respectively, which constitutes the pressure-free expansion space, and which is first filled before macroscopic expansion occurs. In the topochemical mechanism, the initial porosity surrounding the reactive aggregates, *ϕ*_{0,c}, is a fraction of the overall initial porosity *ϕ*_{0} (typically *ϕ*_{0,c}=1%; *ϕ*_{0}=10%). Hence, the pressure-free expansion space of the trans-solution mechanism is roughly 10 times larger than the pressure-free expansion space in a topochemical process, and a macroscopic expansion will only occur after a gel mass formation of in the through-solution mechanism, and in the topochemical reaction occurred. Consequently, for the same gel mass formation *m*_{g}(*t*), the expansion caused by *μ*^{g} according to (2.13), and according to equation (2.32), will occur the later (*t*_{0}≥*t*_{0,c}) and the less intense , the higher the initial porosity (*ϕ*_{0}≥*ϕ*_{0,c}). This is sketched in figure 6. A comparison of this figure with the typical expansion curve of ASR (figure 1) indicates that this initial porosity is a key factor for evaluating the latency time *τ*_{L} of expansive reactions.

#### (ii) Extension to multiple crack families

For purpose of completeness, the previous model of a topochemical reaction that occurs in a single crack family is extended to the case of multiple crack populations, each one characterized by a particular aspect ratio *ω*_{I}, and a specific orientation . The underlying micro–macro-relation between the local strain and the gel pressure prevailing in the crack is given by equation (2.26). Application of a dilute scheme (i.e. negligible interaction between different cracks) yields(2.36)where **S**_{I} is Eshelby's tensor for crack family *I*. Proceeding as before, using Levin's formula for the stress-free expansion yields(2.37)where is the crack porosity of the *I*th crack family of crack density *N*_{I}, aspect ratio *ω*_{I} and crack radius *a*_{I}. For small variations of crack radius *a*_{I} with regard to the average crack radius *a* of all crack families, *ϕ*_{I} can be approximated by(2.38)Finally with equation (2.38), the analogous expression to equation (2.31) is obtained in the form(2.39)Expression (2.39), which establishes the link between the gel mass increase and the local volume variation , is based on the assumption that the gel mass, which forms at the aggregate–cement paste interface, is proportional to this interface surface, i.e. . This comes to assume that the gel mass formed per unit of crack is the same *m*^{g}/*N* irrespective of the aspect ratio *ω*_{I}. Equations (2.36), (2.37) and (2.39) allow the determination of the tensor of chemoelastic expansion coefficients , which relates the macroscopic expansion and the gel mass increase for multiple crack families of different aspect ratios *ω*_{I} and crack orientation.

## 3. Transition from a topochemical to a through-solution expansion

The two cases considered above are limit cases of chemoelastic expansion: one related to a topochemical mechanism, the other to a through-solution mechanism. No doubt, the reality of expansive reactions is expected to be controlled by a progressive transition from a topochemical mechanism in the interface transition zone surrounding the reaction sites, to a through-solution mechanism in the connected pore space. This filling process is the focus of this section.

### (a) Filling of two pore spaces of same morphology

We consider two types of pores, the macropores (characteristic size *D*) and the micropores (characteristic size *d*), of porosity *ϕ*_{1} and *ϕ*_{2}, which constitute the connected porosity of the material:(3.1)In the case of the ASR, the first can be associated with the large pores in the aggregate interface transition zone, and the second with the micropores in the cementitious matrix. The following filling process is considered: first, the macropores are filled. This filling process of the macroporosity *ϕ*_{1,0} occurs pressure free. Once *ϕ*_{1,0} is filled, the continuous gel mass formation leads to a pressure build-up in the macropores. When the gel pressure reaches a threshold swelling pressure *p*_{s}, gel mass is squeezed into the smaller micropores. This threshold pressure can be associated with some specific access radius of the smaller micropores, similar e.g. to a capillary barrier and associated surface tensions, that control the value of the threshold pressure. It may as well represent the threshold pressure at which the matrix cracks under internal pressure, creating a large additional expansion space for the reaction products. This filling process is sketched in figure 7. During this filling process, we distinguish the micropores filled by reaction products (porosity *ϕ*_{r}), and the unfilled part (porosity *ϕ*_{v})(3.2)Once the micropores are filled, the increasing pressure caused by continuous gel formation may lead to micro- to macro-cracking of the surrounding solid material. These irreversible skeleton deformations go beyond the scope of the model presented below, which is dedicated to the chemoelastic regime. As before, we will assume that the surrounding solid material behaves elastic, and the reaction products are assumed incompressible . In addition, in a first approach, we will focus only on spherical pores (*ω*≡1), i.e. a microstructure of same morphology.

For the filling process at hand, macroscopic deformation will occur once the initial porosity *ϕ*_{1,0} is filled. Below the threshold pressure *p*_{s}, the reaction is topochemical, and the results developed in §2*a* apply, i.e.(3.3) represents the dimensionless gel mass in excess of the initial mass filling in a pressure-free manner the initial porosity *ϕ*_{1,0}(3.4) is the chemoelastic coupling tensor defined by(3.5)Here, **C**^{hom} is the elasticity tensor of the overall porous medium composed of the solid matrix and the total porosity *ϕ*=*ϕ*_{1}+*ϕ*_{2}; while represents the elasticity of the matter composed of the same solid matrix, and the partial porosity *ϕ*_{2}. Furthermore, the Biot model parameters read here(3.6)where **S**_{1} is Eshelby's tensor restricted to the specific pore morphology of porosity *ϕ*_{1} and to the reference elasticity . In the case of spherical pores, equations (3.5) and (3.6) simplify to(3.7)(3.8)Then, the first phase is characterized by the following relations linking and *p*^{g} to the dimensionless mass formation (3.9)Once the pressure in the macropores (porosity *ϕ*_{1}) reaches the threshold pressure *p*^{g}=*p*_{s}, the increase in gel mass formation leads to a filling of the microporosity *ϕ*_{2}. The gel volume corresponding to the threshold pressure follows from equation (3.9)(3.10)During this filling process the pressure remains constant, *p*^{g}=*p*_{s}, and appears as a microscopic (or local) eigenstress in the filled pore space(3.11)The connection between the local eigenstress, , and the overall eigenstress follows from Levin's formula(3.12)where 〈.〉_{Ω} and 〈.〉_{α} (*α*=1,*r*) denote the volume averages over, respectively, the entire domain *Ω* and the part of the total porosity effectively filled by the gel products; * A* is still the strain concentration tensor. In addition, in the case of only one pore morphology that represents the ‘natural’ porosity (supscript ‘

*p*’) of the material, there is no morphological difference between the domains defined by

*ϕ*

_{1}or

*ϕ*

_{2}(≡

*ϕ*

_{r},

*ϕ*

_{v}). An immediate consequence of this single morphology assumption concerns the volume average of the concentration tensor(3.13)Hence, use of equation (3.13) in equation (3.12) yields(3.14)where(3.15)The macroscopic state equation thus reads(3.16)The result is remarkable, as it formally identifies the macroscopic stress quantity, , as the effective stress in the sense of Bishop's theory of non-saturated porous media (Bishop & Blight 1963), which is related through a saturation degree and the tensor of Biot coefficients , to the local swelling pressure prevailing in the porosity means that the internal pore faces are stress free, and this is associated with a zero saturation. In turn,

*S*

^{r}=1 corresponds to a complete saturation of the connected porosity by the reaction products. In this case, the gel pressure is uniformly distributed throughout the entire connected porosity.

With equation (3.16) in hand, the micro–macro-approach is (almost) the same as previously applied:

The state equation of stress-free expansion reads(3.17)where

**C**^{hom}is the drained stiffness of the porous medium comprising both pore spaces (porosity*ϕ*)(3.18)**S**_{p}is Eshelby's tensor for the single morphology of the porosity.Application of a dilute estimation scheme allows relating the local strain to and

*p*_{s}(3.19)or equivalently using equation (3.13)(3.20)The local volume strain is related to the gel mass formation by(3.21)where

*ϕ*_{r,0}represents the reference porosity of the micropores in a relaxed (pressure free) configuration.

Finally, by way of application, consider again the simplified mass formation kinetics, with a constant formation rate for times 0<*t*<*t*_{c}, and a zero formation rate beyond *t*>*t*_{c}. For this simplified reaction kinetics, we now distinguish four phases of macroscopic deformation related to the filling of the two pore spaces of same spherical morphology (see figure 8):

The pressure-free filling of the macroporosity

*ϕ*_{1,0}The pressure build-up in the macropores

*p*^{g}and*E*^{g}are given by (3.9). This phase starts at time .The filling process of the micropores at constant pressure

*p*^{g}=*p*_{s}For spherical pores is given by equation (3.16), (3.17) and (3.21). This phase starts at timeEnd of gel mass supply (rest)

Figure 8 is to be compared with figure 3. Initiation and rest defined by horizontal lines are similar, the first corresponding to the pressure-free filling phase of the macroporosity *ϕ*_{1,0}, the second to the end of gel mass supply (chemical equilibrium). In turn, as a result of the threshold pressure and the associated filling process of the micropores *ϕ*_{2}, the development phase is now characterized by a change of concavity of the macroscopic expansion curve. This change of concavity is the more pronounced, and occurs the earlier (time *t*_{s}), the lower the threshold pressure. This is also shown in figure 8.

### (b) Double pore morphology

Finally, we will consider two types of pore morphology: the natural spherical porosity of the concrete, noted *ϕ*_{p}, and the crack porosity, noted *ϕ*_{c}. These two pore spaces are connected, and the total porosity reads(3.22)The crack porosity is assumed to be situated around the reaction site, and is first filled by a topochemical process. Below the pressure threshold we have analogous to equation (3.3)(3.23)where is given by equation (2.33), and is the dimensionless gel mass in excess of *ϕ*_{c,0}(3.24)When the gel pressure *p*^{g} reaches the pressure threshold *p*_{s}, the porosity in the matrix is filled at constant pressure *p*^{g}=*p*_{s}. For this microstructure characterized by two morphologies, the tensor of drained elastic properties reads(3.25)where we made use of 〈* A*〉=

*. Furthermore, equations (2.5), (3.15) and (3.25) suggest the following form of the tensor of Biot coefficients for the double pore morphology(3.26)Proceeding as in §3*

**I***a*, the macroscopic eigenstress, corresponding to an effective stress, is obtained in the form(3.27)where

*ϕ*

_{r}and

*ϕ*

_{v}represent, respectively, the filled and the unfilled part of porosity

*ϕ*

_{p}=

*ϕ*

_{r}+

*ϕ*

_{v}. For a gel saturated pore space,

*ϕ*

_{v}=0 and

*ϕ*

_{r}=

*ϕ*

_{p}, and equation (3.27) reduces to equation (2.3). In the same way, for 〈

*〉*

**A**_{c}=〈

*〉*

**A**_{p}, which corresponds to a single morphology, equation (3.27) with equation (3.26) reduces to equation (3.14). Hence, the micro–macro-relation between the local gel expansion and the macroscopic deformation in a stress-free experiment is obtained from:

The macroscopic stress-free state equation (3.17), which reads here(3.28)with

**C**^{hom}given by equation (3.25).The application of a dilute estimation scheme for the local gel strain in the saturated crack porosity

*ϕ*_{c}(3.29)and for the local strain in the partially saturated natural porosity of spherical form according to equation (3.19)(3.30)The relation between the volume variations in the two pore types, and the gel mass formation, which generalizes equation (3.17) to the case of a double pore morphology(3.31)

Solution of this system of four equations (3.28)–(3.31) yields the relation between the gel mass formation and the macroscopic observable stress-free expansion .

## 4. Conclusions

The micromechanics investigation of expansive reactions reveals some connections between micromechanical properties, morphology and macroscopic stress-free expansion of concrete materials:

The compressibility of the reaction products plays a non-negligible role on the mechanism that damages concrete materials due to chemical expansion. For typical concretes, beyond a compressibility modulus of the reaction products of

*K*^{g}>100 GPa, the reaction products can be considered as incompressible. This is typically the case for steel corrosion products, that develop at the steel–concrete interface due to an electrochemical reaction. Conversely, ice that expands in the pore space of cementitious materials during freeze–thaw cycles, which are known to impair a lot of damage in concrete structures, should rather be considered as compressible (*K*^{g}=10 GPa).At present, no information is available about the ASR-gel compressibility

*K*^{g}. If the ASR-gel was incompressible, it would be expected that the ASR-expansion is primarily a result of microcracking induced by the high pressures that build up in the pore space of the material. For a given number of reaction sites this macroscopic expansion due to gel production is the lower the more the material is cracked. Derived here for the specific case of cracks surrounding the reaction sites, the same is expected to hold for a crack network that is progressively filled when a threshold pressure is reached. In this case, the threshold pressure can be associated with the pressure at which cracks propagate in the cementitious matrix, and could explain the stress-induced anisotropy of ASR-expansion which was found experimentally (Larive 1998). This fracture mechanics problem at the microlevel of the crack network goes beyond the scope of the framework of continuum micromechanics developed in this paper.On first sight, the characteristic

*s*-shaped ‘free’ expansion curve, within the chemoelastic context, seems independent of the specific nature of the chemical reaction and associated locus of occurrence, whether through-solution (TS; see figure 3), topochemical (TP) or both, and with or without a threshold pressure (see figure 8). It is representative of an expansive reaction in a deformable medium characterized by two horizontal asymptotes, one representing a pressure-free filling phase, the other a state of chemical equilibrium at which the reaction stops. Beyond this apparent similarity, the micromechanical analysis, however, reveals some fundamental differences:For a given reaction kinetics, the initiation time

*t*_{i}, that is the duration of the initial filling process of the pores and cracks accessible to the reaction products, is scaled by the volume that is first filled before macroscopic deformation occurs. Therefore, the ratio of initiation times is equal to the pore volume ratio accessible to the gel.The characteristic time of the change of concavity of the

*s*-curve, which characterizes the expansion development, depends on the filling process: in the case of a through-solution mechanism or a topochemical reaction without pressure threshold, the characteristic time is given by the reaction kinetics. In other words, it is of pure chemical origin. In return, when a transition of a topochemical reaction to a through-solution mechanism takes place, the characteristic time is controlled by the threshold pressure (often of non-chemical origin) and by the micromechanical properties of the medium, into which the gel migrates.

A study of the apparent time constants of the macroscopic expansion in the light of the microscopic morphology and micromechanical properties may contribute to the understanding of the chemo-physical origin of expansive reactions, and may provide evidence whether an expansive reaction is of through-solution or topochemical origin. For instance, ASR-expansion tests carried out under controlled chemical kinetics condition on concrete materials of different strength and stiffness should be able to confirm the mixed topochemical–trans-solution mechanisms that have been suggested in the open literature.

For purpose of clarity, the analysis was restricted here to two pore-type populations, and a one-to-one transition process between a topochemical and a trans-solution mechanism defined by a single threshold pressure. This leads to the rough change in concavity in the expansion curve (see figure 8). A smooth concave profile can be obtained, by increasing the number of pores of same or different morphology, which become with increasing threshold pressure accessible to the reaction products. In this case, the increase in threshold pressure may reflect the increase in work required to crack the porous medium or to make the reaction products actually enter the entire pore spectrum of cementitious materials ranging from capillary pores (of size 1 μm) to micropores (of size 10–20 Å).

## Footnotes

One contribution of 7 to a Theme Issue ‘Thermodynamics in solids mechanics’.

- © 2005 The Royal Society