# Modeling Time-Dependent Behavior of Concrete Affected by Alkali Silica Reaction in Variable Environmental Conditions

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## Abstract

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## 1. Introduction

## 2. Multi-Physics Formulation

#### 2.1. Modeling ASR Expansion

#### 2.2. Hygro-Thermo-Chemical (HTC) Model

#### 2.3. Mechanical Behavior

#### 2.3.1. LDPM for Concrete Elastic, Cracking and Damage Behavior

#### 2.4. Microprestress-Solidification Theory for Viscous and Visco-Elastic Deformations

#### 2.5. ASR Induced Deformation

#### 2.6. Thermal and Hygral Deformations

## 3. Numerical Implementation

## 4. Numerical Simulations and Comparison with Experimental Data

#### 4.1. Identification of Cement Hydration Parameters

#### 4.2. Identification of HTC Parameters

#### 4.3. Identification of Shrinkage and Creep Parameters

#### 4.4. Calibration of LDPM Concrete Parameters

#### 4.5. Calibration of ASR Model Parameters

#### 4.6. Validation through Full Scale Beam Simulations

## 5. Discussion of Results

## 6. Conclusions

- ASR progression is a process that takes a few years to multi-decades depending on moisture and temperature conditions as well as cement chemistry and aggregate mineralogy. This makes ASR in full interaction with other aging and deterioration phenomena like creep, shrinkage and thermal expansions. Simple addition of the deformation induced by these phenomena is incorrect because the different phenomena are nonlinearly coupled.
- Meso-scale modeling reveals the sub-scale interactions between coupled phenomena that are not seen at the macroscopic length scale. Namely, for the case of ASR induced free expansion, only modeling of deformations at the meso-scale can capture meso-scale creep deformations and relaxation of meso-scale stress build up that are not seen at the macroscopic scale because the macroscopic stress is zero.
- Relative humidity effect on ASR expansion is essentially a moisture diffusion controlled process that can be modeled similarly to relative humidity effects on moisture diffusivity in concrete.
- Simplified average section models that describe creep and shrinkage can lead to large inaccuracy in predicting ASR deformations for nonsaturated conditions. The humidity profile has a significant effect on ASR expansions that can not be averaged.
- ASR expansions in reinforced concrete elements can lead to large internal forces build up and may lead to reinforcement yielding, reinforcement slippage, and partial bond loss.
- For any complex framework to be predictive, its calibration needs to depend on uncoupled phenomena, then, it must be validated clearly. This was accomplished here by a multi-step calibration procedure on companion specimens with no ASR expansion, followed by ASR expansion calibration, then finally validation on full scale beams. A key factor here is the degree of scatter in the experimental data which is reflected directly in the prediction results of the model.
- To the best knowledge of the authors, this is the only framework in literature that was calibrated on individual lab size specimens and was able to predict structural behavior. Other models are either directly calibrated based on structural response to simulate structural behavior, or are calibrated and validated based on individual lab size specimens.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Model Parameters

Param. (Units) | Modeled Chemical Reaction | Test for Calibration | Value | Source |
---|---|---|---|---|

${A}_{c1}$ (h${}^{-1}$) | Cement hydration | Calorimetric tests | 1.41$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{7}$ | [79] |

${A}_{c2}$ (-) | Cement hydration | Calorimetric tests | 5$\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ | [79] |

${\eta}_{c}$ (-) | Cement hydration | Calorimetric tests | 8 | [79] |

Param. (Units) | Modeled Behavior or Mechanism | Test for Calibration | Value | Source |
---|---|---|---|---|

${E}_{0}$ (GPa) | Elasticity | Any in the linear range | 133 | Calibrated |

$\alpha $ (-) | Poisson’s effect | 0.25 | [67] | |

${\xi}_{1}$ (MPa${}^{-1}$) | Short term visco-elasticity | Basic creep tests with short load duration (≈1 year) | 1.75 $\times {10}^{-5}$ | Calibrated |

${\xi}_{2}$ (MPa${}^{-1}$) | Long term viscocity | Basic creep tests with long load duration (≈10 year) | 7 $\times {10}^{-6}$ | Calibrated |

${n}_{\alpha}$ (-) | Aging visco-elasticity | Basic creep tests at different age of loading | 1.9 | [96] |

${\kappa}_{1}$ (MPa${/}^{\circ}\mathrm{K}$) | Evolution of the microprestress at variable temperature and relative humidity | Drying creep test or transitional thermal creep tests | 19 | [96] |

${\sigma}_{t}$ (MPa) | Tensile fracture | Fracture tests or tensile strength tests | 4.75 | Calibrated |

${\ell}_{t}$ (mm) | Tensile fracture | Fracture tests or size effect tests | 75 | Calibrated |

${\sigma}_{s}/{\sigma}_{t}$ (-) | Cohesive behavior in shear | Unconfined compression test | 3.07 | Calibrated |

${\mu}_{0}$ (-) | Frictional behavior | Triaxial compression tests at low confinement | 0.2 | [67] |

${\sigma}_{N0}$ (MPa) | Frictional behavior | Triaxial tests at high confinement | 600 | [67] |

${\sigma}_{c0}$ (MPa) | Yielding and pore collapse | Hydrostatic compression test | 150 | [67] |

${H}_{c0}/{E}_{0}^{28}$ (-) | Yielding and pore collapse | Hydrostatic compression test | 0.3 | [90] |

${H}_{c1}/{E}_{0}^{28}$ (-) | Yielding and pore distorsion | Passively confined tests | 0.1 | [90] |

${\kappa}_{c0}$ (-) | Material densification after pore collapse | Hydrostatic compression test | 4 | [90] |

${E}_{d}/{E}_{0}^{28}$ (-) | Material densification after pore collapse | Hydrostatic compression test at very high pressure or with unloading | 1 | [67] |

Param. (Units) | Modeled Behavior or Mechanism | Test for Calibration | Value | Source |
---|---|---|---|---|

${c}_{t}$ (J/kg${}^{\circ}$C) | Heat transfer | Thermal conductivity tests | 1100 | [79] |

${\lambda}_{t}$ (W/m${}^{\circ}$C) | Heat transfer | Thermal conductivity tests | 2.3 | [79] |

${g}_{1}$ (-) | Moisture content | Sorption/desorption isotherms relevant to two different values of hydration degree | 1.7 | [79] |

${g}_{2}$ (-) | Moisture content in C-S-H pores | Sorption/desorption isotherms relevant to two different values of hydration degree | 0.2 | [79] |

${D}_{0}$ (kg/mm h) | Moisture transport | Humidity profile during drying tests | 2 $\times {10}^{-9}$ | Calibrated |

${D}_{1}$ (kg/mm h) | Moisture transport | Humidity profile during drying tests | 4 $\times {10}^{-7}$ | Calibrated |

n (-) | Moisture transport | Humidity profile during drying tests | 2.35 | Calibrated |

Param. (Units) | Modeled Behavior or Mechanism | Test for Calibration | Value | Source |
---|---|---|---|---|

${\alpha}_{h}$ (-) | Shrinkage and swelling due to moisture content change | Shrinkage tests | 9$\times {10}^{-4}$ | Calibrated |

${\alpha}_{T}$ (-) | Thermal expansion and contraction | Thermal expansion tests | 1$\times {10}^{-5}$ | [96] |

Param. (Units) | Modeled Behavior or Mechanism | Test for Calibration | Value | Source |
---|---|---|---|---|

${\rho}_{g}$ (kg/m${}^{3}$) | ASR gel density | Free ASR expansion tests at 100 % relative humidity | 689 | [65,68] |

${\kappa}_{z1}$ (cm${}^{5}$kg${}^{-1}$day${}^{-1}$) | ASR gel formation | Free ASR expansion tests at 100 % relative humidity | 2.62 | Calibrated |

${\kappa}_{i}$ (-) | Water imbibition | Free ASR expansion tests at 100 % relative humidity | 1.45 $\times {10}^{-2}$ | Calibrated |

${\tilde{C}}_{i}^{1}$ (mm${}^{2}$/day) | Water imbibition | Free ASR expansion tests at 100 % relative humidity | 7.78 | Calibrated |

${\delta}_{c}$ ($\mathsf{\mu}$m) | ITZ porosity effect on ASR imposed strain | Free ASR expansion tests at 100 % relative humidity | 6.0 | Calibrated |

${c}_{a0}$ (kg/m${}^{3}$) | Alkali content effect | Free ASR expansion tests at 100 % relative humidity and different alkali contents | 2.7 | [65] |

${c}_{a1}$ (kg/m${}^{3}$) | Alkali content effect | Free ASR expansion tests at 100 % relative humidity and different alkali contents | 4.37 | [65] |

${r}_{D}$ (-) | Relative humidity effect | Free ASR expansion at different levels of relative humidity | 3600 | Calibrated |

${n}_{D}$ (-) | Relative humidity effect | Free ASR expansion at different levels of relative humidity | 2 | Calibrated |

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**Figure 1.**(

**a**) Idealization of gel formation in one aggregate; (

**b**) Diffusivity change with relative humidity; (

**c**) One Lattice Discrete Particle Model (LDPM) Cell around an aggregate piece; (

**d**) Equivalent rheological model based on strain additivity; (

**e**) Cylinder and Prism generated LDPM geometries (Aggregate are colored by their relative size); (

**f**) 1/8th of the simulated cylinder showing the discrete facets inside it surrounding the aggregate; (

**g**) 1/8th Hygro-Thermo-Chemical (HTC) cylindrical mesh colored by the values from RH field for the drying case at 420 days; (

**h**) The interpolated values of RH from the HTC mesh into LDPM facets centroids.

**Figure 2.**(

**a**) Experimental and numerically simulated RH values along the depth of the beam at 28 and 448 days; (

**b**) HTC mesh colored by the RH field at 448 days showing the quarter that was simulated; (

**c**) Experimental and numerically simulated average axial expansions of both cylinders and prisms under fully saturated, sealed and 30% RH exposure conditions; (

**d**) Midspan deflections of unreinforced NPC and RPC beams; (

**e**) Midspan deflections of reinforced RRC1 and RRC2 beams; (

**f**) Normalized evolutions of all simulated aggregate diffusion fronts.

**Figure 3.**(

**a**) Beam simulated geometry, showing symmetry boundary conditions, LDPM generated mesh and reinforcements for NRC, RRC1 and RRC2 beams (Aggregate are colored by their relative size); (

**b**) Simulated crack pattern distribution due to ASR with coupling and without coupling with creep and shrinkage deformations; (

**c**) Simulated pure ASR expansion versus coupled ASR, creep and shrinkage expansion; (

**d**) Simulated creep and shrinkage expansions only; (

**e**) Sum of simulated ASR shrinkage and creep expansions versus fully coupled expansion.

**Figure 4.**(

**a**) Simulated crack patterns and crack openings for both RPC and RRC1 beams showing the effects of reinforcement on crack suppression; (

**b**) Simulated rebar internal forces due to beam own weight only; (

**c**) Simulated rebar internal forces due to beam own weight, Alkali Silica Reaction (ASR), creep and shrinkage effects.

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**MDPI and ACS Style**

Alnaggar, M.; Di Luzio, G.; Cusatis, G.
Modeling Time-Dependent Behavior of Concrete Affected by Alkali Silica Reaction in Variable Environmental Conditions. *Materials* **2017**, *10*, 471.
https://doi.org/10.3390/ma10050471

**AMA Style**

Alnaggar M, Di Luzio G, Cusatis G.
Modeling Time-Dependent Behavior of Concrete Affected by Alkali Silica Reaction in Variable Environmental Conditions. *Materials*. 2017; 10(5):471.
https://doi.org/10.3390/ma10050471

**Chicago/Turabian Style**

Alnaggar, Mohammed, Giovanni Di Luzio, and Gianluca Cusatis.
2017. "Modeling Time-Dependent Behavior of Concrete Affected by Alkali Silica Reaction in Variable Environmental Conditions" *Materials* 10, no. 5: 471.
https://doi.org/10.3390/ma10050471