Modeling of alkali-silica reaction in concrete: a review

Abstract

This paper presents a comprehensive review of modeling of alkali-silica reaction (ASR) in concrete. Such modeling is essential for investigating the chemical expansion mechanism and the subsequent influence on the mechanical aspects of the material. The concept of ASR and the mechanism of expansion are first outlined, and the state-of-the-art of modeling for ASR, the focus of the paper, is then presented in detail. The modeling includes theoretical approaches, meso- and macroscopic models for ASR analysis. The theoretical approaches dealt with the chemical reaction mechanism and were used for predicting pessimum size of aggregate. Mesoscopic models have attempted to explain the mechanism of mechanical deterioration of ASR-affected concrete at material scale. The macroscopic models, chemo-mechanical coupling models, have been generally developed by combining the chemical reaction kinetics with linear or nonlinear mechanical constitutive, and were applied to reproduce and predict the long-term behavior of structures suffering from ASR. Finally, a conclusion and discussion of the modeling are given.

Keywords

alkali-silica reaction (ASR) / modeling / concrete / mesoscopic / macroscopic

Introduction

Numerous concrete structures, such as dams and bridges, have been suffering from deterioration due to alkali-silica reaction (ASR). ASR is a deleterious chemical reaction between alkalis in Portland cement paste and certain amorphous silica in a variety of natural aggregate. The ASR production expands in time and exerts pressures on the surrounding matrix, causing extensive cracking in concrete. Thus, the stiffness and strength of concrete may be reduced and the safety of structures may also be affected. This problem was first reported by Stanton in 1940 [1]. Since then, many efforts have been made to investigate the ASR on the material and structural scales, including physical chemistry of ASR, structural response to ASR effects, accelerated test methods, and ASR-reducing measures, etc.

The chemical processes behind ASR are very complex and not fully understood yet. To identify the reactive aggregate and examine the structure of ASR products, Scanning Electron Microscopy (SEM) and Energy-Dispersive X-ray (EDX) techniques are widely used for microscopic analysis in laboratory [2–4]. ASR in the realistic structures usually needs a very long period of more than 10 years to take place, and it may hamper the corresponding researches. Accelerated test methods have been conducted ranging from Accelerated Mortar Bar Test (AMBT) [5–7] to Concrete Prism Test (CPT) [8,9]. These experiments take a very short time for ASR ranging from several days to several months depending on the conditions where the specimens are located. Therefore, investigation of ASR kinetics and its effects on mechanics become possible with the application of the accelerated tests. Numerous experiments have been carried out to examine the degradation of material properties due to ASR [10,11] and the influence of ASR on the mechanics of prisms and concrete beams [12–18]. In these experimental programs, a general conclusion was found that the mechanical properties of reactive concrete, including elastic modulus and tensile strength, decrease as ASR develops [19]. However, it cannot find a relationship between the changes in mechanical properties and the expansion since the changes depend on the materials and chemical mechanisms. Several factors may significantly influence the ASR process and the effects on the mechanical properties of concrete. These factors include mineralogy of the rock, the size of the aggregate, the alkali content in pore solution, the relative humidity, temperature, and the confining stress, etc. A number of experimental and numerical investigations have conducted concerning the influence factors [20–30]. Among these researches, theoretical approaches [31–33] are widely developed to predict the pessimum size of aggregate [10,34] which induces the maximum expansion of ASR-affected concrete compared to smaller or larger size of aggregate. The theoretical approach generally examines a representative volume element (RVE) which is comprised of a spherical aggregate particle and the surrounding cement paste. Chemical reaction kinetics and gel swelling are considered in the RVE.

ASR plays a vital role for the durability of concrete structures, thus ASR-avoiding or reducing measures are required. The common measures are to identify the potential reactivity of aggregate [35,36] and/or add admixtures such as fly ash to the concrete [37-39]. To seek a more effectively ASR-mitigating measure, the mechanism of chemical reaction and the ASR influencing mechanics should be fundamentally understood. Petrographic analysis and SEM are used for experimental investigation into the microstructure of ASR gel and cracks initializing and propagating in concrete. An alternative approach, mesoscopic model in numerical method is available for the analysis of ASR mechanism. These mesoscopic models take into account multiphase of aggregate, cement paste, void and ASR gel, thus anisotropy can be explicitly represented in concrete. In the exiting mesoscopic models, ASR swelling is simply treated by equivalent expansion of aggregate particles [40] or dilatation of gels randomly distributed in aggregate [41].

Regarding the large number of operating structures made of reactive concrete with high risk of ASR, the only way to reduce its hazards is by taking remedial measures such as slot cutting or grouting of cracks [42]. The effectiveness of remedial measures depends strongly on the prediction of stress and strain field in the reactive concrete. Therefore, macroscopic models at structural scale are required for the realistically simulating the ASR-induced expansion of the structures. In recent years, many macroscopic models have been developed and they can successfully reproduce the displacement field of ASR-affected structures [43]. Most of the macroscopic models are formulated in the framework of finite element method combining the chemical reaction kinetics with mechanical constitutive. The mechanical constitutive is assumed as linear elastic for expansion [44,45] or dealt with isotropic/anisotropic damage models for both expansion and cracking [46-48]. An even sophisticated constitutive may be introduced considering creep and shrinkage effects during ASR process [49,50]. The reaction kinetics either adopts the kinetics proposed by Larive [51] or uses the experimental findings [22,27,51]. Several ASR-depending functions have been introduced in the reaction kinetics for describing the effects of temperature, relative humidity and confining stress.

The purpose of this paper is to give a comprehensive review of modeling for ASR in concrete developed by many researchers. The modeling includes theoretical approach, meso- and macroscopic models. A brief description of ASR gel formation and swelling is first provided. The state-of-the-art of the modeling for ASR is then summarized. Finally, a conclusion and discussion are given.

Mechanism of ASR

Alkali-silica gel formation

The alkali-silica reaction in concrete is a chemical reaction that occurs between amorphous or poorly crystalline silica present in reactive aggregate and alkali and hydroxyl ions in pore solution. The chemical reaction produces an amorphous gel, which expands with the absorption of water.

The ASR process is complicated, and it consists of several stages. A simplified description for ASR, proposed by Dent Glasser and Kataoka [52,53], was summarized in two steps as shown in Fig. 1. The first step is rupture of the aggregate siloxane networks caused by the attack of hydroxyl ions to produce alkali silicate and silicic acid,

(1)\equiv \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{S}\mathrm{i}\equiv +{\mathrm{R}}^{+}+\mathrm{O}{\mathrm{H}}^{-}\to \equiv \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{R}+\mathrm{H}-\mathrm{O}-\mathrm{S}\mathrm{i}\equiv ,

where {\mathrm{R}}^{+} denotes an alkali ion such as sodium and potassium ions (\mathrm{N}{\mathrm{a}}^{+} and {\mathrm{K}}^{+}). The weak acid production of silicic acid immediately reacts with further hydroxyl ions,

(2)\equiv \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{H}+{\mathrm{R}}^{+}+\mathrm{O}{\mathrm{H}}^{-}\to \equiv \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{R}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\mathrm{.}

The alkali silicate, so called alkali-silica gel, resulting from (1) and (2) is amorphous and hygroscopic.

The second step is the expansion of the alkali-silica gel by absorption of free water,

(3)\equiv \mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{R}+n{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\to \equiv \mathrm{S}\mathrm{i}-{\mathrm{O}}^{-}-{({\mathrm{H}}_{\mathrm{2}}\mathrm{O})}_n+{\mathrm{R}}^{+},

in which n is the hydration number.

For the ASR to occur in concrete, it is generally known that three conditions must be satisfied: the presence of reactive aggregate, a high level of alkalinity, and sufficient moisture (say not less than 80% relative humidity in the concrete pores). Without any of these conditions, the ASR will not take place. Once initiated, the reaction is highly sensitive to the three conditions (as shown in Fig. 2).

When the hydrated alkali-silica gel is generated, it is diffused away from aggregate into pores, and then acts with calcium ions (\mathrm{C}{\mathrm{a}}^{\mathrm{2}+}) in the cement paste to form an alkali-calcium-silicate hydrate gel [55-57]. The ASR products expand by absorption of moisture and induce cracking in the aggregate and the surrounding cement paste, resulting in deterioration of concrete. The chemical composition of the gel determines the swelling and mechanics characterization, thus it has significant influence on the deterioration mechanism of concrete.

Many researchers have investigated the chemical composition of ASR gel using the qualitative analyses carried out by SEM with Energy Dispersive Spectrometry (EDS) [57–60]. Thaulow et al. [59] found that gel chemical compositions fell within the range 53 to 63% silica, 20 to 30% calcium and relative constant about 15% sodium and potassium. The chemical composition of ASR gel varies strongly depending on the position of the gel. It is reported that the gel within pores and cracks in the cement paste has higher calcium contents compared with the gel close to and in the reactive aggregates [59–61]. Gel forming exudations in a viscous state consists of silicon and alkalis, but calcium is absence [58,61]. The absence of calcium can be attributed to a short transport within the cement paste. Gel composition would also vary with time according to Thaulow et al. [59] and Šachlová et al. [62], but it is found similar, though in different locations inside the concrete of two structures of a very different age [61].

Many researchers [57,59,63,64] have tried to establish the relationship between the gel composition and expansibility but the role of chemical composition in the expansivity is not yet totally understood. Powers and Steinour [65,66] stated that ASR gel is composed of two components, which are an alkali-calcium-silicate hydrate gel and an alkali-silicate hydrate gel. Reactions with highly reactive aggregates could be safe or unsafe depending on the contents of calcium in the reaction product. High calcium content gel, i.e., alkali-calcium-silicate hydrate gel, does not swell, and it will not cause the cracking problem in concrete. However, if calcium ion concentrations are quite low, both swollen and non-swollen gels are formed, resulting in huge expansion and severe cracking. Helmuth and Stark [67] supported this view. Jun and Jin [68] applied Electron Probe Micro-analyzer (EPMA) to investigate ASR products and also found that there was a correlation between the CaO/SiO₂ ratios of reactive products and expansion values. However, Diamond [57] pointed out the composition and properties of ASR gel are complex and Powers and Steinour’s model may be reasonable but not universal. Scrivener and Monteiro [69] found a type of ASR gel with higher calcium content absorbed a similar amount of water compared to a gel with less calcium. Fernandes and coworkers [61,70] pointed out that expansion due to ASR was verified in the concrete even with low contents of calcium found in the gel composition.

Swelling and cracking mechanism

It is widely accepted that deterioration of ASR-affected concrete is due to water imbibition by the gel and its expansion. The swelling of gel creates an increasing internal stress, causing the opening and propagation of cracks when the tensile strength of the concrete skeleton is exceeded. However, the mechanism of expansion and cracking is still not clearly understood. Many attempts have been made to explain the mechanism of expansion and cracking due to ASR.

Hansen [71] first proposed the osmotic pressure theory to describe the mechanism of expansion. This theory suggests that the cement paste surrounding the reactive aggregate behaves as a semipermeable membrane. As presence of the membrane, alkali-silicate ions cannot diffuse from the reactive sites into the surrounding cement paste, but water can pass through from the pore solution. Thus, an osmotic pressure cell forms and the alkali-silica gel swells with increasing hydrostatic pressure, thereby inducing cracking in the cement paste. Dent Glasser [72] and Poole [73] also applied the osmotic pressure theory to the mechanism of expansion due to ASR.

McGowan and Vivian [74] proposed a mechanical theory to explain the expansion of the ASR gel. In the theory, a solid alkali-silicate layer forms on the reactive aggregate surface, and absorbs moisture from the pore solution and transforms from a solid to a gel. The gel then swells and induces the cement paste to crack.

The osmotic pressure theory and mechanical expansion theory are fundamentally the same [65]. The alkali-silica gel can exist in the forms of solid or colloidal fluid, strongly depending on the amount of imbibed water. The expansion of concrete is generated primarily through the osmotic pressure cell of fluid gel, or through the solid gel swelling.

Bazănt and Steffens [31] proposed that the concrete expansion due to ASR is caused by the swelling pressure accumulated in the interfacial transition zone between the aggregate and the surrounding cement paste. The imbibition of water generates a pressure in the gel, which is initially released by pushing the gel to permeate the micro pores in the cement paste located near the surface of the aggregate. Once the pores around the reactive aggregate are fully filled by the gel, further formation of the gel induces the increasing pressure, thereby cracking of the cement paste. Dron and Brivot [56] assumed that the dissolved silica is diffused far away from the reactive aggregates into the connected pores in the cement paste and swelling reactions then occur to induce the cracking of the concrete.

According to the aforementioned swelling mechanisms, cracking takes place only in the cement paste and the aggregate subjected to a compression pressure would not fracture. However, cracking patterns of ASR-affected concrete have been shown by Ponce and Batic [75] to depend on the mineralogical nature of the aggregate. Ponce and Batic carried out the petrographic examination with a stereobinocular and a polarizing microscope, and they concluded that aggregates such as opal and vitreous volcanic rocks with gel formation and expansion at the aggregate-paste interface caused cracking of cement paste; whereas mixed mineralogy aggregates, which are more common in the field, with interior reaction generated cracks in both the aggregates and cement paste (as shown in Fig. 3) Based on the realistic cracking patterns of concrete due to ASR, Idorn [76] stated that the expansive pressure due to the formation of hydrate gel develops directly inside the reactive aggregate, thereby causing tensile stress in the aggregate. The increasing expansive pressure is finally released by cracking both the aggregate and the surrounding cement paste. Garcia-Diaz et al. [77] also suggested that formation of ASR gel causes a swelling and a micro-cracking of the reactive aggregate. Ichikawa [78] et al. proposed another model to describe cracking of aggregate due to ASR. This model supposed that the reactive aggregate is tightly packed with an insoluble and rigid reaction rim, which does not permit diffusion of viscous alkali-silicate gel, but allows penetration of alkaline solution. Therefore, the alkali-silica gel generated after formation of the rim is stored inside the aggregate and the resultant expansive pressure cracks the aggregate and the surrounding cement paste when the strength of material is exceeded.

Modeling for ASR

ASR affects many concrete structures and causes permanent deformation and cracking, which reduces the durability and safety of the structures. A modeling is needed to analyze the behavior of ASR-affected structures, thus evaluating the safety level and reinforcement costs of the degraded structures. There are two basic aspects involved when simulating the behaviors of the ASR-affected concrete: 1) the modeling of the chemical reaction kinetics and diffusion processes, and 2) the modeling of mechanical fracture which induces expansion and deterioration of concrete [79]. The first aspect, the reaction kinetics, determines the degree of ASR and potential expansion, and the second aspect of mechanical fracture describes the material and structural degradation. The modeling should be developed combining the reaction kinetics with the mechanical fracture. Unfortunately, ASR is influenced by many factors, among them humidity, reactivity of silica, amount of alkali and aggregate size, resulting in very complicated description of the problem. It seems nearly impossible to make a comprehensive model that gives realistic response of structures based on the present qualitative knowledge of the ASR chemistry. Therefore, theoretical, semi-empirical models and numerical models have been mainly developed depending on the observed response of ASR-affected concrete at a material scale or a structural scale. In other words, the most recently developed ASR models follow either a mesoscopic or a macroscopic approach.

Theoretical models

Theoretical models are generally developed based on a description of interaction between the ASR gel and the concrete matrix in a representative volume element (RVE). The RVE consists of a single spherical aggregate particle surrounding by a shell of arbitrary thickness of cement paste. This approach takes several assumptions to simplify the modeling of ASR expansion. The theoretical models deal with chemical mechanism of ASR including diffusion process, kinetics of dissolution and gel formation and its swelling which induces deterioration of concrete. The objective of the models primarily focuses on predicting the pessimum expansion of ASR-affected concrete and the corresponding pessimum size of aggregate.

Hobbs [80] presented a theoretical model based on that the reaction rate was directly proportional to the quantity of reactive aggregate. The model could predict the time to cracking and expansion, and the predictions were in good agreement with the observed behavior of mortar specimens containing accessible opaline silica. This model was limited to mortar, and was not available for predicting concrete cracking and expansion behavior due to more energy being required to initiate cracking in concrete than in mortar.

Groves and Zhang [81] proposed a quantitative dilatation theory for the expansion of a mortar composing of silica glass particles in ordinary Portland cement paste. This dilatation model assumed that the ASR product was mainly generated at the transition interface between the particles and cement paste, forming a gel layer. The mortar expansion was predicted by the increase in volume of the glass particle plus the ASR layer using the elasticity theory of a misfitting sphere in the matrix.

Furusawa et al. [82] later developed a model by combining the diffusion theory with the dilatation model to characterize the expansion of ASR-affected mortar. In this model, the expansion process contained two stages, i.e. the diffusion of the hydroxide and alkali ions into aggregate and the subsequent chemical reaction; and the expansion induced by ASR. In addition, it predicted the ASR-induced expansion of concrete with a hypothesis that there existes a porous zone around the aggregates and that the expansion is initiated only when the volume of the reaction products exceeds the available volume of the porous zone.

Bazănt and Steffens [31] proposed a mathematical model for quantitative prediction of ASR-induced expansion. The model was developed based on the analysis of a characteristic unit cell of concrete containing one spherical glass particle as shown in Fig. 4. The radial growth of the spherical layer of ASR gel into the glass particle was assumed to be controlled by the diffusion of water toward the reaction front. Expansion of the gel was partly accommodated by its expulsion into the capillary pores in the cement paste surrounding the particle. It was reasonable to assume imbibition of additional water from the adjacent cement paste causes swelling of the gel and represented another diffusion process. Parametric study of the model clarified the effects of particle size and the pessimum size of aggregate was given in terms of the effect of internal pore relative humidity on ASR expansion. The model was not available for regular concrete, but for the tests conducted by Meyer and Baxter [83] to incorporate ground waste glass into concrete.

Xi et al. [84,85] proposed a mathematical model with the purpose of predicting the pessimum expansion of concrete and the pessimum size of aggregate. The model took into account the chemo-mechanical coupling of the ASR expansion process, in which the mechanical part of the model was formulated based on the multiphase generalized self-consistent method [86] and the chemical part was composed of two opposing diffusion processes: 1) the diffusion of chemical ions from pore solution into aggregate, and 2) the permeation of ASR gel from the aggregate surface into the surrounding porous cement matrix. In the mechanical part, a two-phase composite theory was applied to characterize the expansion and internal pressure generated by ASR with reactive aggregate of different sizes. In the chemical part, the micro-diffusion of hydroxide and alkali ions penetrating from pore solution into aggregate particles was described by Fick’s law [87] and the ASR gel permeation through porous cement paste was characterized by Darcy’s law. Thus, the pessimum size of aggregate was obtained by combining the composite and diffusion models with the micro-structural features of concrete.

The aforementioned theoretical models primarily emphasized the diffusion mechanism during ASR expansion. In contrast, Bazănt el al [79]. developed a fracture mechanics theory to predict the pessimum size of aggregate. In this model, a regular array of cubical cells, each containing one spherical glass particle imbedded in the cement paste matrix, was considered for the fracture analysis, as shown in Fig. 5. Restricting to a single cubical cell and realizing that the stress field caused by the gel-induced pressure p on the crack and void surfaces was a superposition of the stress field caused by the hydrostatic pressure p and the stress field caused by the externally applied tensile stress \sigma =-p (Fig. 6). The stress intensity factor at the preexisting flaws at the surface of the particles was then obtained as

(4)K_{\mathrm{I}}=(\sigma +p)\sqrt{s}k(\omega ),

where k(\omega ) is the dimensionless stress intensive factor, and \omega is defined as the dimensionless damage variable

(5)\omega ={\displaystyle \frac{A_{\mathrm{c}}}{s^{\mathrm{2}}}},

in which A_c = the area of the crack, and s = the size of spherical void in one cubical.

The dimensionless stress intensive factor k(\omega ) was approximated through a matching approach, thus the stress intensity factor K_I at the preexisting flaws was determined and used to evaluate the crack propagation.

Mesoscopic models

Relatively few research efforts have focused on ASR-induced degradation mechanism with micro-cracking initialization and propagation within aggregate particles and cement paste. Very few mecoscopic models for the degradation mechanism, only two references [40,41] found in recent years, are developed based on a description of the material level interaction between the gel and the concrete matrix. The mesoscopic approach takes into account the particle size distribution and the reactive aggregate location, which are explicitly presented with refined finite elements at an experimental scale.

Comby-Peyrot et al. [40] presented a three-dimensional (3D) mesoscopic model that could describe the heterogeneities of concrete for ASR-induced damage. The model used a take-and-place method [88] to generate an aggregate-mortar structure of concrete based on a 3D existing finite element mesh, and the heterogeneous sample generated is shown in Fig. 7. The two-phase mesoscopic model was composed of aggregate particles and cement paste, in which the aggregate was assumed purely linear elastic and the cement paste behaved as an elastic-damage material. The ASR expansion mechanism was simplified based on an isotropic dilatation phenomenon of reactive aggregate. The reactive aggregate particles distributed in the model were randomly selected, and their swelling was assumed to follow the volume variation of the granular skeleton over time as shown in Fig. 8, which was evaluated by Riche et al. [89]. This model reproduced the ASR-induced crack map on concrete and progressive degradation of material with certain accuracy.

Dunant and Scrivener [41] confirmed the relationship between the void and crack content of aggregate and the macroscopic free expansion of ASR-affected mortar, which was presented by Ben Haha et al. [90]. They made a hypothesis that concrete damage was induced by growing gel pockets in the aggregates according to the connection between microstructural observation and macroscopic measurement. A finite element framework of meso-scale model was proposed, where spherical aggregate was assumed and was generated using a random packing algorithm. Gel pockets within the aggregates were explicitly described with an enrichment function [91] that simulated a perfect contact between the gel pocket and the aggregate. The gel pockets were assumed to be linear elastic and the aggregate and the surrounding cement paste were considered quasi-brittle and followed a damage law. The geometry of gel swelling could be accurately represented by updating the enrichment function during the simulation of ASR process, as shown in Fig. 9. The gel was grown by steps in each aggregate with an imposed strain, and its expansion was stop when a preset percentage (say 3%) of the aggregate had reacted. The gel swelling generated sufficient stress to damage the aggregate and the surrounding cement paste. This approach showed that the ASR-induced cracking pattern in the concrete was in good agreement with the experimental observation [92]. Regarding the stiffness of the concrete, degradation was predicted as shown in Fig. 10. It was explained by the model that the mechanism of stiffness loss was due to the aggregate cracking as the cement paste mostly remained in compression with stress levels below the strength limit (Fig. 11).

Shin and colleagues [93,94] developed a microstructure-based finite element approach to analyze the mechanical response of concrete subjected to ASR. In this approach, scanning electron microscopy (SEM) was used to obtain the microstructural images of specimens suffering from ASR, and then the technique of backscattered electron intensity in conjunction with EDX data was applied to identify the constituents of concrete. Finite element model, including microstructure of aggregates, voids, paste, gels and cracks, was implemented based on the image analysis. An example of microstructural finite element mesh prepared for a sample specimen, compared with the corresponding image, is shown in Fig. 12. The ASR effect was considered through the swelling pressure caused by the gel [95] and it was simulated by assigning the gel a fictitious thermal expansion coefficient and temperature change. Two-dimensional linear elastic finite element modeling was employed to predict ASR expansion and the results were in good agreement with the experimental data.

Macroscopic models

Several macroscopic models have been recently developed to analyze the global behavior of ASR-affected concrete at a structural scale. This approach focuses on the displacement field, stress field and damage cracking of concrete structures attacked by ASR. It is beneficial to properly reproduce observed behavior of structures and investigate the deterioration mechanism. The macroscopic model is also applicable to predicting the long-term effects of ASR on the durability and assessing the stability of structure, thus assisting the corresponding repair measures. Most of the existing models were formulated in the framework of finite element method combining the chemical aspect with its mechanical effects. The chemical reaction kinetics was based on the experimental findings or in situ measurements. The mechanical behavior of concrete structures could be simulated with linear elastic constitutive or nonlinear models including creep, shrinkage and cracking.

Parametric model

One of the earliest models was a phenomenological model presented by Charlwood et al. [96] and Thompson et al. [97], in which the chemical reaction kinetic was not considered, and anisotropic expansion was dependent on the stress state and was defined as a function of the stress tensor. Assuming the principal direction of the expansion coincide with the principal stress directions, the anisotropic expansion rate was given by

(6){\dot{ϵ}}_i^{\mathrm{a}\mathrm{s}\mathrm{r}}=\{\begin{array}{ll}{\dot{ϵ}}^u\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{0}\le {\sigma }_i\le {\sigma }_{\mathrm{L}};\hfill \\ {\dot{ϵ}}^u-K\log ({\sigma }_i/{\sigma }_{\mathrm{L}})\hfill & \mathrm{f}\mathrm{o}\mathrm{r}{\sigma }_{\mathrm{L}}\le {\sigma }_i\le {\sigma }_{\mathrm{M}\mathrm{A}\mathrm{X}}\hfill \\ \mathrm{0}\hfill & \mathrm{f}\mathrm{o}\mathrm{r}{\sigma }_i>{\sigma }_{\mathrm{M}\mathrm{A}\mathrm{X}}.\hfill \end{array};

Where {\dot{ϵ}}^u denotes free expansion rate, {\sigma }_{\mathrm{L}} has been taken about 0.3 MPa, {\sigma }_{MAX} is the compression stress above which there is no ASR expansion and has been taken approximate 5 to 10 MPa, K is the slope of the line defining the growth rate vs log-stress, and i=\mathrm{1},\mathrm{2},\mathrm{3} correspond to the principal stress directions. The expansion rate evolution is shown in Fig. 13.

The expansion in these models was generally treated as an initial strain induced by an equivalent temperature increase. The model was very simple and effective, but it did not pay attention to the detailed chemical mechanism of ASR.

Subsequently, a more refined model has been proposed by Leger et al. [43], which distributed the observed concrete expansion for the numerical modeling of ASR concrete swelling of dams. This procedure was formulated in proportion to the compressive stress state, temperature, moisture, and reactivity of the concrete constituents. Consider a particular zone m of n finite elements where expansion characteristics are expected. The ASR induced strain according to this procedure is given by

(7){ϵ}_{\mathrm{a}\mathrm{s}\mathrm{r}}^m={\beta }^m(t){[F_{\mathrm{C}}({\sigma }_{\mathrm{c}},t)·F_{\mathrm{T}}(t)·F_{\mathrm{M}}(t)·F_{\mathrm{R}}(t)]}^m={\beta }^m(t)·CTM{R}^m,

where F_{\mathrm{C}}, F_{\mathrm{T}}, F_{\mathrm{M}}, and F_{\mathrm{R}} are normalized expansion factors (between zero to one) corresponding to compressive stress state, temperature, moisture, and reactivity. CTM{R}^m is the normalized ASR induced strain. {\beta }^m is a calibration factor to adjust the numerical results to the field observation.

In the procedure, the normalized expansion factors were first determined according to some empirical methods. The mechanical constitutive considered cracking and the effect of ASR expansion by assuming a reduction of the tensile strength and elastic modulus due to ASR, and the reduction law was estimated from the literature or from the laboratory tests [98]. The procedure was then implemented in the finite element frame and used to analyze the displacement of concrete dams suffering from ASR. Herrador et al. [99] combined creep effects in this procedure to analyze the core-drilled specimens extracted from Belesar Dam attacked by ASR in Lugo, Spain. It should be emphasized that the procedure is the most successful models used in the analysis of ASR-affected concrete dams to date [100]. However, this type of procedure has been based on the trial-and-error basis to reproduce the observed deformation, and it does not reflect the physical mechanism of the phenomenon.

Chemo-mechanical coupling model

Huang and Pietruszczak [44,101] proposed a nonlinear continuum model to describe the mechanical effects of ASR in concrete structures. The model assumed that the progress of ASR is coupled with the degradation of mechanical properties, which is described using the framework of elastoplasticity. It neglected the influence of temperature and humidity on the ASR expansion. The model has been later extended to include the effects of temperature in the evolution laws governing the rate of ASR [46]. The expansion rate was thus controlled by the alkali content in the cement, the magnitude of confining stress as well as the time history of the temperature.

In the elastic range, the constitutive law according to the model by Huang and Pietruszczak [46] was written as

(8)ϵ=C^e\sigma +{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{ϵ}_{\mathrm{a}\mathrm{s}\mathrm{r}}\delta +{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\alpha }_{\mathrm{T}}\left(T-\overline{T}\right)\delta ,

where ϵ and \sigma are the principal strains and stresses, C^{\mathrm{e}} is the elastic compliance matrix, {ϵ}_{\mathrm{a}\mathrm{s}\mathrm{r}} is the ASR induced volumetric strain, \delta is the Kronecker’s delta, {\alpha }_{\mathrm{T}} is the coefficient of thermal expansion, and T and \overline{T} represent the absolute and the reference temperature, respectively.

To introduce the ASR evolution law, the extent of the reaction is defined as

(9)\zeta (\tau )={\displaystyle \frac{\xi (\tau )}{\overline{\xi }}},\overline{\xi }=\xi (\tau \to \infty ),

where \tau is so-called the thermal activation time, which is considered as a local property influenced by the temperature history. \xi (\tau ) is the free volume expansion, \overline{\xi } is the material constant specifying the maximum value of \xi for a given conditions (i.e., alkali content, reactive particle size, etc.). Thus the extent of reaction \zeta evolves from 0 to 1 during the ASR process under free stress state.

The extent of the reaction is chosen as a simple form according to Eq. (9)

(10)\zeta (\tau )={\displaystyle \frac{\tau }{A_{\mathrm{1}}+\tau }},\zeta =\zeta (\tau \to \infty )=\mathrm{1.}

Relating the thermal activation time \tau and the actual physical time t via the following relationship

(11)\mathrm{d}\tau =g_{\mathrm{1}}(T)\mathrm{d}t,

where the function g_{\mathrm{1}}(T) specifies the temperature effects and a hyperbolic form is employed

(12)g_{\mathrm{1}}(T)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left(\mathrm{1}+\tanh \left({\displaystyle \frac{T-\overline{T}}{A_{\mathrm{2}}}}\right)\right),\mathrm{0}\le g_{\mathrm{1}}\le \mathrm{1.}

Introducing a function g_{\mathrm{2}}(tr\sigma ) corresponding to constraining effects of hydrostatic pressure

(13)g_{\mathrm{2}}(tr\sigma )={\mathrm{e}}^{A_{\mathrm{3}}tr\sigma /f_{\mathrm{c}\mathrm{0}}}.

The expansion due to ASR is then given by

(14){ϵ}_{\mathrm{a}\mathrm{s}\mathrm{r}}=g_{\mathrm{2}}(tr\sigma )\zeta (\tau )\overline{\xi }.

Therefore, the evolution law for ASR-induced expansion is obtained as

(15){\dot{ϵ}}_{\mathrm{a}\mathrm{s}\mathrm{r}}=g_{\mathrm{2}}(tr\sigma )\overline{\xi }{\displaystyle \frac{(\mathrm{1}-\zeta )}{A_{\mathrm{3}}}}{g}_{\mathrm{1}}(T).

The degradation of mechanical properties due to continuing ASR was considered. The degradation functions for Young’s modulus and compression strength were defined by

(16)E=E_{\mathrm{0}}(\mathrm{1}-(\mathrm{1}-B_{\mathrm{1}})\zeta ),f_{\mathrm{c}}=f_{\mathrm{c}\mathrm{0}}(\mathrm{1}-(\mathrm{1}-B_{\mathrm{2}})\zeta ).

In the above equations, E_{\mathrm{0}} and f_{\mathrm{c}\mathrm{0}} are the initial values of Yong’s modulus and compression strength, respectively. A_{\mathrm{1}}, A_{\mathrm{2}}, A_{\mathrm{3}}, B_{\mathrm{1}}, and B_{\mathrm{2}} are the material constants.

The above linear elastic framework combined with the chemical reaction kinetics can be extended to elastoplastic regime by invoking the additivity of elastic and plastic strain, and the constitutive relation can be found in detail in the reference [46].

The model was applied to analyze the deformation history in the junction of the right wing gravity dam and intake structure of the Beauharnois complex in Canada [101]. Figure 14 shows the vertical displacement of the dam after 25 years of reaction. The prediction was in good agreement with the field measurements. The model was also used to assess the dynamic stability of ASR-affected hydraulic structures subjected to earthquakes [102]. This model can be easily extended to analyze ASR-affected reinforced concrete by replacing the plain concrete material with a composite medium comprising the concrete matrix and reinforcement [103].

Ulm et al. [45] later developed another chemo-elastic model that accounted for ASR kinetics and the swelling effects. They assumed that the ASR products filled the pores and microcracks of the cementitious matrix, and exerted a pressure on the surrounding concrete skeleton. Under stress-free conditions, the internal swelling pressure p_{\mathrm{g}} of the expansive gel is self-balanced by the tension stress {\sigma }_{\mu } in the concrete skeleton. This can be idealized by the one-dimensional (1D) model of Fig. 15, where \sigma is the macroscopic stress due to external forces, and ϵ is the corresponding overall strain. The stress equilibrium in the 1D chemoelastic device is given by

(17)\sigma ={\sigma }_{\mu }-p_{\mathrm{g}}=E_{\mathrm{s}}ϵ+E_{\mathrm{g}}(ϵ-\kappa \zeta ).

in which E_{\mathrm{s}} and E_{\mathrm{g}} are the spring modulus of the chemoelastic device, and \zeta =\xi (t)/\xi (\infty ) is the extent of the ASR.

The model is then extended to the 3D case using an energetic approach. For the 1D chemoelastic device, the free energy is obtained as

(18)\psi ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}[E_{\mathrm{s}}{ϵ}^{\mathrm{2}}+E_{\mathrm{g}}{(ϵ-\kappa \zeta )}^{\mathrm{2}}]+g(\zeta ).

The first term in brackets represents the free energy of the spring, and function g(\zeta ) corresponds to the free energy in the chemical pressure cell. Use of Eq. (18) in the expression of dissipation \phi \mathrm{d}t=\sigma \mathrm{d}ϵ-\mathrm{d}\psi \ge \mathrm{0} yields

(19)\phi \mathrm{d}t=[\sigma -(E_sϵ+E_{\mathrm{g}}(ϵ-\kappa \zeta ))]\mathrm{d}ϵ+\left[\kappa E_{\mathrm{g}}(ϵ-\kappa \zeta )-{\displaystyle \frac{\partial g}{\partial \zeta }}\right]\mathrm{d}\zeta \ge \mathrm{0.}

From Eqs. (17) and (19), it yields that

(20)\sigma ={\displaystyle \frac{\partial \psi }{\partial ϵ}}.

The extension to the 3D is thus obtained by straightly replacing the scalar with the tensor in the isotropic case

(21){\sigma }_{ij}={\displaystyle \frac{\partial \psi }{\partial {ϵ}_{ij}}}=\left(K-{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}G\right)\xi {\delta }_{ij}+\mathrm{2}G{ϵ}_{ij}-\mathrm{3}\beta K\zeta {\delta }_{ij},

where \xi ={ϵ}_{ii}, K=E/\mathrm{3}(\mathrm{1}-\mathrm{2}\nu ) and G=E/\mathrm{2}(\mathrm{1}+\nu ) are overall bulk modulus and the shear modulus, respectively; E=E_{\mathrm{s}}+E_{\mathrm{g}} is the overall Young’s modulus; \nu is the Poisson’s ratio; and \beta =\kappa E_{\mathrm{g}}/E is the chemical dilatation coefficient.

Considering a stress-free experiment, the only unknown variable is the reaction extent \zeta. The time evolution of reaction extent is defined by a kinetics law, a rate equation relating the reaction affinity A_m=A_m(\zeta ) to the reaction rate \dot{\zeta }=\mathrm{d}\zeta /\mathrm{d}t [104]. A linear law is adopted for the stress-free condition

(22)A_m(\zeta )=k_d{\displaystyle \frac{\mathrm{d}\zeta }{\mathrm{d}t}},

where k_d>\mathrm{0} is a coefficient. The affinity A_m(\zeta ) expresses the local imbalance driving the ASR gel formation, and it defines the reaction order following physical chemistry. A first-order reaction kinetics is assumed, thus

(23)\mathrm{1}-\zeta =t_{\mathrm{c}}({\theta }_{\mathrm{0}},\zeta ){\displaystyle \frac{\mathrm{d}\zeta }{\mathrm{d}t}}\mathrm{o}\mathrm{r}\xi (\infty )-\xi (t)=t_{\mathrm{c}}({\theta }_{\mathrm{0}},\zeta ){\displaystyle \frac{\mathrm{d}\xi }{\mathrm{d}t}},

where t_{\mathrm{c}} defines the characteristic time of the reaction, and it can be determined from a stress-free expansion test. According to the experimental program by Larive [51], The characteristic time t_{\mathrm{c}} has been found to depend on both temperature {\theta }_{\mathrm{0}} and reaction extent

(24)t_{\mathrm{c}}={\tau }_{\mathrm{c}}({\theta }_{\mathrm{0}})\lambda (\zeta ,{\theta }_{\mathrm{0}});\lambda (\zeta ,{\theta }_{\mathrm{0}})={\displaystyle \frac{\mathrm{1}+\exp [-{\tau }_{\mathrm{L}}({\theta }_{\mathrm{0}})/{\tau }_{\mathrm{c}}({\theta }_{\mathrm{0}})]}{\zeta +\exp [-{\tau }_{\mathrm{L}}({\theta }_{\mathrm{0}})/{\tau }_{\mathrm{c}}({\theta }_{\mathrm{0}})]}}.

Substituting Eq. (24) into Eq. (23), it yields

(25)\xi (t)=\xi (\infty )\zeta (t);\zeta (t)={\displaystyle \frac{\mathrm{1}-\exp (-t/{\tau }_{\mathrm{c}})}{\mathrm{1}+\exp (-t/{\tau }_{\mathrm{c}}+{\tau }_{\mathrm{L}}/{\tau }_{\mathrm{c}})}}.

The reaction extent in terms of normalized time t/{\tau }_{\mathrm{c}} is shown in Fig. 16. In the above equations, {\tau }_{\mathrm{c}} and {\tau }_{\mathrm{L}} stand for the characteristic time and the latency time of ASR swelling, respectively. They can be directly determined from stress-free ASR swelling tests. There is a temperature dependence of the time constants {\tau }_{\mathrm{c}} and {\tau }_{\mathrm{L}} [105]. Larive’s experiments [51] have shown that the time constants vary according to the Arrhenius relation

(26)\{\begin{array}{l}{\tau }_{\mathrm{c}}(\theta )={\tau }_{\mathrm{c}}({\theta }_{\mathrm{0}})\exp [U_{\mathrm{c}}(\mathrm{1}/\theta -\mathrm{1}/{\theta }_{\mathrm{0}})],U_{\mathrm{c}}=\mathrm{5400}\pm \mathrm{500}K;\hfill \\ {\tau }_{\mathrm{L}}(\theta )={\tau }_{\mathrm{L}}({\theta }_{\mathrm{0}})\exp [U_{\mathrm{L}}(\mathrm{1}/\theta -\mathrm{1}/{\theta }_{\mathrm{0}})],U_{\mathrm{c}}=\mathrm{9400}\pm \mathrm{500}K.\hfill \end{array}

This reaction kinetics proposed by Ulm et al. [45], based on the Larive’s kinetics according to the experimental program carried out at Laboratoires des Ponts et Chaussees in Paris [51], is essentially the same as that proposed by Huang and Pietruszczak [46] in the general concept [103]. The reaction kinetics was broadly adopted to combine with various nonlinear mechanical constitutive models.

Among these chemo-mechanical coupling models based on the reaction kinetics proposed by Ulm et al. [45], Farage et al. [106] developed a smeared crack finite element approach to analyze the ASR effects in concrete structures under certain loading and boundary conditions. Fairbairn et al. [107] also presented an ASR chemo-mechanical model, in which the stress-induced anisotropy was represented by the classical smeared crack model. This model was used to predict the expansion of a concrete dam, and the computed displacements had good agreement with the field observations.

Saouma and Perotti [108] further considered the effects of stress state on the ASR expansion of concrete. They assumed relatively high compressive or tensile stresses inhibit ASR expansion due to the formation of microcracks or macrocracks that absorb the expanding gel, and introduced two functions in terms of tensile and compression stresses to account for the reduction. Therefore ASR expansion strains in the three principal directions are determined based on the stress state, resulting in anisotropic ASR expansion. Deterioration of material properties during continuing ASR is considered as Huang and Pietruszczak model [46], and follows Eq. (16). This model was applied to a two-dimensional analysis of an arch gravity dam.

Multon et al. [109] used the chemoelastic model to investigate the effects of moisture distribution and stress state on the ASR-induced expansion in concrete. The numerical results were calibrated by the experimental observation [17]. It concluded that cracking and compressive stresses have a large influence on the anisotropy of the ASR swellings, and the anisotropy should be taken into consideration for predicting the behavior of ASR-damaged structures. Furthermore, the effect of stresses on ASR expansion anisotropy can be precisely quantified [22].

Comi et al. [47] combined the reaction kinetics with an isotropic damage model to develop a chemo-thermo-damage model. The model was first calibrated and validated on the basis of test data [22,51]. The damage pattern due to ASR in Fontana dam [110] was then simulated and compared to the actual cracking pattern (Fig. 17). Later, Comi and Perego [48] further extended this model into an anisotropic damage approach.

Besides the models based on Ulm and colleague’s work, Bangert et al. [111] developed an ASR chemo-hygro-mechanical model within the framework of Theory of Porous Media. Concrete was treated as a mixture of three superimposed and interacting components: the skeleton, the pore liquid and the pore gas. The skeleton consists of the unreacted, unswollen and the reacted, swollen materials. The mixture is illustrated in Fig. 18. The ASR was described using the first-order reaction kinetics. It was found that even a slight non-uniform distribution of moisture contents would produce stress gradient which may result in cracking in the structure.

Poyet et al. [112] proposed another law of reaction kinetics for modeling ASR expansion considering the influence of water and temperature. A dimensionless variable A is defined to monitor the progress of ASR. It depends on the absolute temperature T, time t and saturation ratio S_r. The evolution of ASR advancement A is given by

(27)\dot{A}(S_{\mathrm{r}},T,t)={\alpha }_{\mathrm{0}}\exp \left[{\displaystyle \frac{Ea}{R}}\left({\displaystyle \frac{\mathrm{1}}{\overline{T}}}-{\displaystyle \frac{\mathrm{1}}{T}}\right)\right]{\displaystyle \frac{{(S_{\mathrm{r}}-S_{\mathrm{r}}^{\mathrm{0}})}^{+}}{(\mathrm{1}-S_{\mathrm{r}}^{\mathrm{0}})}}{[S_{\mathrm{r}}-A(S_{\mathrm{r}},T,t)]}^{+},

where {(\cdot )}^{+} is the positive part of the equation, {\alpha }_{\mathrm{0}} is a constant that characteristic ASR kinetics, Ea is the activation energy of the ASR, R is the gas constant, \overline{T} is the reference temperature, T is the current temperature, and S_{\mathrm{r}}^{\mathrm{0}} is the smallest saturation ratio necessary to allow the evolution of the chemical reaction.

The free expansion strain {ϵ}_{\mathrm{a}\mathrm{s}\mathrm{r}} is linked to the advancement A by the following relation

(28)

where A_{\mathrm{0}} denotes the minimal ASR advancement necessary to initiate swelling.

Later, Grimal et al. [49] combined the above ASR advancement with a visco-elastic-plastic orthotropic damage model to simulate long-term behavior of concrete, considering creep and drying shrinkage phenomena. The rheological model is illustrated in Fig. 19. The mechanical effects of ASR are introduced by gel pressure P_{\mathrm{g}}, and its formulation is chosen as

(29)P_{\mathrm{g}}=K_{\mathrm{g}}{[A{V}_{\mathrm{g}}-{(A_{\mathrm{0}}{V}_{\mathrm{g}}+b_{\mathrm{g}}trϵ)}^{+}]}^{+},

where K_{\mathrm{g}} is the gel bulk modulus, b_{\mathrm{g}} is a coefficient that allows a decrease of gel pressure according to the volumetric expansion trϵ, V_{\mathrm{g}} is the maximum gel volume fraction that can be created by ASR.

The rheological model was calibrated and used to reproduce an experimental program comprising various reinforced reactive concrete beams [50,113]. The results showed that the model gave accurate midspan deflection and crack pattern of the beams.

The aforementioned macroscopic models all belong to the deterministic analysis, Capra and Sellier [114] have attempted to develop an orthotropic model for swelling of concrete subjected to ASR based on a probabilistic description of the main parameters of concrete and ASR. The reaction is modeled by a global kinetics including temperature and humidity effects. A probability is defined to relate to the damage rate of the material, thus the constitutive model describes an orthotropic decreasing of the elastic properties and the residual swelling under both ASR and mechanical loadings.

Conclusions

This paper provides a comprehensive review of the state-of-the-art of modeling for ASR in concrete. The mechanism of alkali-silica reaction (ASR) gel formation and swelling is first outlined, and then theoretical approaches, meso- and macroscopic models for ASR analysis are summarized.

The theoretical approaches mainly focus on the reaction kinetics and ASR-induced swelling in a representative volume element which is composed of a spherical aggregate particle and the surrounding cement paste. These approaches characterize the reaction at the local level and the expansion on the material level. Several hypotheses are generally considered to simplify the approach based on the empirical knowledge of ASR. The application of these approaches is successful in predicting the pessimum expansion of ASR-affected concrete and the corresponding pessimum size of aggregate. However, the reaction mechanisms adopted in most of the existing theoretical approaches are based on the experiments performed on opal and highly reactive aggregates. These aggregates are not of common use in structures, thus extending the reaction mechanism to normal structures needs further verification. Besides, it is unable to describe the micro-cracks in the matrix and the degradation mechanism of the materials.

Very few mesocopic models have been developed for ASR-induced degradation mechanism within the framework of finite element method. Aggregate particles, cement paste and ASR products are explicitly represented using the mesh refinement technique. ASR gel expands and exerts pressures on the matrix to crack the concrete. A damage constitutive model is introduced in aggregate and cement paste, thereby micro-cracks propagating. The ASR gel expansion is generally determined by an inverse analysis, thus the chemical reaction kinetics is not well considered. In addition, these models are formulated based on the continuum approach, in which simulation of multi-crack development is very difficult. Discontinuous approach such as the discrete element method can easily solve this problem due to its adequacy for modeling multi-crack initialization and propagation in concrete. Development of micromechanical discrete element models for investigating mechanisms of mechanical degradation induced by ASR may be promising.

The macroscopic models have been developed in terms of chemo-mechanical coupling, and analyze ASR effects on structural scale. These models generally consist of two parts, i.e., mechanical constitutive and reaction kinetics. The mechanical constitutive relation ranges from linear elastic assumption to isotropic/anisotropic damage models. Sophisticated constitutive is also introduced considering even creep and shrinkage effects during ASR process. The reaction kinetics either adopts Larive’s kinetics or uses experimental observations. The validation of the macroscopic models is to compare the predicted deformation with the experimental or field measurements. The macroscopic models are applicable to predicting the structural behavior and assisting the remedial measures of structures impacted by ASR. However, the predicted stress field cannot be assured to be sufficiently accurate due to the experimental based reaction kinetics. In addition, the macroscopic models cannot provide an explanation to the failure mechanisms of concrete due to the cracking process associated with ASR. A possible way to solve these problems may be the development of an approach that allows bridging the scale from the local chemo-mechanical mechanisms to the macroscopic response of the structures subjected to ASR.

References

[1] Stanton T E. Expansion of concrete through reaction between cement and aggregate. Proceeding American Society of Civil Engineers, 1940, 66: 1781–1811

[2] Fernandes I, Noronha F, Teles M. Microscopic analysis of alkali-aggregate reaction products in a 50-year-old concrete. Materials Characterization, 2004, 53(2-4): 295–306

[3] Ramyar K, Çopuroglu O, Andiç Ö, Fraaij A L A. Comparison of alkali-silica reaction products of fly-ash- or lithium-salt-bearing mortar under long-term accelerated curing. Cement and Concrete Research, 2004, 34(7): 1179–1183

[4] Peterson K, Gress D, Vandam T, Sutter L. Crystallized alkali-silica gel in concrete from the late 1890s. Cement and Concrete Research, 2006, 36(8): 1523–1532

[5] 1260-94 AC. Standard test method for determining the potential alkali reactivity of combinations of cementious materials and aggregate (accelerated mortar-bar method). Annual Book of ASTM Standards 2002, American Society for Testing and Materials, 1916 Race Street, Philadelphia, PA 19103. 2002; 04.02

[6] A23.2-25A-00 C. Detection of alkali-silica reactive aggregate by accelerated expansion of mortar bars. CSA A23.2-00: Methods of Test for Concrete, Canadian Standards Association, Mississauga

[7] TC191-ARP-AAR02 R. Detection of potential alkali-reactivity of aggregates-the ultra-accelerated mortar-bar test. Materials and Structures, 2000, 33: 283–293

[8] A23. 2-14A-00 C. Potential expansivity of aggregates (procedure for length change due to alkali-aggregate reaction in concrete prisms). CSA A23.2-00: Methods of Test for Concrete, Canadian Standards Association, Mississauga (ON), 2000, 207–216

[9] 1293-01 AC. Standard test method for concrete aggregates by determination of length change of concrete due to alkali-silica reaction. Annual Book of ASTM Standards, vol. 04.02 (Concrete and Aggregates), Philadelphia (PA), 2002

[10] Hobbs D W. Alkali-silica reaction in concrete. London: Thomas Telford, 1988

[11] Swamy R N, Al-Asali M M. Engineering properties of concrete affected by alkali-silica reaction. ACI Materials Journal, 1988, 85: 367–374

[12] Swamy R N, Al-Asali M M. Effect of alkali-silica reaction on the structural behavior of reinforced concrete beams. ACI Materials Journal, 1989, 86: 451–459

[13] Clark L A. Modeling the structural effects of alkali-silica reac-tions on reinforced concrete. ACI Materials Journal, 1991, 88: 271–277

[14] Ahmed T, Burley E, Rigden S, Abu-Taira A I. The effect of alkali reactivity on the mechanical properties of concrete. Construction & Building Materials, 2003, 17(2): 123–144

[15] Fan S F, Hanson J M. Length expansion and cracking of plain and reinforced concrete prisms due to alkali-silica reaction. ACI Materials Journal, 1998, 95: 480–487

[16] Monette L J, Gardner N J, Grattan-Bellew P E. Re-sidual strength of reinforced concrete beams damaged by alkali-silica reaction—Examination of damage rating index method. ACI Materials Journal, 2002, 99: 42–50

[17] Multon S, Multon S, Seignol J F, Toutlemonde F. Structural behavior of concrete beams affected by alkali-silica reaction. ACI Materials Journal, 2005, 102: 67–76

[18] Giaccio G, Zerbino R, Ponce J, Batic O. Mechanical behavior of concretes damaged by alkali-silica reaction. Cement and Concrete Research, 2008, 38(7): 993–1004

[19] Marzouk H, Langdon S. The effect of alkali-aggregate reactivity on the mechanical properties of high and normal strength concrete. Cement and Concrete Composites, 2003, 25(4-5): 549–556

[20] Berra M, Mangialardi T, Paolini A E. Rapid evaluation of the threshold alkali level for alkali-reactive siliceous aggregates in concrete. Cement and Concrete Composites, 1999, 21(4): 325–333

[21] Smaoui N, Berube M, Fournier B, Bissonnette B, Durand B. Effects of alkali addition on the mechanical properties and durability of concrete. Cement and Concrete Research, 2005, 35(2): 203–212

[22] Multon S, Toutlemonde F. Effect of applied stresses on alkali-silica reaction-induced expansions. Cement and Concrete Research, 2006, 36(5): 912–920

[23] Multon S, Cyr M, Sellier A, Leklou N, Petit L. Coupled effects of aggregate size and alkali content on ASR expansion. Cement and Concrete Research, 2008, 38(3): 350–359

[24] Puertas F, Palacios M, Gil-Maroto A, Vázquez T. Alkali-aggregate behaviour of alkali-activated slag mortars: Effect of aggregate type. Cement and Concrete Composites, 2009, 31(5): 277–284

[25] Berra M, Faggiani G, Mangialardi T, Paolini A E. Influence of stress restraint on the expansive behaviour of concrete affected by alkali-silica reaction. Cement and Concrete Research, 2010, 40(9): 1403–1409

[26] Multon S, Cyr M, Sellier A, Diederich P, Petit L. Effects of aggregate size and alkali content on ASR expansion. Cement and Concrete Research, 2010, 40(4): 508–516

[27] Multon S, Toutlemonde F. Effect of moisture conditions and transfers on alkali silica reaction damaged structures. Cement and Concrete Research, 2010, 40(6): 924–934

[28] Smaoui N, Bissonnette B, Bérubé M A, Fournier B. Stresses induced by alkali-silica reactivity in prototypes of reinforced concrete columns incorporating various types of reactive aggregates. Canadian Journal of Civil Engineering, 2007, 34(12): 1554–1566

[29] Steffens A, Li K, Coussy O. Aging approach to water effect on alkali-silica reaction degradation of structures. Journal of Engineering Mechanics of Materials, 2003, 129: 50–59

[30] Poyet S, Sellier A, Capra B, Thèvenin-Foray G, Torrenti J M, Tournier-Cognon H, Bourdarot E. Influence of water on alkali-silica reaction: Experimental Study and Numerical Simulations. Journal of Materials in Civil Engineering, 2006, 18(4): 588–596

[31] Bazǎnt Z P, Steffens A. Mathematical model for kinetics of alkali-silica reaction in concrete. Cement and Concrete Research, 2000, 30(3): 419–428

[32] Suwitoa A, Jinb W, Xia Y, Meyerc C. A Mathematical Model for the Pessimum Effect of ASR in Concrete. 2000

[33] Multon S, Sellier A, Cyr M. Chemo-mechanical modeling for prediction of alkali silica reaction (ASR) expansion. Cement and Concrete Research, 2009, 39(6): 490–500

[34] Vivian H E. The effects on mortar expansion of reactive component in the aggregate. Studies in cement-aggregate reactions, part 10. CSIRO Bull, 1947, 256: 13–20

[35] Mather B. How to make concrete that will not suffer deleterious alkali-silica reaction. Cement and Concrete Research, 1999, 29(8): 1277–1280

[36] Alasali M M, Malhotra V, Soles J A. Performance of various test methods for assessing the potential alkali reactivity of some Canadian aggre-gates. ACI Materials Journal, 1991, 88: 613–619

[37] Shehata M H, Thomas M D A. The effect of fly ash composition on the expansion of concrete due to alkali-silica reaction. Cement and Concrete Research, 2000, 30(7): 1063–1072

[38] Bleszynski R F, Thomas M D A. Microstructural Studies of Alkali-Silica Reaction in Fly Ash Concrete Immersed in Alkaline Solutions. Advanced Cement Based Materials, 1998, 7(2): 66–78

[39] Karakurt C, Topçu İ B. Effect of blended cements produced with natural zeolite and industrial by-products on alkali-silica reaction and sulfate resistance of concrete. Construction & Building Materials, 2011, 25(4): 1789–1795

[40] Comby-Peyrot I, Bernard F, Bouchard P O, Bay F, Garcia-Diaz E. Development and validation of a 3D computational tool to describe concrete behaviour at mesoscale. Application to the alkali-silica reaction. Computational Materials Science, 2009, 46(4): 1163–1177

[41] Dunant C F, Scrivener K L. Micro-mechanical modelling of alkali-silica-reaction-induced degradation using the AMIE framework. Cement and Concrete Research, 2010, 40(4): 517–525

[42] Amberg F. Performance of Dams affected by Expanding Concrete. V-0009401EN/1021-R-237, 2011

[43] Léger P, Côté P, Tinawi R. Finite element analysis of concrete swelling due to alkali-aggregate reactions in dams. Computers & Structures, 1996, 60(4): 601–611

[44] Pietruszczak S. On the mechanical behaviour of concrete subjected to alkali-aggregate reaction. Computers & Structures, 1996, 58(6): 1093–1097

[45] Ulm F J, Coussy O, Kefei L, Larive C. Thermo-chemo-mechanics of ASR expansion in concrete structures. Journal of Engineering Mechanics, 2000, 126(3): 233–242

[46] Huang M, Pietruszczak S. Modeling of Thermomechanical Effects of Alkali-Silica Reaction. Journal of Engineering Mechanics, 1999, 125(4): 476–485

[47] Comi C, Fedele R, Perego U. A chemo-thermo-damage model for the analysis of concrete dams affected by alkali-silica reaction. Mechanics of Materials, 2009, 41(3): 210–230

[48] Comi C, Perego U. Anisotropic Damage Model for Concrete Affected by Alkali-Aggregate Reaction. International Journal of Damage Mechanics, 2011, 20(4): 598–617

[49] Grimal E, Sellier A, Pape Y L, Bourdarot E. Creep, Shrinkage, and Anisotropic Damage in Alkali-Aggregate Reaction Swelling Mechanism-Part I: A Constitutive Model. ACI Materials Journal, 2008, 105: 227–235

[50] Grimal E, Sellier A, Pape Y L, Bourdarot E. Creep, Shrinkage, and Anisotropic Damage in Alkali-Aggregate Reaction Swelling Mechanism-Part II: Identification of Model Parameters and Application. ACI Materials Journal, 2008, 105: 236–242

[51] Larive C. Apports Combin'es de l’Experimentation et de la Modélisation à la Comprehension del Alcali-Réaction et de ses Effets Mécaniques. Dissertation for the Doctoral Degree, Laboratoire Central des Ponts et Chaussées, Paris (in France). 1998

[52] Dent Glasser L S, Kataoka N. N. K. The chemistry of alkali-aggregate reaction. Cement and Concrete Research, 1981, 11(1): 1–9

[53] Dent Glasser L S, Kataoka N. On the role of Calcium in the Alkali-aggregate reaction. Cement and Concrete Research, 1982, 12(3): 321–331

[54] Deschenes D J, Bayrak O, Folliard K J. ASR/DEF-damaged bent caps: shear tests and field implications. Technical Report No 12-8XXIA006, Texas Department of Transportation. 2009

[55] Dron R, Brivot F. Thermodynamic and kinetic approach to the alkali-silica reaction. Part 1: Concepts. Cement and Concrete Research, 1992, 22(5): 941–948

[56] Dron R, Brivot F. Thermodynamic and kinetic approach to the alkali-silica reaction. Part 2: Experiment. Cement and Concrete Research, 1993, 23(1): 93–103

[57] Diamond S. Chemistry and Other Characteristics of ASR Gels. Proceedings of the 11th International Conference on Alkali-Aggregate Reaction in Concrete, Quebec City, Canada. 2000, 31–40

[58] Knudsen T, Thaulow N. Quantitative microanalyses of alkali-silica gel in concrete. Cement and Concrete Research, 1975, 5(5): 443–454

[59] Thaulow N, Jakobsen U H, Clark B. Composition of alkali silica gel and ettringite in concrete railroad ties SEM-EDX and X-ray diffraction analyses. Cement and Concrete Research, 1996, 26(2): 309–318

[60] Regourd M, Hornain H, Poitevin O. Alkali-aggregate reaction-concrete microstructural evolution. Proceedings of the 5l International Conference on Alkali-Aggregate Reaction in Concrete, Cape Town, South Africa. 1981, S252/35

[61] Fernandes I. Composition of alkali-silica reaction products at different locations within concrete structures. Materials Characterization, 2009, 60(7): 655–668

[62] Šachlová Š, Přikryl R, Pertold Z. Alkali-silica reaction products: Comparison between samples from concrete structures and laboratory test specimens. Materials Characterization, 2010, 61(12): 1379–1393

[63] Kawamura M, Arano N, Terashima T. Composition of ASR gels and expansion of mortars. In: Cohen M, Mindess S, Skalny J, editors Materials Science of Concrete: Special Volume-The Sidney Diamond Symposium Westerville, OH: American Ceram-ic Society. 1998, 261–76

[64] Thomas M. The role of calcium in alkali -silica reaction. In: Cohen M, Mindess S, Skalny J, editors Materials Science of Concrete — The Sidney Diamond Symposium Westerville, OH: American Ceramic Society Bulletin. 1998, 325–37

[65] Powers T C, Steinour H H. An interpretation of some published researchers on alkali-aggregate reaction. Part 1-the chemical reactions and mechanism of expansion. ACI Journal Proceedings., 1955, 51: 497–516

[66] Powers T C, Steinour H H. An interpretation of some published researchers on alkali-aggregate reaction. Part 2-a hypothesis concerning safe and unsafe reactions with reactive silica in concrete. ACI Journal Proceedings, 1955, 51: 785–812

[67] Helmuth R, Stark D. Alkali-silica reactivity mechanisms. Ed Skalny, J. Materials Science of Concrete, 1992, III: 131–208

[68] Jun S S, Jin C S. ASR products on the content of reactive aggregate. KSCE Journal of Civil Engineering, 2010, 14(4): 539–545

[69] Scrivener K L, Monteiro P J M. The alkali-silica reaction in a monolithic opal. Journal of the American Ceramic Society, 1994, 77(11): 2849–2856

[70] Fernandes I, Noronha F, Teles M. Examination of the concrete from an old Portuguese dam: Texture and composition of alkali-silica gel. Materials Characterization, 2007, 58(11-12): 1160–1170

[71] Hansen W C. Studies relating to the mechanism by which the alkali-aggregate reaction produces expansion in concrete. Journal of the American Concrete Institute, 1944, 15: 213–227

[72] Dent Glasser L S. Osmotic pressure and the swelling of gels. Cement and Concrete Research, 1979, 9(4): 515–517

[73] Poole A B. Alkali-silica reactivity mechanisms of gel formation and expansion. In: Proceedings of the 9th International Conference on Alkali-Aggregate Reaction, London (England). 1992, 104: 782–9

[74] McGowan J K, Vivian H E. Studies in cement-aggregate reaction: correlation between crack development and expansion of mortars. Australian Journal of Applied Science, 1952, 3: 228–232

[75] Ponce J, Batic O. Different manifestations of the alkali-silica reaction in concrete according to the reaction kinetics of the reactive aggregate. Cement and Concrete Research, 2006, 36(6): 1148–1156

[76] Idorn G M. A discussion of the paper 'Mathematical model for kinetics of alkali-silica reaction in concrete' by Zdenek P. Bazant and Alexander Steffens. Cement and Concrete Research, 2001, 31(7): 1109–1110

[77] Garcia-Diaz E, Riche J, Bulteel D, Vernet C. Mechanism of damage for the alkali-silica reaction. Cement and Concrete Research, 2006, 36(2): 395–400

[78] Ichikawa T, Miura M. Modified model of alkali-silica reaction. Cement and Concrete Research, 2007, 37(9): 1291–1297

[79] Bazant Z P, Zi G, Meyer C. Fracture mechanics of asr in concretes with waste glass particles of different sizes. Journal of Engineering Mechanics, 2000, 126(3): 226–232

[80] Hobbs D W. The alkali-silica reaction—a model for predicting expansion in mortar. Magazine of Concrete Research, 1981, 33(117): 208–220

[81] Groves G W, Zhang X. A dilatation model for the expansion of silica glass/OPC mortars. Cement and Concrete Research, 1990, 20(3): 453–460

[82] Furusawa Y, Ohga H, Uomoto T. An analytical study concerning prediction of concrete expansion due to alkali-silica reaction. Third International Conference on Durability of Concrete, Nice, France. 1994, 757–80

[83] Meyer C, Baxter S. Use of recycled glass and fly ash for precast concrete. Report NYSERDA 98-18 (4292-IABR-IA-96) to New York State Energy Research and Developement Authority, Department of Civil Engineering and Engineering Mechanics, Columbia University. 1998

[84] Suwito A, Jin W, Xi Y, Meyer C. A mathematical model for the pessimum effect of ASR in concrete. Concrete Science and Engineering, RILEM., 2002, 4: 23–34

[85] Xi Y, Suwito A, Wen X, Meyer C, Jin W. Testing and modeling alkali-silica reaction and the associated expansion of concrete. Mechanics of Quasi-Brittle Materials and Structures, Proceedings of International Workshop in honor of Prof ZP Bazant 60th birthday, Hermes Science Publications, Paris. 1998

[86] Xi Y, Jennings H M. Shrinkage of cement paste and concrete modelled by a multiscale effective homogeneous theory. Materials and Structures, 1997, 30(6): 329–339

[87] Fick A. Ueber Diffusion. Annalen der Physik., 1855, 170(1): 59–86

[88] Wang Z M, Kwan A K H, Chan H C. Mesoscopic study of concrete I: generation of random aggregate structure and finite element mesh. Computers & Structures, 1999, 70(5): 533–544

[89] Riche J, Garcia-Diaz E, Bulteel D, Siwak J M. Mechanism of damage for the alkali-silica reaction: relationship between swelling and reaction degree. in: Dhir R, Roderick Jones M, Li Z (Eds), Repair, Rejuvenation and Enhancement of Concrete, Proceedings of International Conference of Dundee. 2002, 94–102

[90] Haha M B, Gallucci E, Guidoum A, Scrivener K L. Relation of expansion due to alkali silica reaction to the degree of reaction measured by SEM image analysis. Cement and Concrete Research, 2007, 37(8): 1206–1214

[91] Moës N, Cloire M, Cartra P, Remacle J. A computational approach to handle complex microstructure geometries. Communications in Numerical Methods in Engineering, 2003, 192: 3163–3177

[92] Ben Haha M. Mechanical effects of alkali silica reaction in concrete studied by SEM-image analysis. DissertationTip. 2006

[93] Shin J h, Struble L J, Kirkpatrick R J. Modeling alkali-silica reaction using image analysis and finite element analysis. Proceedings of the 6th International Symposium on Cement & Concrete and Canmet/Aci International Symposium on Concrete Technology for Sustainable Development, 2006, 3: 1669–75

[94] Shin J h. Modeling alkali-silica reaction using image analysis and finite element analysis. DissertationTip, University of Illinois at Urbana-Champaign. 2009

[95] Struble L J, Diamond S. Swelling properties of synthetic alkali silica gels. Journal of the American Ceramic Society, 1981, 64(11): 652–655

[96] Charlwood R G, Solymar S V, Curtis D D. A review of alkali aggregate reactions in hydroelectric plants and dams. Proceedings of the International Conference of Alkali-Aggregate Reactions in Hydroelectric Plants and Dams, Fredericton, Canada. 1992, 129

[97] Thompson G, Charlwood R, Steele R, Curtis D. Mactaquac generating station Intake and spillway remedial measures. Proceedings for the Eighteenth International Congress on Large Dams, Durban, South Africa. 1994, V. 1, Q-68, R.24: 347–68

[98] Engineering IoS. Effects of alkali-silica reaction SETO, London. 1992

[99] Herrador M F, Martínez-Abella F, Hoyo Fernández-Gago R. Mechanical behavior model for ASR-affected dam concrete under service load: formulation and verification. Materials and Structures, 2009, 42(2): 201–212

[100] Saouma V, Xi Y. Literature review of alkali aggregate reactions in concrete dams. Swiss Federal Office for Water and Geology, Bienne Switzerland, Report No CU/SA-XI-2004/001. 2004

[101] Huang M, Pietruszczak S. Numerical analysis of concrete structures subjected to alkali-aggregate reaction. Mechanics of Cohesive-Frictional Materials, 1996, 1(4): 305–319

[102] Parvini M, Pietruszczak S, Gocevski V. Seismic analysis of hydraulic structures affected by alkali-aggregate reaction: a case study. Canadian Journal of Civil Engineering, 2001, 28(2): 332–338

[103] Winnicki A, Pietruszczak S. On mechanical degradation of reinforced concrete affected by alkali-silica reaction. Journal of Engineering Mechanics, 2008, 134(8): 611–627

[104] Coussy O. Mechanics of porous continua. Wiley, Chichester, UK. 1995

[105] Atkins P W. Physical Chemistry. 5th ed, Oxford: Oxford University Press, 1994

[106] Farage M, Alves J L D, Fairbairn E M R. Macroscopic model of concrete subjected to alkali-aggregate reaction. Cement and Concrete Research, 2004, 34(3): 495–505

[107] Fairbairn E M R, Ribeiro F L B, Lopes L E, Toledo-Filho R D, Silvoso M M. Modelling the structural behaviour of a dam affected by alkali-silica reaction. Communications in Numerical Methods in Engineering, 2005, 22(1): 1–12

[108] Saouma V, Perotti L. Constitutive Model for Alkali-Aggregate Reactions. ACI Materials Journal, 2006, 103: 194–202

[109] Multon S, Seignol J F, Toutlemonde F. Chemomechanical assessment of beams damaged by alkali-silica reaction. Journal of Materials in Civil Engineering, 2006, 18(4): 500–509

[110] Newell V A, Wagner C D. Fontana Dam A Crack in the Curve. Waterpower, 1999, 1–10

[111] Bangert F, Kuhl D, Meschke G. Chemo-hygro-mechanical modelling and numerical simulation of concrete deterioration caused by alkali-silica reaction. International Journal for Numerical and Analytical Methods in Geomechanics, 2004, 28(78): 689–714

[112] Poyet S, Sellier A, Capra B, Foray G, Torrenti J M, Cognon H, Bourdarot E. Chemical modelling of Alkali Silica reaction: Influence of the reactive aggregate size distribution. Materials and Structures, 2007, 40(2): 229–239

[113] Grimal E, Sellier A, Multon S, Le Pape Y, Bourdarot E. Concrete modelling for expertise of structures affected by alkali aggregate reaction. Cement and Concrete Research, 2010, 40(4): 502–507

[114] Capra B, Sellier A. Orthotropic modelling of alkali-aggregate reaction in concrete structures: numerical simulations. Mechanics of Materials, 2003, 35(8): 817–830

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