# A Review on the Transport-Chemo-Mechanical Behavior in Concrete under External Sulfate Attack

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Chemical Reaction Products of Sulfate Attack

^{+}/K

^{+}[27,28,29], as shown in Figure 2 [30]. Besides gypsum and ettringite, the insoluble brucite, $\mathrm{Mg}{\left(\mathrm{OH}\right)}_{2}$, is also formed to cause a decrease in the pH value of pore solutions and the decalcification of C-S-H gel, which becomes the cohesionless silica gel, ${\mathrm{SiO}}_{2}\cdot x{\mathrm{H}}_{2}\mathrm{O}$, or magnesium silicate, $3\mathrm{MgO}\cdot 2{\mathrm{SiO}}_{2}\cdot 2{\mathrm{H}}_{2}\mathrm{O}$. In short, sulfate products are different in different corrosion environments.

## 3. Formation Mechanism of Reaction Products

## 4. Failure Forms Caused by Chemical Reaction Attack

^{+}/K

^{+}, and brucite type CSA in the case of Mg

^{2+}. The different types of CSA present different failure forms of concrete, including softening, cohesiveness, volume expansion, and its-induced cracking/spalling [37,38]. Table 2 shows the relationship between cation types of sulfate solution, CSA classification, and failure forms of concrete. It should be pointed out that the third failure form is the most common among them. At the initial stage of CSA, there are two major disputes about the degradation of concrete exposed to the sulfate environment, namely “degradation cause” and “degradation mechanism”. The former refers to which reaction product (gypsum or ettringite) is the main cause of concrete expansion/cracking, while the latter indicates how CSA causes concrete degradation.

#### 4.1. Degradation Cause of Concrete

_{3}S cement with low C

_{3}A content was used to reduce the source of the aluminum phase required for ettringite formation, thereby minimizing the amount of ettringite formation. The results showed that the C

_{3}S cement specimen soaked in sulfate solution for a long time occurred macroscopic damage phenomena, such as volume expansion and surface cracking, mass, and strength loss. Additionally, the expansion deformation of the C

_{3}S cement specimen is obviously greater than that of the Portland cement specimen. Therefore, the formation of ettringite and gypsum is the main cause of the volume expansion or cracking of concrete subjected to CSA.

#### 4.2. Degradation Mechanism of Concrete

_{3}A is irrelevant to that of ettringite, which means that it is difficult to achieve the above transformation at room temperature [37,50]. In reality, SEM observation shows that ettringite is not found at the location of C

_{3}A hydrated products and on the surface of unhydrated C3A particles. Water swelling refers to the fact that, in the condition of the saturated lime solution, the gel-like and small ettringite colloid can be formed in concrete. This kind of colloid with a large specific surface area can absorb a large number of water molecules, resulting in the volume expansion of concrete [43]. However, Scrivener [50] believes that ettringite generated in concrete subjected to a sulfate attack is a typical crystal, and it is impossible to absorb a lot of water.

## 5. Model for Chemical Sulfate Attack

#### 5.1. Diffusion-Reaction Model of Sulfate

#### 5.2. Volume Expansion or Crystallization Pressure

#### 5.2.1. Free Volume Expansion of Concrete Caused by Sulfate Products

_{3}A). q is the equivalent reaction coefficient of ettringite produced by gypsum consumption, which is equal to 8/3. In reality, the gypsum formation from the sulfate ion and calcium ion, provided by the dissolution of solid calcium (calcium hydroxide and C-S-H gel), also causes the volume expansion of concrete.

Chemical Reactions | $\mathit{\Delta}{\mathit{V}}_{\mathit{i}}/{\mathit{V}}_{\mathit{i}}$ |
---|---|

$\mathrm{Equation}\left(\mathrm{a}\right):{\mathrm{C}}_{4}{\mathrm{AH}}_{13}+3{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+14{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}+\mathrm{CH}$ | 0.48 |

$\mathrm{Equation}\left(\mathrm{b}\right):{\mathrm{C}}_{4}{\mathrm{A}\overline{\mathrm{S}}\mathrm{H}}_{12}+2{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+16{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}$ | 0.51 |

$\mathrm{Equation}\left(\mathrm{c}\right):{\mathrm{C}}_{3}\mathrm{A}+3{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+26{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}$ | 1.26 |

_{CA}is the concentration of CA in the concrete; ${\phi}_{0}$ is the initial porosity in the concrete.

#### 5.2.2. Equivalent Expansive Force Generated by Crystallization Pressure

#### 5.3. Chemo-Mechanical Model of Sulfate Attack

#### 5.4. Damage Characterization of Concrete

_{c}. The second and third one is both for the stress-induced microcracks, while the sources of stress generated in the concrete are different. In this review, the stress-induced damage related to the interaction between the expansion of sulfate products and the constraint of the cement matrix is called mechanical damage d

_{m}. The stress-induced damage caused by the external loading on the concrete surface is called load damage d

_{l}.

_{c}, d

_{m,}and d

_{l}is proposed according to the theory of continuum damage mechanics as:

#### 5.4.1. Chemical Damage

_{c}, Saetta et al. [100,101] earlier carried out research on the mechanical response of concrete structures under coupled load action and environmental corrosion at 1998 and 1999. They assumed that the chemical damage of concrete caused by an environmental attack (not only for sulfate attack, but also for chloride corrosion, calcium dissolution, etc.) mainly depends on the pollutant concentration of the corrosion ion C and the residual strength of degraded concrete ${f}_{\mathrm{dam}}$, defined as Equation (19). It is an empirical equation, and the ${f}_{C-C\mathrm{ref}}$ reflects the degree of chemical reaction, calculated by the ratio of ion concentration C to the reference concentration C

_{ref}(${f}_{C-C\mathrm{ref}}=C/{C}_{\mathrm{ref}}$). The ${\eta}_{\mathrm{R}-\mathrm{re}}$ reflects the relative residual strength of the concrete caused by chemical reactions, calculated by the ratio of ${f}_{\mathrm{dam}}$ to the initial strength ${f}_{\mathrm{ini}}$ (${\eta}_{\mathrm{R}-\mathrm{re}}={f}_{\mathrm{dam}}/{f}_{\mathrm{ini}}$).

_{c}is introduced, which is only related to the extent of the chemical reaction ξ, expressed as Equation (20). Other similar expressions of d

_{c}are shown in Table 7.

#### 5.4.2. Mechanical Damage

_{m}, the work performed by Tixier and Mobasher [78] is the earliest and most meaningful, and was developed by many later studies, such as Sarkar et al. [88,89], Yu et al. [64,98,105], Qin et al. [106], Wang et al. [107], Yin et al. [59,96] and Li et al. [108] and is expressed as Equation (22). In this work, a simplified uniaxial tensile stress–strain law was adopted to quantitatively describe the evolution of the damage degree in the process of CSA. The d

_{m}is associated with the free expansive strain ${\epsilon}_{V}$ in Equation (6) caused by the formation of sulfate products, namely ${d}_{\mathrm{m}}\sim {\epsilon}_{V}$. The uniaxial tensile stress–strain curve is divided into three stages, including the linear ascending stage (LAS), pre-peak non-linear ascending stage (PNAS), and post-peak non-linear descending stage (PNDS). In the CSA process, the initial ${\epsilon}_{V}$ is in LAS, meaning that the concrete is undamaged, namely ${d}_{\mathrm{m}}=0$. When the increasing ${\epsilon}_{V}$ is into PNAS, the microcracks begin to form and spread in the concrete, and the ${d}_{\mathrm{m}}$ increases linearly. Later, the ${\epsilon}_{V}$ increases into PNDS, the microcracks converge into macrocracks, and the ${d}_{\mathrm{m}}$ has a non-linear rapid increase.

_{m}(d

_{m-I}) and II type d

_{m}(d

_{m-II}) is whether the calculation can be decoupled. d

_{m-I}can be directly calculated by the free volume expansion ${\epsilon}_{V}$ from the growth of sulfate products, namely ${\epsilon}_{V}\left(\xi \right)\to {d}_{\mathrm{m}-\mathrm{I}}$. d

_{m-II}needs to be numerically solved by coupling with stress and strain, and they all need to meet the unified equations, including the equilibrium differential equation, constitutive equation, and geometric equation, as shown in Figure 5.

#### 5.4.3. LOAD DAMAGE

_{m-II}) is based on load damage d

_{l}. The achievements of d

_{l}have been quite rich, and they have proposed along with the plasticity or damage constitutive model for the mechanical response in concrete under an external load. The typical constitutive models include Mazars’ damage model [109], Lubliner’s damage model [110], Faria’s rate-independent plasticity damage constitutive model [111], Grassl’s damage-plastic model [112,113], and Wu’s plastic damage model based on the energy release rate [114]. In these models, the load damage is determined by the damage loading function, the evolution law for the damage variable, and the loading-unloading conditions, as Equations (16)–(18).

Author | Basic Mechanical Equation | Number | |
---|---|---|---|

Chemical damage d_{c} | |||

Saetta et al. [110,111] | ${d}_{\mathrm{c}}=\left(1-{\eta}_{\mathrm{R}-\mathrm{re}}\right)\underset{\mathrm{Related}\mathrm{to}\mathrm{ion}\mathrm{concentration}C}{\underset{\u23df}{\left[1-\frac{1}{1+{\left(2{f}_{C-C\mathrm{ref}}\right)}^{2}}\right]}}$ | (19) | |

Cefis and Comi [92] | ${d}_{\mathrm{c}}=\underset{\mathrm{Related}\mathrm{to}\mathrm{reaction}\mathrm{extent}}{\underset{\u23df}{\frac{1-\mathrm{exp}\left(-{A}_{1}\xi \right)}{1+\mathrm{exp}\left(-{A}_{1}\xi +{A}_{2}\right)}}}{A}_{3}$ | (20) | |

Sun et al. [73] Zhang et al. [115] | ${d}_{\mathrm{c}}=\left(1-{\phi}_{0}\right)\underset{\mathrm{Related}\mathrm{to}\mathrm{diffuse}\mathrm{time}}{\underset{\u23df}{\left[1-\mathrm{exp}\left(-{A}_{4}\frac{t}{{t}_{0}}\right)\right]}}\underset{\mathrm{Related}\mathrm{to}\mathrm{ion}\mathrm{concentration}C}{\underset{\u23df}{\left[1-\frac{1}{1+{A}_{5}{\left(\frac{C}{{C}_{\mathrm{es}0}}\right)}^{{A}_{6}}}\right]}}$ | (21) | |

$t/{t}_{0}$ is related to the diffuse time or hydrated time. $C/{C}_{0}$ is the ratio of ion concentration in the concrete to that in the external solution. | |||

Mechanical damage d_{m} | |||

I | Tixier and Mobasher [78] | ${d}_{\mathrm{m}}={r}_{7}\cdot \underset{\mathrm{Related}\mathrm{to}\mathrm{volume}\mathrm{expansion}}{\underset{\u23df}{{\left(1-\frac{{\epsilon}_{\mathrm{th}}}{\epsilon}\right)}^{{r}_{8}}}},\hspace{1em}\epsilon =\frac{1}{3}{\epsilon}_{V}$ | (22) |

${\epsilon}_{\mathrm{th}}$ is a threshold strain for the initiation of microcracks or damage. | |||

Wang et al. [116] | ${d}_{\mathrm{m}}=1-\frac{1}{1+\left(\varsigma -1\right)\cdot \underset{\mathrm{Related}\mathrm{to}\mathrm{volume}\mathrm{expansion}}{\underset{\u23df}{{\left(\epsilon /{\epsilon}_{\mathrm{m}}\right)}^{\varsigma /\left(1-\varsigma \right)}}}},\hspace{1em}\varsigma =\frac{{E}_{0}}{{f}_{\mathrm{m}}/{\epsilon}_{\mathrm{m}}}$ | (23) | |

${E}_{0}$ is the initial elastic modulus. ${\epsilon}_{\mathrm{m}}$ is the ultimate tensile strain corresponding to the ultimate tensile stress ${f}_{\mathrm{m}}$ at the peak of the stress–strain curve. | |||

II | Bary et al. [84,87] | ${d}_{\mathrm{m}}=1-\frac{1-{A}_{9}}{\tilde{\epsilon}/{\epsilon}_{\mathrm{th}}}-\frac{{A}_{9}}{\mathrm{exp}\left[{A}_{10}\left(\tilde{\epsilon}-{\epsilon}_{\mathrm{th}}\right)\right]}$ | (24) |

$\tilde{\epsilon}$ is the equivalent strain. | |||

Yin et al. [96,102] | $\left\{\begin{array}{l}{d}_{\mathrm{m}}=\frac{\u2308{\overline{\mathsf{\sigma}}}^{+}\u2309{d}_{\mathrm{m}-\mathrm{t}}+\u2308{\overline{\mathsf{\sigma}}}^{-}\u2309{d}_{\mathrm{m}-\mathrm{c}}}{\u2308\overline{\sigma}\u2309}\\ {d}_{\mathrm{m}-\mathrm{c}\mathrm{or}-\mathrm{t}}=1-\frac{\left(1-{A}_{11-\mathrm{c}\mathrm{or}-\mathrm{t}}\right)}{1+f\left(\tilde{\epsilon}\right)/{\epsilon}_{0-\mathrm{c}\mathrm{or}-\mathrm{t}}}-\frac{{A}_{11-\mathrm{c}\mathrm{or}-\mathrm{t}}}{\mathrm{exp}\left[{A}_{12-\mathrm{c}\mathrm{or}-\mathrm{t}}f\left(\tilde{\epsilon}\right)/{\epsilon}_{0-\mathrm{c}\mathrm{or}-\mathrm{t}}\right]}\end{array}\right.$ | (25) | |

${\overline{\mathsf{\sigma}}}^{+}$$\mathrm{and}{\overline{\mathsf{\sigma}}}^{-}$ are the positive and negative spectral decomposition parts of the effective stress tensors $\overline{\mathsf{\sigma}}$. ${d}_{\mathrm{m}-\mathrm{c}\mathrm{or}-\mathrm{t}}$ is the compressive or tensile stress-induced mechanical damage. | |||

Load damage d_{l} | |||

Grassl et al. [112] | ${d}_{\mathrm{l}}=1-\mathrm{exp}\left(-\frac{{\kappa}_{\mathrm{d}}}{{\epsilon}_{\mathrm{f}}}\right)$ | (26) | |

${\epsilon}_{\mathrm{f}}$ is the parameter that controls the slope of the softening curve. | |||

Wu et al. [114] | ${d}_{\mathrm{l}}{}^{+}=1-\frac{{r}_{0}^{+}}{{r}^{+}}\mathrm{exp}\left[{A}_{13}\left(1-\frac{{r}^{+}}{{r}_{0}^{+}}\right)\right]$$,{d}_{\mathrm{l}}{}^{-}=1-\frac{{r}_{0}^{-}}{{r}^{-}}\left(1-{A}_{14}\right)-{A}_{14}\mathrm{exp}\left[{A}_{15}\left(1-\frac{{r}^{-}}{{r}_{0}^{-}}\right)\right]$ | (27) | |

${d}_{\mathrm{l}}{}^{+}$$\mathrm{and}{d}_{\mathrm{l}}{}^{-}$ are the tensile damage and shear damage corresponding to positive and negative stress components. | |||

Mazars et al. [109] Zheng et al. [117] | $\left\{\begin{array}{l}{d}_{\mathrm{m}}=1-\left(1-{s}_{\mathrm{t}}{d}_{\mathrm{l}-\mathrm{t}}\right)\left(1-{s}_{\mathrm{c}}{d}_{\mathrm{l}-\mathrm{c}}\right)\\ {d}_{\mathrm{l}-\mathrm{c}\mathrm{or}-\mathrm{t}}=1-\frac{\left(1-{A}_{16-\mathrm{c}\mathrm{or}-\mathrm{t}}\right)}{1+{\kappa}_{\mathrm{d}-\mathrm{c}\mathrm{or}-\mathrm{t}}/{\epsilon}_{0-\mathrm{c}\mathrm{or}-\mathrm{t}}}-\frac{{A}_{16-\mathrm{c}\mathrm{or}-\mathrm{t}}}{\mathrm{exp}\left({A}_{17-\mathrm{c}\mathrm{or}-\mathrm{t}}{\kappa}_{\mathrm{d}-\mathrm{c}\mathrm{or}-\mathrm{t}}/{\epsilon}_{0-\mathrm{c}\mathrm{or}-\mathrm{t}}\right)}\end{array}\right.$ | (28) | |

${s}_{\mathrm{t}}$ and ${s}_{\mathrm{c}}$ represent the stiffness recovery effects of tension and compression. |

## 6. Challenges

## 7. Conclusions

^{+}/K

^{+}basic ions type CSA, ettringite, gypsum, and thaumasite are the major reaction products. Additionally, the formation mechanism of ettringite and thaumasite is controversial, namely the dispute between the ion–ion reaction and solid–solid reaction.

^{+}/K

^{+}basic ions and brucite type in the case of Mg

^{2+}basic ions. The different types of CSA present different failure forms of the concrete, including softening, cohesiveness, volume expansion, and its-induced cracking/spalling.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Chemical reactions in concrete under different cation sulfates [30].

**Figure 4.**Filling of sulfate products in pore [70].

**Figure 5.**Expressions of damage degree in concrete caused by CSA. Note: ${f}_{\mathrm{cons}}$, ${f}_{\mathrm{geom}}$, and ${f}_{\mathrm{dama}}$ are the functions of stress–strain relationship, strain–displacement relationship, and damage degree. F is an external load; u is the displacement.

Cation Type | Sulfate Product | Formation Mechanism | Chemical Reaction | |
---|---|---|---|---|

Na^{+}/K^{+} | Gypsum | ${\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}$ | Ion–ion reaction | ${\mathrm{Ca}}^{2+}+{\mathrm{SO}}_{4}^{2-}+2{\mathrm{H}}_{2}\mathrm{O}\to \mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_{2}$ |

Ettringite | ${\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}$ | Topochemical mechanism (Solid–solid reaction) | $\left\{\begin{array}{l}{\mathrm{Equation}}{\left(}{\mathrm{a}}{\right)}{:\mathrm{C}}_{4}{\mathrm{AH}}_{13}+3{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+14{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}+\mathrm{CH}\\ {\mathrm{Equation}}{\left(}{\mathrm{b}}{\right)}{:\mathrm{C}}_{4}{\mathrm{A}\overline{\mathrm{S}}\mathrm{H}}_{12}+2{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+16{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}\\ {\mathrm{Equation}}{\left(}{\mathrm{c}}{\right)}{:\mathrm{C}}_{3}\mathrm{A}+3{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+26{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}\end{array}\right.$ | |

Through-solution mechanism (Ion–ion reaction) | $\left\{\begin{array}{l}\mathrm{Al}{\left(\mathrm{OH}\right)}_{4}^{-}+2{\mathrm{OH}}^{-}\to {\left[\mathrm{Al}{\left(\mathrm{OH}\right)}_{6}\right]}^{3-}\\ {\left[\mathrm{Al}{\left(\mathrm{OH}\right)}_{6}\right]}^{3-}+3{\mathrm{Ca}}^{2+}+12{\mathrm{H}}_{2}\mathrm{O}\to {\left[{\mathrm{Ca}}_{3}\mathrm{Al}{\left(\mathrm{OH}\right)}_{6}\cdot 12{\mathrm{H}}_{2}\mathrm{O}\right]}^{3+}\\ 2{\left[{\mathrm{Ca}}_{3}\mathrm{Al}{\left(\mathrm{OH}\right)}_{6}\cdot 12{\mathrm{H}}_{2}\mathrm{O}\right]}^{3+}+3{\mathrm{SO}}_{4}^{2-}+2{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}\end{array}\right.$ | |||

Thaumasite | ${\mathrm{C}}_{3}\mathrm{S}\cdot {\overline{\mathrm{C}}\overline{\mathrm{S}}\mathrm{H}}_{15}$ | Direct reaction (Ion–ion reaction) | ${\mathrm{SO}}_{4}^{2-}+3{\mathrm{Ca}}^{2+}+{\mathrm{CO}}_{3}^{2-}+{\mathrm{SiO}}_{3}^{2-}+15{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{C}}_{3}\mathrm{S}\cdot {\overline{\mathrm{C}}\overline{\mathrm{S}}\mathrm{H}}_{15}$ | |

Indirect reaction (Solid–solid reaction) | $\begin{array}{l}{\mathrm{C}}_{3}{\mathrm{S}}_{2}{\mathrm{H}}_{3}+{\mathrm{C}}_{6}{\mathrm{A}\overline{\mathrm{S}}}_{3}{\mathrm{H}}_{32}+2\mathrm{C}\overline{\mathrm{C}}+4\mathrm{H}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to 2\left({\mathrm{C}}_{3}\mathrm{S}\cdot {\overline{\mathrm{C}}\overline{\mathrm{S}}\mathrm{H}}_{15}\right)+{\mathrm{C}\overline{\mathrm{S}}\mathrm{H}}_{2}+{\mathrm{AH}}_{3}+4\mathrm{CH}\end{array}$ | |||

Mg^{2+} | Brucite | $\mathrm{Mg}{\left(\mathrm{OH}\right)}_{2}$ | Ion–ion reaction | $\left\{\begin{array}{l}{\mathrm{Mg}}^{2+}+{\mathrm{SO}}_{4}^{2-}+{\mathrm{Ca}}^{2+}+2{\mathrm{OH}}^{-}+2{\mathrm{H}}_{2}\mathrm{O}\to \mathrm{MH}+\mathrm{C}\overline{\mathrm{S}}{H}_{2}\\ a{\mathrm{Mg}}^{2+}+a{\mathrm{SO}}_{4}^{2-}+a\mathrm{CaO}\cdot \mathrm{S}\mathrm{i}{\mathrm{O}}_{2}\cdot x{\mathrm{H}}_{2}\mathrm{O}+2{\mathrm{H}}_{2}\mathrm{O}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to a\mathrm{MH}+a\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_{2}+\mathrm{S}{\mathrm{H}}_{x}\\ 2a{\mathrm{Mg}}^{2+}+2a{\mathrm{SO}}_{4}^{2-}+2\left(a\mathrm{CaO}\cdot \mathrm{Si}{\mathrm{O}}_{2}\cdot x{\mathrm{H}}_{2}\mathrm{O}\right)+\left(6a-1-2x\right){\mathrm{H}}_{2}\mathrm{O}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to \left(2a-3\right)\mathrm{MH}+2a\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_{2}+{\mathrm{M}}_{3}{\mathrm{S}}_{2}{\mathrm{H}}_{2}\end{array}\right.$ |

Cation Type | CSA Type/Sulfate Product | ||||
---|---|---|---|---|---|

Thaumasite | Gypsum | Ettringite | Brucite | ||

Na^{+}/K^{+} | √$({\mathrm{CO}}_{2}\mathrm{or}\mathrm{C}{\mathrm{O}}_{3}^{2-}$) | √ (main) | √ (main) | ||

Mg^{2+} | √ | √ | √ (main) | ||

Failure form | Cohesiveness | ◯ | ◯ | ||

Softening | ◯ | ||||

Volume expansion and its-induced cracking/spalling | ◯ | ◯ |

Basic Form | $\frac{\partial \mathit{C}}{\partial \mathit{t}}=\underset{\mathbf{Ion}\mathbf{tan}\mathbf{sform}}{\underset{\u23df}{-\nabla \mathit{J}}}+\underset{\mathbf{Chemical}\mathbf{consumption}}{\underset{\u23df}{{\mathit{k}}_{\mathit{r}}\mathit{C}}}$ | |
---|---|---|

Ion flow | Diffusion behavior | $J=D\nabla C$ |

Mutual restriction effect between charged ions | $J=DC\nabla \Psi $ | |

Influence of ionic chemical activity | $J=DC\nabla \left(\mathrm{ln}\gamma \right)$ | |

Ion migration effect with solution convection | $J=Cv$ | |

Temperature effect | $J=\frac{DC}{T}\nabla T$ | |

Coupled by the above factors | $J=D\nabla C+DC\nabla \Psi +DC\nabla \left(\mathrm{ln}\gamma \right)+Cv$ | |

Effective diffusivity | Temperature effect | $D=\left\{\begin{array}{l}{D}_{\mathrm{c}-\mathrm{min}}+\left(\underset{\mathrm{Tempeture}}{\underset{\u23df}{{D}_{\mathrm{c}0}\left(T\right)}}-\underset{\mathrm{Product}\mathrm{filling}}{\underset{\u23df}{{D}_{\mathrm{c}-\mathrm{min}}}}\right)\underset{\mathrm{Porosity}}{\underset{\u23df}{f\left(\phi \right)}}\\ \mathrm{or}\underset{\mathrm{Crack}\mathrm{effect}\left(\mathrm{stress}\right)}{\underset{\u23df}{{D}_{\mathrm{cr}}\left(\sigma \right)}}\\ \mathrm{or}{D}_{\mathrm{w}}\cdot \underset{\mathrm{Load}}{\underset{\u23df}{\phi \left(\sigma \right)}}\cdot \underset{\mathrm{Load},\mathrm{tortuosity}}{\underset{\u23df}{\tau {\left(\sigma ,\phi \right)}^{-1}}}\\ \mathrm{or}{D}_{\mathrm{c}0}\left(\underset{\mathrm{hydration}}{\underset{\u23df}{\phi \left(w\right)}}+\underset{\mathrm{ESA}-\mathrm{induced}\mathrm{damage}}{\underset{\u23df}{d\left(C,t\right)}}\right)\end{array}\right.$ |

Sulfate products filling in pore | ||

ESA-induced microcrack effect | ||

Damage degree | ||

Cement hydration |

Author | Published Time | Degradation Cause | Degradation Mechanism |
---|---|---|---|

Tixier and Mobasher [78,86] | 2003 | Ettringite | Volume increase theory |

Bary [84] | 2008 | Ettringite, Gypsum | Crystallization pressure theory |

Bary et al. [87] | 2014 | Ettringite | Volume increase theory, crystallization pressure theory |

Basista and Weglewski [79] | 2009 | Ettringite | Volume increase theory |

Sarkar et al. [88] | 2010 | Ettringite | Volume increase theory |

Sarkar et al. [89] | 2012 | Ettringite, Gypsum | Volume increase theory |

Idiart et al. [80] | 2011 | Ettringite | Volume increase theory |

Ikumi et al. [81] | 2014 | Ettringite | Volume increase theory |

Ikumi et al. [90] | 2016 | Ettringite | Volume increase theory |

Cefis and Comi [91] | 2014 | Ettringite | Volume increase theory |

Cefis and Comi [92] | 2017 | Ettringite | Volume increase theory |

Zuo et al. [93] | 2009 | Ettringite | Volume increase theory |

Zuo et al. [94] | 2012 | Ettringite | Volume increase theory |

Nie et al. [95] | 2015 | Ettringite | Volume increase theory |

Yin et al. [96] | 2017 | Ettringite, Gypsum | Volume increase theory |

Yin et al. [97] | 2019 | Ettringite | Crystallization pressure theory |

Yu et al. [64] | 2018 | Ettringite | Volume increase theory |

Yu et al. [98] | 2021 | Ettringite | Volume increase theory |

Yi et al. [99] | 2019 | Ettringite | Volume increase theory |

Authors | Basic Mechanical Equation | Number |
---|---|---|

Saetta et al. [100,101] | $\sigma =\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{c}}\right)}}\cdot \underset{\mathrm{Load}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{l}}\right)}}\cdot {\u2102}_{0}:\epsilon $ | (8) |

d_{c} is the CSA-induced chemical damage. ${d}_{\mathrm{l}}$ is the loading damage caused by external loading. This model can be unsed to analyze the stress in concrete under the external load and environment corrosion. | ||

Sarkar et al. [88] | $\sigma =\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{{\displaystyle \u23df}}{\left(1-{d}_{\mathrm{c}}\right)}}\cdot {E}_{0}\cdot \underset{\stackrel{\u23df}{\mathrm{Related}\mathrm{to}\mathrm{free}\mathrm{expansion}{\epsilon}_{V}}}{\epsilon},\hspace{1em}{d}_{\mathrm{c}}=f\left(\epsilon \right)$ | (9) |

In sraker’s model, the $\epsilon $ is calclulated by the free expnasion and ${\epsilon}_{V}$ is caused by the growth of sulfate products. | ||

Bary et al. [87] | $\sigma =\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{c}}\right)}}\cdot \left[{\u2102}_{0}:\left(\epsilon -\underset{\mathrm{Related}\mathrm{to}\mathrm{free}\mathrm{expansion}{\epsilon}_{V}}{\underset{\u23df}{{\epsilon}_{\mathrm{es}}}}\right)-\underset{\mathrm{Crystallization}\mathrm{pressure}}{\underset{\u23df}{{\alpha}_{\mathrm{AFm}}{p}_{\mathrm{c}}\mathbf{1}}}\right]$ | (10) |

The effects of crystallization pressure and free expansion are considered together in the model. | ||

Cefis and Comi [92] | $\sigma =\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{c}}\right)}}\cdot \underset{\mathrm{Mechanical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{m}}\right)}}\cdot \left[2Ge+K\mathrm{tr}\epsilon 1\right]-\underset{\mathrm{Hydrostatic}\mathrm{pressure}}{\underset{\u23df}{b{p}_{\mathrm{w}}1}}$ | (11) |

In this model, the CSA-induced mechanical and chemical damages are considered. Additionally, the hydrostatic pressure p_{w} is introduced in the basic constitutive equation. So, this model is applicable to the case of sulfate attack on unsaturated concrete. | ||

Ikumi et al. [81] | $\sigma =\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{c}}\right)}}\cdot {E}_{0}\cdot \left(\epsilon -\underset{\mathrm{Related}\mathrm{to}\mathrm{free}\mathrm{expansion}{\epsilon}_{V}}{\underset{\u23df}{{\epsilon}_{\mathrm{non}-\mathrm{mech}}}}\right)$ | (12) |

Ikumi’s model is similar to that of Sarkar, but he considered the influence of pore size on ${\epsilon}_{V}$. | ||

Yin et al. [96] | $\mathbf{\sigma}=\underset{\mathrm{Load}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{l}}\right)}}\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{c}}\right)}}{\u2102}_{0}:\left(\mathsf{\epsilon}-{\mathsf{\epsilon}}_{\mathrm{p}}\right)$ | (13) |

Yin et al. [102] | $\mathbf{\sigma}=\underset{\mathrm{Chemical}\mathrm{damage}}{\underset{\u23df}{\left(1-{d}_{\mathrm{c}}\right)}}{\u2102}_{0}:\left(\mathsf{\epsilon}-{\mathsf{\epsilon}}_{\mathrm{p}}-\underset{\mathrm{Related}\mathrm{to}\mathrm{free}\mathrm{expansion}{\epsilon}_{V}}{\underset{\u23df}{{\mathsf{\epsilon}}_{\mathrm{EG}}}}\right)$ | (14) |

In Yin’s models, the chemical damage d_{c} is analyzed by using the elastoplastic damage mechanics, not determined by the empirical formula related to the content of sulfate ion or the reaction product. |

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**MDPI and ACS Style**

Yin, G.-J.; Wen, X.-D.; Miao, L.; Cui, D.; Zuo, X.-B.; Tang, Y.-J.
A Review on the Transport-Chemo-Mechanical Behavior in Concrete under External Sulfate Attack. *Coatings* **2023**, *13*, 174.
https://doi.org/10.3390/coatings13010174

**AMA Style**

Yin G-J, Wen X-D, Miao L, Cui D, Zuo X-B, Tang Y-J.
A Review on the Transport-Chemo-Mechanical Behavior in Concrete under External Sulfate Attack. *Coatings*. 2023; 13(1):174.
https://doi.org/10.3390/coatings13010174

**Chicago/Turabian Style**

Yin, Guang-Ji, Xiao-Dong Wen, Ling Miao, Dong Cui, Xiao-Bao Zuo, and Yu-Juan Tang.
2023. "A Review on the Transport-Chemo-Mechanical Behavior in Concrete under External Sulfate Attack" *Coatings* 13, no. 1: 174.
https://doi.org/10.3390/coatings13010174