In the pore solution, alkali metal ions are generally balanced with hydroxide ions at a 1:1 ratio, so the alkalinity represents the alkali metal hydroxide concentration. Therefore, it is sufficient to only model the alkalinity of the solution. By considering the system as closed without any supply or leaching of alkali, the total amount of alkali (Na + K) per unit volume of mortar C_mortar can be expressed by the following equation from Kawabata et al. (30):

where C_cp is the total alkali content per unit volume of cement paste (mol/m³-paste), A_cp is the cement paste content in the mortar (m³-paste/m³-mortar), C_ag is the alkali content instantaneously consumed by the aggregate (mol/g-aggregate), A_ag is the total reactive aggregate content in the mortar (g-aggregate/m³-mortar), γ_Na/Si is the (Na+K)/Si molar ratio of the ASR gel, and V_gel is the total amount of ASR gel in the mortar (mol/m³-mortar). Note that the alkalis supplied from aggregates are not considered since alkalis do not necessarily increase the pH of the pore solution (31).

Considering the equilibrium between the solid C-S-H gel and pore solution in the cement paste system, C_cp may be written as:

where C_CSH is the amount of C-S-H per unit volume of cement paste (g/m³-paste), C_fw is the free water per unit volume of cement paste (ml/m³-paste), R_s is the alkali content in the solid C-S-H (mol/g), and R_l is the alkali concentration in solution (mol/mL). The distribution ratio, R_d (mL/g), may be described by the next Equations:

where Ca/Si is the Ca/Si molar ratio, and γ and δ are experimental constants, which were 2.5 and −3.1, respectively (30).

Following the work of Furusawa et al. (26), the reaction was modeled one-dimensionally from the surface of the aggregate towards the interior. Assuming that the alkali profile within the reaction layer is linear, the reaction ratio can be expressed as:

where t is the duration of the reaction (hr), x is the thickness of the reaction layer (cm), k is the reaction rate constant (cm²/hr), C_cp is the concentration of alkali metal hydroxide in the paste (mol/L), and C_th is the threshold concentration of alkali metal hydroxide (mol/L).

The initial condition in which x is zero when t is zero gives:

Assuming that the aggregate particle is spherical and the particle is in contact with the pore solution, the reaction ratio of the aggregate i with radius R_i can be formulated as:

where R_i is the radius of the particle (cm) and α_i is the reaction ratio of the particle with radius R_i.

After calculating the reaction ratio for each particle of each particle size, the total amount of ASR gel is given by Equation [8]:

where β_i is the ratio of radius R_i to the total aggregate weight.

The stress σ_ASR imposed on the aggregates due to ASR gel formation can be described as:

where σ_ASR is the stress on the aggregates (MPa for gel), and S_gel is the stiffness per mol of ASR gel in the mortar (GPa/mol/m³-mortar).

Since the elastic modulus of aggregate is large, the elastic deformation of aggregate before cracking is negligible and is thus assumed to be zero for simplification. When the stress is greater than the critical tensile stress of the aggregate, ASR gel formed in the aggregates start to expand the mortar, giving:

where ε is expansion of the mortar, σ_cr is the critical tensile stress for aggregates to be cracked (MPa), E_agg is a modulus of the damaged aggregate (GPa), E_d is a variable for the conversion of the stress imposed on the aggregates to the expansion of the mortar (-), and <X>⁺ is the positive part of X.

Hence, when E_d is constant, ASR expansion is linearly related to the reaction ratio. However, this expansion is not realistic. Aggregate and paste are damaged by ASR expansion and this damage simultaneously reduces the restraints on the ASR gel with the increase in expansion. Therefore, the expansion pressure on the ASR gel is released and reaction kinetics no longer control the expansion. A damage model is introduced as a simplifying assumption that expresses the microstructural damage and resultant reduced expansion pressure of the ASR gel as:

where α_th is the threshold expansion above which the viscoelasticity of the ASR gel dominates the expansion due to reduced confinement from damage, E₀ is the initial value without damage, and ω is a parameter representing the interaction between the rate of damage evolution and viscosity of ASR gel. Note that this is a simplified model to represent the above factors. The model assumes an exponential decline in the stiffness, although the aggregate or paste will split apart in reality. This is consistent with commonly used damage models.

Ordinary Portland cement with an alkali content of 0.55 wt% was used as the reference material. Two reactive coarse aggregates were used for testing: a highly reactive andesite (designated “TO”) with trydimite reactive phase, and andesite from a different source (designated “SI”) containing cristobalite and volcanic glass as the reactive minerals. Both aggregates have caused ASR damage in real structures. Considering the proportional pessimum effect, the reactive aggregate proportion was 30 wt% for both reactive aggregates. For comparison, published data (highly reactive andesite “NT” from (34)) were also included. Pure limestone “LSG” was found to be a non-reactive coarse aggregate by various test methods such as CPT, the chemical method, mortar bar method, and field experiences. Non-reactive limestone sand “LSS” was also used. The properties of the aggregates are summarized in Table 1.

The mix proportions of the concrete specimens are summarized in Table 2. The water content was 160 kg/m³, water-to-cement ratio was 0.50, and air content was 4.5 %. This mixture is commonly used in Japan. The grading of aggregate was in accordance with JIS but not controlled precisely like RILEM or CSA. The total alkali content (Na₂O_eq) of concrete was boosted to 5.50 kg/m³ by adding NaOH solution to the mixing water.

AW-CPT was applied as an expansion test for the concrete specimens (75 × 75 × 250 mm³). The general protocols are nearly identical to those of RILEM AAR-3 (38 °C) and AAR-4 (60 °C). The only difference is that concrete specimens are wrapped in a wet cloth containing an alkaline solution (1.5 mol/L NaOH). After the concrete specimens were demolded, they were submerged in water for 30 mins. This short submersion process was used despite 3-4% of the alkali leaching from the concrete during this step (8).

The concrete specimen was then wrapped with a nonwoven polypropylene cloth containing 50 g of 1.50 mol/L NaOH solution, which approximated the alkali metal ion concentration of the pore solution of concrete with 5.50 kg-Na₂O_eq/m³ (30). The wet cloth was placed on top of a thin plastic film on a table. Then, the specimen was placed on top of the cloth, and the cloth and film were wrapped around the specimen. The cloth was secured by rubber bands to prevent detachment. Plastic film (or a plastic bag) was used to minimize alkaline solution losses from the cloth and to prevent it from drying. Three wrapped specimens were placed over water in storage containers, which were then placed in temperature-controlled chambers at 20, 40, or 60 °C.

To measure length changes, the storage container was placed in a room at 20 °C for one day to let the specimens cool. After cooling and carefully separating the cloth from the specimens, the length and weight of the specimens and the weight of the cloth were measured. Ion-exchanged water was added after the measurements to keep the solution mass in the cloth at 50 g. Note that alkaline solution was not used as an additional water supply to avoid an excessive supply of alkali to the concrete specimen. This process was necessary to supply enough moisture for the ASR. The same cloths were used throughout the testing process.

The AW-CPT results for different temperatures are presented in Figure 1. For the TO aggregate, the early-stage expansion was higher at a higher temperature. At 60 °C, the expansion appeared to plateau after 50 weeks. At 38 °C, although the initial expansion onset was later than at 60 °C, the expansion continued over time and exceeded that at 60 °C after 78 weeks. At 20 °C, the expansion began at 15 weeks and was almost linear with time. For the SI aggregate, the expansion rates at the early and late stages were higher at higher temperatures. However, while the early-stage expansion rate at 38 °C was 42% of that at 60 °C, the late-stage expansion rate at 38 °C was 65% of that at 60 °C. Regarding chemical reaction kinetics, the expansivity of ASR gel reduced for late-stage expansions at high temperatures. The early- and late-stage expansion rates are summarized in Figure 2. Note that early-stage expansion is defined as expansions smaller than 0.15%, since the expansions showed strong nonlinearity above approximately 0.15%. Also, the published data from our previous study are plotted as “NT aggregate” in the figure. The early-stage expansion rates depended strongly on temperature, while late-stage rates were almost constant and irrespective of temperature. These trends are consistent with the previous study.

The experimental results suggest that the reduction in the expansivity of ASR gel may be pronounced at higher temperatures, which could be strongly affected by the aggregate through the location of gel pockets and aggregate texture. A numerical simulation is required to separate the chemical reaction from the expansion.

In the previous study (23), the parameters used in the damage model (ω and α_th in Eq. 11) were determined by fitting the numerical results to the experimental data. In this paper, the early-stage expansion rate was assumed to be a function of the stiffness of the gel (S_gel), and the late-stage expansion rate was a nonlinear parameter ω. The parameters were determined by fitting the numerical results to the experimental plots while fixing the other parameters for each aggregate.

The parameters (k, γ_Na/Si) of the reaction models and the activation energy of k were determined by chemical testing. Then, for the expansion models, σ_cr was determined from the expansion results for the TO aggregate at 20 °C. For S_gel×E₀, the expansion results until 0.15% expansion for the TO aggregate at 38 °C were used. Above 0.15% expansion, ω was obtained to fit the late-stage expansion while fixing α_th for each aggregate. The same approach was applied to the SI and NT aggregates. Note that an early-stage expansion threshold of 0.05% was used for NT.

The results of the numerical simulation are presented in Figure 3. Temperature dependencies of the parameters are displayed as Arrhenius plots in Figure 4 (a). Below 0.10-0.15% expansion, the calculated expansion curves were almost consistent with the experimental results, indicating that early-stage expansion almost conforms to reaction kinetics. In contrast, S_gel×E₀ (representing early-stage expansion) depended strongly on temperature and reduced with increasing temperature. The activation energy was different for each aggregate type. This suggests that the stiffness of ASR gel can be reduced even in the early stage, and the effect of temperature on the stiffness varies with aggregate type.

For late-stage expansion, the numerical expansion curves were fitted to the experimental plots by changing ω. Note that according to our previous study, there were large discrepancies between the numerical and experimental results when ω was zero at expansions above 0.10-0.15%. This means that expansion behavior is not equivalent to reaction kinetics. Thus, late-stage expansion cannot be simulated by only considering the chemical reaction. Arrhenius plots of ω are shown in Figure 4 (b). The parameter ω depended on temperature for TO and NT, but not for SI. This indicates that the reduction in the expansion ability of ASR gel varies for different aggregate characteristics such as reactive minerals and texture, which is strongly related to how ASR gel exerts expansion pressure inside the aggregate. For TO and NT, higher temperatures increased ω. This may indicate that excessively high temperatures cause considerable amounts of ASR gel to exude from the reaction site (gel pocket) and lose its expansion ability in the late stage.