A time−space porosity computational model for concrete under sulfate attack

Abstract

The deterioration of the microscopic pore structure of concrete under external sulfate attack (ESA) is a primary cause of degradation. Nevertheless, little effort has been invested in exploring the temporal and spatial development of the porosity of concrete under ESA. This study proposes a mechanical–chemical model to simulate the spatiotemporal distribution of the porosity. A relationship between the corrosion damage and amount of ettringite is proposed based on the theory of volume expansion. In addition, the expansion strain at the macro-scale is obtained using a stress analysis model of composite concentric sphere elements and the micromechanical mean-field approach. Finally, considering the influence of corrosion damage and cement hydration on the diffusion of sulfate ions, the expansion deformation and porosity space−time distribution are obtained using the finite difference method. The results demonstrate that the expansion strains calculated using the suggested model agree well with previously reported experimental results. Moreover, the tricalcium aluminate concentration, initial elastic modulus of cement paste, corrosion damage, and continuous hydration of cement significantly affect concrete under ESA. The proposed model can forecast and assess the porosity of concrete covers and provide a credible approach for determining the residual life of concrete structures under ESA.

Graphical abstract

Keywords

expansion deformation / porosity / internal expansion stress / external sulfate attack / mechanical–chemical coupling model

1 Introduction

External sulfate attack (ESA) is a complex physical–chemical–mechanical interaction process that hampers the durability of concrete [1,2]. Invasive sulfate ions react with hydration products to produce chemical products such as ettringite and gypsum [3,4], and such chemical reactions have two main effects on concrete. First, the dissolution of calcium hydroxide (CH) breaks the equilibrium of calcium ions in the pore solution, resulting in continuous decalcification of the calcium silicate hydrate (CSH) gel. Subsequently, the microstructure of the CSH gel becomes loose, and the bonding force between the aggregate and cement paste is weakened. Second, the chemical products form expansive stress in the pores, leading to expansion deformation of the concrete and the generation of cracks, which eventually cause concrete failure [5–8].

Several attempts have been made to assess the effects of the cement paste composition and corrosion method on the expansion deformation and porosity using experimental methods [9–19]. Experimental approaches have the benefit of being simple and capable of producing reliable data. However, ESA is a lengthy process with outcomes determined by the concrete mix and corrosion environment. Furthermore, most of the results are macroscopic averages, which cannot characterize the distributions of the expansion, deformation, and porosity. The degradation of concrete stems from the evolution of microscopic pores, and solving such problems using experimental approaches is a huge task. Consequently, it is difficult to develop a corrosion layer in concrete. Thus, it is necessary to develop an optimal and reliable prediction model to evaluate the durability.

Many numerical models have been proposed to analyze concrete expansion based on corrosion mechanisms and quantify the expansion deformation behavior. Ettringite accumulation is recognized as the most significant cause of expansion [20], whereas gypsum plays only a minor role [21]. Expansion deformation and corrosion damage originate from the internal expansion stress caused by ettringite [22–24]. However, there is no unified conclusion regarding the exact mechanism of the internal expansion stress. Presently, most models use the crystallization pressure theory [25,26] or the volume accumulation method [27]. The crystallization pressure has a rigorous theoretical thermodynamic foundation and can explain the degradation mechanism. To form any crystal, the solution must be under hydrostatic pressure and supersaturated [27–29]. However, corrosion cracks may disrupt the hydrostatic pressure state required to form crystals. Meanwhile, ettringite generated during the initial expansion stage mainly fills the pores without causing expansion deformation [30–32]. Additionally, to calculate the crystallization pressure, it is necessary to determine the composition of the solution at different positions on the sample, which is a challenge for experimental methods [28].

The volume expansion method is another common method for characterizing expansion deformation. The concept of this method is to attribute the expansion deformation to the volume increase caused by chemical products [33–35]. Existing models usually represent the volume change before and after the chemical reactions as the expansion deformation at each spatial point [36,37]. Although the expansion deformation is clearly related to the expansion products [38], it has not been proven experimentally that the expansion deformation can be completely determined by the volume deformation caused by ettringite [27]. In these models, the stiffness differential between the cement paste and expansion products is not considered, thereby neglecting the local force produced at the contact surface. That increasing local stress causes local damage to the cement paste and deterioration of the concrete [22–24]. Furthermore, in addition to the porosity gradient distribution in the concrete, the corrosion damage cracks formed by the expansion stress may accelerate corrosion by ions. The difference in ion transport at different locations makes it difficult to predict the durability of structures, thus necessitating the development of a more accurate and efficient method for analyzing the deterioration of concrete [39]. The porosity evolution is related to the internal expansion stress and the damage induced by the increase in volume of the chemical products [40]. Therefore, two aspects should be considered when simulating the degradation progress: one is establishing a quantitative relationship between the expansion products and expansion stress, and the other is considering the influence of corrosion damage on ion diffusion.

In this study, a mechanical model based on local stress is proposed to quantify the expansion deformation and porosity evolution of concrete under ESA. First, a quantitative relationship between the chemical products, local expansion stress, and corrosion damage is established. Then, the local micro-expansion strain under the action of micro-expansion stress is obtained based on the elastic theory. Second, considering the influence of corrosion damage on the cement paste and sulfate ion diffusion, an unsteady diffusion equation for sulfate ions is established according to Fick’s second law of diffusion. This equation is solved using the finite difference method, and the spatiotemporal evolution of the expansion force, expansion deformation, and porosity of concrete related to the diffusion process are obtained. Finally, the theoretical model is validated using experimental data for the macroscopic expansion deformation of concrete, and the impact of the elastic modulus and concrete composition on the expansion deformation is examined.

2 Critical molar concentration of calcium aluminate

2.1 Chemical reaction and expansion process

The expansion force induced by delayed ettringite generation in the pores is the main reason for the deterioration of the mechanical properties of concrete, while the accumulation of gypsum has a negligible effect on the expansion deformation of concrete [20,21]. Therefore, the model established in this study assumes that the expansion deformation of concrete is primarily caused by the volume expansion of delayed ettringite, and the effect of gypsum is ignored.

Sulfate ions react with CH to form dihydrate gypsum during the transportation process. The relevant chemical reactions are expressed as follows:

(1)\mathrm{C}\mathrm{H}+{\text{Na}}_2{\mathrm{S}\mathrm{O}}_4+2\mathrm{H}=\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_2+2\mathrm{N}\mathrm{a}\mathrm{O}\mathrm{H}

The generated gypsum is unstable and continues to combine with aluminum in the hydration products to form delayed ettringite. The main chemical reactions are as follows.

(2)\begin{array}{c}{\mathrm{C}}_4\mathrm{A}\overline{\mathrm{S}}{\mathrm{H}}_{12}+2\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_2+16\mathrm{H}={\mathrm{C}}_6\mathrm{A}{\overline{\mathrm{S}}}_3{\mathrm{H}}_{32}\\ {\mathrm{C}}_3\mathrm{A}{\mathrm{H}}_6+3\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_2+20\mathrm{H}={\mathrm{C}}_6\text{A}{\overline{\mathrm{S}}}_3{\mathrm{H}}_{32}\\ {\mathrm{C}}_4\mathrm{A}{\mathrm{H}}_{13}+3\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_2+14\mathrm{H}={\mathrm{C}}_6\mathrm{A}{\overline{\mathrm{S}}}_3{\mathrm{H}}_{32}+\text{CH}\\ {\mathrm{C}}_3\mathrm{A}+3\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_2+26\mathrm{H}={\mathrm{C}}_6\mathrm{A}{\overline{\mathrm{S}}}_3{\mathrm{H}}_{32}\end{array}

For convenience, Eq. (2) is simplified as follows.

(3)\begin{array}{c}\mathrm{C}\mathrm{A}+q\mathrm{C}\overline{\mathrm{S}}{\mathrm{H}}_2\to {\mathrm{C}}_6\mathrm{A}{\overline{\mathrm{S}}}_3{\mathrm{H}}_{32}\\ \mathrm{C}\mathrm{A}={\gamma }_1{\text{C}}_{\text{3}}\text{A}+{\gamma }_2{\text{C}}_{\text{4}}\text{A}{\text{H}}_{\text{6}}+{\gamma }_3{\text{C}}_{\text{4}}\text{A}{\text{H}}_{\text{13}}+{\gamma }_4{\text{C}}_{\text{4}}\text{A}{\text{H}}_{\text{12}}\\ q=3\left({\gamma }_1+{\gamma }_2+{\gamma }_3\right)+2{\gamma }_4,\end{array}

where {\gamma }_i\left(i=1\text{−}4\right) is the percentage content of the aluminum phases ({\text{C}}_{\text{3}}\text{A}, {\text{C}}_4\text{A}{\text{H}}_{\text{6}}, {\text{C}}_4\text{A}{\text{H}}_{\text{13}}, and {\text{C}}_4\mathrm{A}{\text{H}}_{\text{12}}) in the total aluminum phase.

2.2 Free volume expansion deformation of ettringite

The rate of volume change caused by each aluminum phase before and after the chemical reaction can be obtained from Eq. (2) as follows:

(4)\frac{\mathrm{\Delta }{V}_i}{V}=\frac{m_{\mathrm{V}\mathrm{A}\mathrm{F}\mathrm{t}}}{m_{\text{VCA}i}+a_i{m}_{\text{VC}\overline{\mathrm{S}}{\text{H}}_2}}-1,

where ΔV_i and V is the volume change and initial volume of each aluminium phase; a_i is the chemical reaction coefficient of gypsum in Eq. (2); m_{\text{VCA}i} is the molar volume of {\text{C}}_{\text{3}}\text{A}, {\text{C}}_4\text{A}{\text{H}}_{\text{13}}, {\text{C}}_4\text{A}{\text{H}}_{\text{6}}, and {\text{C}}_4\text{A}{\text{H}}_{\text{12}}, which are 88.41, 317.46, 276.24, and 150.00 mol/cm³, respectively; and m_{\text{VAFt}} and m_{\mathrm{V}\mathrm{C}\overline{\text{S}}{\text{H}}_2} are the molar volumes of ettringite and gypsum, which are 704.22 and 74.12 mol/cm³, respectively [41].

The free bulk expansion strain of ettringite formed by various aluminum phases can be expressed as follows:

(5){\epsilon }_{\text{V}}=C_{\mathrm{R}i}\sum \frac{\mathrm{\Delta }{V}_i\cdot {\gamma }_i}{V\cdot m_{\text{VCA}i}},

where C_{\mathrm{R}i} is the molar concentration of calcium aluminate ({\text{C}}_{\text{3}}\text{A}, {\text{C}}_4\text{A}{\text{H}}_{\text{6}}, {\text{C}}_4\text{A}{\text{H}}_{\text{13}}, and {\text{C}}_4\text{A}{\text{H}}_{\text{12}}) consumed in the chemical reaction.

2.3 Critical concentration of calcium aluminate

The initial porosity, f₀, of cement mortar can be expressed as follows [42]:

(6)f_0={\varphi }_{\mathrm{c}}\frac{w/c-0.39\alpha }{w/c+0.32},

where {\varphi }_{\mathrm{c}} is the volume fraction of cement, w/c is the water-to-cement ratio, and \alpha is the hydration degree of the cement after standard curing [41]:

(7)\alpha =1-\exp (-3.15(w/c)).

Yu and Chen [43] monitored the porosity evolution of cement mortar using mercury intrusion tests. The results showed that the porosity decreased with time owing to the continuous hydration of cement. Scherer [44] and Flatt and Scherer [45] showed that the smaller the pore diameter, the larger the compressive stress formed in the pores would be. Optimization of the pore structure caused by continuous hydration can accelerate the corrosion process. In addition to the reduction in porosity, the available pore spaces of ettringite may also be reduced, which significantly increases the expansion deformation of more ettringite crystals. Therefore, the influence of the continuous hydration of cement on the porosity is considered in developing the corrosion degradation model. The relationship between the capillary porosity, f_w, and cement hydration time t is given as follows [43]:

(8)f_{\mathrm{w}}=f_0-f_0{C}_{\mathrm{j}}\left(1-\frac{1}{1+k_{\mathrm{h}}t}\right),

where C_{\mathrm{j}} is the hydration reaction coefficient, and k_h is the continuous hydration constant.

The confined stress, P′, is generated after ettringite partially fills the pores of the cement [46,47]. It is assumed that the filling volume fraction corresponding to the expansion stress on the pore wall represents the capillary volume filling coefficient, \psi. Hence, the critical free expansion volumetric strain of ettringite corresponding to the generation of expansion stress is the product of the capillary porosity, f_{\mathrm{w}}, and the capillary volume filling coefficient, \psi.

The residual calcium aluminate concentration of a cement-based material corresponding to the internal expansion stress, P, caused by ettringite is called the critical molar concentration, C_cr (mol/m³), which can be expressed as follows:

(9)C_{\text{cr}}=C_{\mathrm{C}\mathrm{A}}-\frac{\psi \cdot f_{\mathrm{w}}}{{\displaystyle \sum {\displaystyle \frac{\mathrm{\Delta }{V}_i{\gamma }_i}{V\cdot m_{\text{VCA}i}}}}},

where C_CA (mol/m³) is the initial molar concentration of calcium aluminate.

The critical molar concentration, C_cr, is related to the porosity and the concentration of each aluminum phase. When the calcium ion concentration in the residual tricalcium aluminate is less than the critical molar concentration after corrosion, an internal expansion stress is generated. If only the effect of the pore-filling factor is considered and the cement paste constrained is ignored, the free volume expansion strain of ettringite can be expressed as follows:

(10)

3 Expansion deformation model

3.1 Expansion force model of a composite concentric spherical element

We assume that the cement mortar is composed of a cement mortar matrix and pores to calculate the spatial and temporal evolution of its deformation under internal expansion forces. Cement mortar is regarded as a material composed of numerous small composite spherical units. A composite spherical unit is a concentric cavity unit composed of spherical pores and cement mortar shells around the pores. The ettringite crystals are constrained by the outer sphere of the cement mortar as they expand in volume, and the outer sphere is also affected by the internal expansion force, P. Simultaneously, the outer sphere is constrained by the confined stress, F, of the infinite cement mortar. A schematic of the stress analysis of the composite concentric spherical unit and its components is shown in Fig.1.

The inner diameter of the unit volume hole of the composite concentric sphere is a, and the outer diameter is b. Parameter b can be approximated as the pore diameter with the greatest distribution in the cement mortar. At a certain corrosion time, the unit porosity can be expressed as follows:

(11)f_{\mathrm{w}}={\left(\frac{a}{b}\right)}^3.

The internal stress, P, depends on the free volume expansion of ettringite, elastic constant of the ettringite crystals, and elastic constant of the cement mortar matrix. For simplicity, we assume that the internal expansion stress is positively correlated with the free volume expansion of ettringite and the bulk modulus of the cement mortar matrix after removing the filling effect.

(12)

where K^m is the volume modulus of cement mortar, and β is a parameter reflecting the influence of corrosion patterns that is determined experimentally.

3.2 Corrosion damage

Tensile stress can be generated under the action of internal expansion and constraint forces. Corrosion damage occurs when the tensile stress produced by the inner section of the outer shell exceeds the strength limit of the material. The inhomogeneity of internal defects results in a different strength of each microelement in concrete. The Weibull strength distribution is often used to characterize the cumulative failure probability of brittle materials under stress [48,49]. If the corrosion damage is defined as the cumulative failure probability of concrete under internal expansion stress, the corrosion damage can be expressed as follows:

(13)D_{\mathrm{M}}=1-\exp \left[-{\left(\frac{P-P_{\mathrm{t}\mathrm{h}}}{P_0}\right)}^n\right],

where n is the shape parameter, P_th is the expansion stress corresponding to corrosion damage, and P₀ is the normalized expansion stress.

Substituting Eqs. (10) and (11) into Eq. (13) yields the following:

(14)\begin{array}{l}{D}_{\mathrm{M}}=\{\begin{array}{ll}0,& {\epsilon }^{\ast }\text{lt 3}{\epsilon }_{\text{th}},\\ 1-\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{{\epsilon }^{\ast }/3-{\epsilon }_{\text{th}}}{{\epsilon }_{\text{0}}}}\right)}^n\right],& {\epsilon }^{\ast }\ge \text{3}{\epsilon }_{\text{th}},\end{array}\\ {\epsilon }_{\text{th}}={\displaystyle \frac{P_{\text{th}}}{\beta K^{\text{m}}}},{\epsilon }_0={\displaystyle \frac{P_0}{\beta K^{\text{m}}}}.\end{array}

The viscosity has a critical influence on the long-term mechanical properties of concrete. Neglecting the viscosity will result in a greater calculated expansion deformation [22], which is safer for engineering structure design. Therefore, the ettringite crystals and cement mortar matrix are assumed to be elastic materials to simplify the calculations and facilitate engineering applications. Without considering the influence of physical forces, the displacement u_{\mathrm{r}}^{\mathrm{p}} and stress {\sigma }_{\mathrm{r}}^{\text{p}} of the ettringite crystal under the constraint of the cement mortar matrix are given as follows in spherical coordinates:

(15)u_{\mathrm{r}}^{\mathrm{p}}=C_{1}r+\frac{C_2}{r^2},

(16)\begin{array}{c}{\sigma }_{\mathrm{r}}^{\text{p}}=\frac{E^{\text{p}}}{1-2{\nu }^{\text{p}}}\left(C_1-{\epsilon }^{\ast }\right)-\frac{2E^{\text{p}}}{\left(1+{\nu }^{\text{p}}\right)r^3}{C}_2,\\ {\sigma }_{\varphi }^{\text{p}}=\frac{E^{\text{p}}}{1-2{\nu }^{\text{p}}}\left(C_1-{\epsilon }^{\ast }\right)+\frac{E^{\text{p}}}{\left(1+{\nu }^{\text{p}}\right)r^3}{C}_2,\end{array}

where E^{\text{p}} and {\nu }^{\text{p}} are the elastic modulus and Poisson’s ratio of ettringite, respectively, and C₁ and C₂ are integral constants.

Here, the ettringite crystal is a solid sphere; thus r=0, and u_r^{\text{p}}=0. Therefore, C₂ = 0. Substituting r=a and radial stress {\sigma }_{\mathrm{r}}^{\text{p}}=-P into Eq. (15), we obtain the following:

(17)C_1=-\frac{P\left(1-2{\nu }^{\text{p}}\right)}{E^{\text{p}}}+{\epsilon }^{\ast }.

Substituting Eq. (17) into Eqs. (15) and (16), we obtain

(18)u_{\mathrm{r}}^{\mathrm{p}}=\left(-\frac{P\left(1-2{\nu }^{\mathrm{p}}\right)}{E^{\mathrm{p}}}+{\epsilon }^{\ast }\right)r,

(19){\epsilon }_{\mathrm{r}}^{\mathrm{p}}={\epsilon }_{\varphi }^{\mathrm{p}}={\epsilon }_{\theta }^{\mathrm{p}}={\epsilon }^{\ast }-\frac{P\left(1-2{\nu }^{\mathrm{p}}\right)}{E^{\mathrm{p}}}.

The inner surface of the hollow spherical shell cement mortar is subjected to the compressive stress, P, of the ettringite crystal as well as the confining stress, F, of the surrounding medium, as shown in Fig.1. The hollow spherical shell of cement mortar is regarded as a linear elastic material with damage. According to the elastic theory, the displacement of the hollow spherical shell of cement mortar is expressed as follows in spherical coordinates.

(20)u_{\mathrm{r}}^{\mathrm{m}}={\displaystyle \frac{\left(1+{\nu }^{\mathrm{m}}\right)r}{\left(1-D_{\mathrm{M}}\right)E^{\mathrm{m}}}}\left[{\displaystyle \frac{{\displaystyle \frac{b^3}{2r^3}}+{\displaystyle \frac{1-2{\nu }^{\mathrm{m}}}{1+{\nu }^{\mathrm{m}}}}}{{\displaystyle \frac{b^3}{a^3}}-1}}P-{\displaystyle \frac{{\displaystyle \frac{a^3}{2r^3}}+{\displaystyle \frac{1-2{\nu }^{\mathrm{m}}}{1+{\nu }^{\mathrm{m}}}}}{1-{\displaystyle \frac{a^3}{b^3}}}}F\right],

where E^{\mathrm{m}} and {\nu }^{\mathrm{m}} are the elastic modulus and Poisson’s ratio of the hollow spherical shell of cement mortar, respectively.

The strain of the hollow spherical shell of cement mortar can be obtained from the displacement as follows.

(21){\epsilon }_{\mathrm{r}}^{\mathrm{m}}={\displaystyle \frac{\left(1+{\nu }^{\mathrm{m}}\right)}{\left(1-D_{\mathrm{M}}\right)E^{\mathrm{m}}{r}^3}}\left[-{\displaystyle \frac{b^3}{{\displaystyle \frac{b^3}{a^3}}-1}}P+{\displaystyle \frac{a^3}{1-{\displaystyle \frac{a^3}{b^3}}}}F\right],

(22){\epsilon }_{\varphi }^{\mathrm{m}}={\displaystyle \frac{\left(1+{\nu }^{\mathrm{m}}\right)}{\left(1-D_{\mathrm{M}}\right)E^{\mathrm{m}}}}\left[{\displaystyle \frac{{\displaystyle \frac{b^3}{2r^3}}+{\displaystyle \frac{1-2{\nu }^{\mathrm{m}}}{1+{\nu }^{\mathrm{m}}}}}{{\displaystyle \frac{b^3}{a^3}}-1}}P-{\displaystyle \frac{{\displaystyle \frac{a^3}{2r^3}}+{\displaystyle \frac{1-2{\nu }^{\mathrm{m}}}{1+{\nu }^{\mathrm{m}}}}}{1-{\displaystyle \frac{a^3}{b^3}}}}F\right].

Substituting r = a and u_{\mathrm{r}}^{\mathrm{p}}=u_{\mathrm{r}}^{\mathrm{m}} into Eqs. (18) and (20), we obtain the following:

(23)\begin{array}{rl}F=& [[\left(\left(1+v^{\mathrm{m}}\right)+2\left(1-2v^{\mathrm{m}}\right)\cdot f_{\mathrm{w}}\right)\\ & +2(1-f_{\mathrm{w}})\left(1-D_{\mathrm{M}}\right)E^{\text{m}}\frac{1-2v^{\mathrm{p}}}{E^{\mathrm{p}}}]P\\ & -2(1-f_{\mathrm{w}})E^{\mathrm{m}}{\epsilon }^{\ast }]{\left(3\left(1-v^{\mathrm{m}}\right)\right)}^{-1}.\end{array}

The most probable pore size in cement mortar is approximately 320 nm [43]. Hence, the microscale of the composite concentric sphere unit can be approximated by a point in infinite space in relation to the macro-scale of the cement mortar. According to the homogenization method at different macro- and micro-scales, the strain of the cement mortar samples can be expressed as the equivalent strain of the composite concentric spherical unit at the same position in the macro-scale, i.e.,

(24)\begin{array}{rl}{\epsilon }_{\text{r}}=& \frac{1}{V_{\text{REV}}}\left({\displaystyle \underset{{\mathrm{\Omega }}_{１}}{\iiint }{\epsilon }_{\text{r}}^{\mathrm{p}}}{r}^2\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{d}r\mathrm{d}\beta \mathrm{d}\theta +{\displaystyle \underset{{\mathrm{\Omega }}_2}{\iiint }{\epsilon }_{\text{r}}^{\mathrm{m}}}{r}^2\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{d}r\mathrm{d}\beta \mathrm{d}\theta \right)\\ =& \left({\epsilon }^{\ast }-\frac{P\left(1-2v^{\mathrm{p}}\right)}{E^{\mathrm{p}}}\right)f_{\text{w}}\\ & -\frac{1+{\nu }^{\mathrm{m}}}{\left(1-D_{\mathrm{M}}\right)E^{\text{m}}}\left[\frac{f_{\text{w}}}{\left(1-f_{\text{w}}\right)}\left(P-F\right)\right]\mathrm{l}\mathrm{n}{f}_{\text{w}},\end{array}

(25)\begin{array}{rl}{\epsilon }_{\mathit{\text{θ}}}=& {\epsilon }_{\varphi }=\frac{1}{V_{\text{REV}}}\left({\displaystyle \underset{{\mathrm{\Omega }}_1}{\iiint }{\epsilon }_{\varphi }^{\mathrm{p}}}{r}^2\mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{d}r\mathrm{d}\beta \mathrm{d}\theta +{\displaystyle \underset{{\mathrm{\Omega }}_2}{\iiint }{\epsilon }_{\varphi }^{\mathrm{m}}}{r}^2\mathrm{s}\mathrm{i}\mathrm{n}\beta \mathrm{d}r\mathrm{d}\beta \mathrm{d}\theta \right)\\ =& \frac{1+v^{\text{m}}}{\left(1-D_{\mathrm{M}}\right)E^{\mathrm{m}}}\left[\frac{3\mathrm{l}\mathrm{n}\left({\displaystyle \frac{b}{a}}\right)a^3}{2\left(b^3-a^3\right)}\left(P-F\right)+\frac{\left(1-2v^{\mathrm{m}}\right)\left(a^{3}P-b^{3}F\right)}{\left(1+v^{\mathrm{m}}\right)b^3}\right]\\ & +\left({\epsilon }^{\ast }-\frac{P\left(1-2v^{\mathrm{p}}\right)}{E^{\text{p}}}\right){\varphi }_0\\ =& \frac{1+v^{\text{m}}}{\left(1-D_{\text{M}}\right)E^{\text{m}}}\left[-\frac{\gamma \mathrm{l}\mathrm{n}{f}_{\text{w}}}{2\left(1-f_{\text{w}}\right)}\left(P-F\right)+\frac{\left(1-2v^{\text{m}}\right)\left(f_{\text{w}}P-F\right)}{\left(1+v^{\text{m}}\right)}\right]\\ & +\left({\epsilon }^{\ast }-\frac{P\left(1-2v^{\text{p}}\right)}{E^{\text{p}}}\right)f_{\text{w}},\end{array}

where 0\le \theta \le 2\text{π},0\le \beta \le \text{π}.

Therefore, the volume expansion strain of a composite spherical volume element of the sample at a given spatial point can be expressed as follows:

(26){\epsilon }_{\text{EV}}={\epsilon }_{\mathrm{r}}+{\epsilon }_{\varphi }+{\epsilon }_{\mathit{\text{θ}}}.

4 Unsteady diffusion equation of sulfate ions

4.1 Diffusion equation of sulfate ions

The modified Fick’s second law is adopted to simulate the transport process of sulfate ions in concrete. The chemical reaction between sulfate radicals and hydration products is considered in the migration process. According to Eqs. (1) and (4), the chemical reaction rate can be expressed as follows:

(27)\begin{array}{c}\frac{\mathrm{d}{C}_{\text{CA}}}{\mathrm{d}t}=-\frac{k_{\mathrm{d}}{C}_{\text{CA}}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{q},\\ \frac{\mathrm{d}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{\mathrm{d}t}=-k_{\mathrm{d}}{C}_{\text{CA}}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}},\end{array}

where k_d is the chemical reaction constant, and k_d = 3.05 × 10⁻⁸ mol/(m³∙s) [26]; C_{\text{CA}} (mol/m³) and C_{{\text{SO}}_{\text{4}}^{\text{2}-}} (mol/m³) are the concentrations of calcium aluminate and sulfate ions, respectively.

Through mass conservation, the nonlinear diffusion column coordinate control equation for sulfate ions can be expressed as follows:

(28)\begin{array}{rl}& r\frac{\mathrm{\partial }{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{\mathrm{d}t}=D_{\mathrm{u}}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{\mathrm{\partial }r}\right)+D_{\mathrm{u}}\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(r\frac{\mathrm{\partial }{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{\mathrm{\partial }x}\right)-k_{\mathrm{d}}r{C}_{\text{CA}}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}},\\ & \frac{\mathrm{d}{C}_{\text{CA}}}{\mathrm{d}t}=-\frac{k_{\mathrm{d}}{C}_{\text{CA}}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{q},\end{array}

where r and z are cylindrical coordinates, t is the time, and D_{\mathrm{u}} is the sulfate ion diffusion coefficient. The initial and boundary conditions of Eq. (28) are as follows:

\begin{array}{}\text{(29a)}& t=0,C_{{\text{SO}}_{\text{4}}^{2-}}=0,C_{\text{CA}}=C_{\text{CA0}},\end{array}

\begin{array}{c}r=R_0,C_{{\text{SO}}_{\text{4}}^{2-}}=U_0,C_{\text{CA}}=0,\\ x=0,C_{{\text{SO}}_{\text{4}}^{2-}}=U_0,C_{\text{CA}}=0,\\ x=L,C_{{\text{SO}}_{\text{4}}^{2-}}=0,C_{\text{CA}}=0,\end{array}

where R₀ and L are the radius and length of the cylindrical specimen, respectively, U₀ is the concentration of sulfate ions on the sample surface, and C_CA0 is the initial concentration of calcium aluminate.

4.2 Effect of continued hydration and damage on the diffusion process

The porosity of concrete affects the ion permeation process. The continuous hydration of unhydrated cement particles in concrete reduces its porosity. Garboczi and Bentz [50] established a model to determine the relationship between the sulfate ion diffusion coefficient and capillary porosity. Considering the continuous hydration of cement, this relationship can be expressed as follows:

(30)D_{\mathrm{u}0}=1.8\cdot D_0\cdot {\left(\frac{f_{\mathrm{w}}}{{\varphi }_{\mathrm{c}}}-0.18\right)}^2,

where D_{０}=1.07\times {10}^{-9}{\text{m}}^{\text{2}}/\text{s} is the sulfate ion diffusion coefficient in water.

The corrosion damage to the cement paste caused by the internal expansion stress is equivalent to the increase in the internal porosity of concrete, which implies that corrosion damage can also affect the diffusion rate of ions. The diffusion of ions from the outside to the inside causes corrosion damage to distribute in a gradient manner within the concrete along the diffusion direction; therefore, the diffusion coefficients at each point inside the cement mortar are different during the corrosion process. The relationship between the ion diffusion coefficient and damage degree can be expressed as follows:

(31)D_{\mathrm{u}}=(1+\lambda \cdot D_{\text{M}})D_{{\mathrm{u}}_0},

where \lambda is the damage impact factor, which is determined by the experimental results.

Substituting Eq. (31) into Eq. (30), the diffusion coefficient related to the continuous hydration and corrosion damage of cement can be obtained as follows:

(32)D_{\mathrm{u}}=\left(1+\lambda \cdot D_{\text{M}}\right)1.8\cdot D_0\cdot {\left(\frac{f_{\mathrm{w}}}{{\varphi }_{\mathrm{c}}}-0.18\right)}^2.

4.3 Solution of the diffusion equation

Considering the diffusion of ions along the radial direction, the diffusion equation (Eq. (28)) can be simplified as

(33)\begin{array}{c}r\frac{\mathrm{\partial }{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{\mathrm{d}t}=D_{\mathrm{u}}\frac{\mathrm{\partial }}{\mathrm{\partial }r}\left(r\frac{\mathrm{\partial }{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{\mathrm{\partial }r}\right)-k_{\mathrm{d}}r{C}_{\text{CA}}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}},\\ \frac{\mathrm{d}{C}_{\text{CA}}}{\mathrm{d}t}=-\frac{k_{\mathrm{d}}{C}_{\text{CA}}{C}_{{\text{SO}}_{\text{4}}^{\text{2}-}}}{q}.\end{array}

The intermediate variable Z=C_{{\text{SO}}_{\text{4}}^{\text{2}-}}-q{C}_{\text{CA}} is introduced to solve the equation, and the control equation is obtained as follows:

(34)\frac{\mathrm{\partial }Z}{\mathrm{\partial }t}=D_{\mathrm{u}}\frac{{\mathrm{\partial }}^{2}Z}{\mathrm{\partial }{r}^2}+\frac{1}{r}\frac{\mathrm{\partial }Z}{\mathrm{\partial }r}.

The boundary and initial conditions are

\begin{array}{}\text{(35a)}& r=R_0,Z=U_0,\end{array}

\begin{array}{}\text{(35b)}& t=0,Z=-q\cdot C_{\text{CA0}}.\end{array}

The diffusion equation (Eq. (34)) is solved using the finite difference method, and the Crank–Nicolson difference scheme is adopted. For K={\displaystyle \frac{h^2}{\mathrm{\Delta }x}}, the difference equation is obtained as follows:

(36)\begin{array}{rl}& -D_{{\mathrm{u}}_{j+\frac{1}{2}}}^{k+1}{Z}_{j+1}^{k+1}+\left(2K+D_{{\mathrm{u}}_{j+\frac{1}{2}}}^{k+1}+D_{{\mathrm{u}}_{j-\frac{1}{2}}}^{k+1}\right)Z_j^{k+1}-D_{{\mathrm{u}}_{j-\frac{1}{2}}}^{k+1}{Z}_{j-1}^{k+1}\\ =& \left(D_{{\mathrm{u}}_{j+\frac{1}{2}}}^k+\frac{1}{j}\right)Z_{j+1}^k-\left(D_{{\mathrm{u}}_{j+\frac{1}{2}}}^k+D_{{\mathrm{u}}_{j-\frac{1}{2}}}^k-2K\right)Z_j^{k+1}\\ & +\left(D_{{\mathrm{u}}_{j-\frac{1}{2}}}^k-\frac{1}{j}\right)Z_{j-1}^k,\end{array}

where

\begin{array}{c}{D}_{{\mathrm{u}}_{j+\frac{1}{2}}}^k=\frac{1}{2}\left(D_{{\mathrm{u}}_j}^k+D_{{\mathrm{u}}_{j+1}}^k\right),D_{{\mathrm{u}}_{j+\frac{1}{2}}}^{k+1}=\frac{1}{2}\left(D_{{\mathrm{u}}_j}^{k+1}+D_{{\mathrm{u}}_{j+1}}^{k+1}\right),\\ D_{{\mathrm{u}}_{j-\frac{1}{2}}}^k=\frac{1}{2}\left(D_j^k+D_{j-1}^k\right),D_{{\mathrm{u}}_{j-\frac{1}{2}}}^{k+1}=\frac{1}{2}\left(D_{{\mathrm{u}}_j}^{k+1}+D_{{\mathrm{u}}_{j-1}}^{k+1}\right).\end{array}

The nonlinear diffusion equation (Eq. (33)) is also solved using the finite difference method. The Douglas difference scheme is adopted, as follows:

(37)\begin{array}{rl}& C_{j+1}^{k+1}-2\left(1+\frac{K}{D_{{\mathrm{u}}_j}^k}+\frac{k_{\mathrm{d}}\cdot h^2\cdot U{H}_j}{2q{D}_{{\mathrm{u}}_j}^k}\right)C_j^{k+1}+C_{j-1}^{k+1}\\ =& -\left(1+\frac{1}{j}\right)C_{j+1}^k+2\left(1-\frac{K}{D_{{\mathrm{u}}_j}^k}+\frac{k_{\mathrm{d}}\cdot h^2\cdot U{H}_j}{2q{D}_{{\mathrm{u}}_j}^k}\right)C_j^k\\ & -\left(1-\frac{1}{j}\right)C_{j-1}^k-\frac{2k_{\mathrm{d}}{Z}_j^k{h}^{2}U{H}_j}{q{D}_{\mathrm{u}j}^k},\end{array}

(38)U{H}_1=\frac{D_{{\text{u}}_1}^k}{K}\left(1+\frac{1}{j}\right)C_{j+1}^k+\left(1-\frac{D_{{\text{u}}_1}^k}{K}\right)C_1^k+\frac{k_{\text{d}}\mathrm{\Delta }t{C}_1^k}{2q}\left(Z_1^k-C_1^k\right).

The iterative solution of the difference equation for intermediate variable Z and the sulfate ion concentration is calculated using MATLAB.

5 Validation of the calculation model and parametric analysis

5.1 Model verification

The spatial distributions of the sulfate ion concentration, calcium aluminate concentration, internal expansion force, expansion deformation, and porosity are evaluated using MATLAB. To verify the rationality and accuracy of the proposed model, experimental expansion deformation results are compared with the calculation results. The experimental results for the macroscopic linear expansion deformation are obtained from the studies of Ferraris et al. [51] and Lagerblad [52]. Tab.1 summarizes the main parameter values of the model, which are selected from the existing literature.

According to the material parameters in Tab.1, the calculated expansion deformations of cement mortar specimens with different water–cement ratios are shown in Fig.2 and Fig.3.

As shown in Fig.2 and Fig.3, the calculation results based on the proposed model are in good agreement with the experimental results. However, the relative error between the calculated and experimental values is higher during the initial stage of corrosion. The main reason for this may be that the effect of calcium precipitation of cementitious materials during sulfate ion intrusion is neglected. Therefore, the concrete stiffness is overestimated in the calculation of the expansion deformation at this stage, resulting in a smaller calculated result for the expansion deformation than the experimental value. It should be noted that the proposed model is mainly intended for concrete in a constant-temperature and constant-pressure ESA environment, and the temperature and pressure in the corrosive environment also have a crucial influence on the transport of harmful ions. The effects of temperature and pressure changes on the durability of concrete need to be studied further.

The determination coefficient R-squared value is used to evaluate the accuracy of the calculation method. When the R-squared value is close to one, the calculation outcomes are in better agreement with the experimental values. According to the statistical method, if the test value is Y_i and the corresponding calculation result of the model is y_i, then the R-squared value can be expressed as follows:

(39)R^{\text{2}}=\frac{{\displaystyle \sum _{i=1}^n{y}_i^2-{\displaystyle \sum _{i=1}^n{\left(y_i-Y_i\right)}^2}}}{{\displaystyle \sum _{i=1}^n{y}_i^2}}.

Compared with the experimental results reported by Ferraris et al. [51], the R-squared values for the Tixier−Mobasher model [34,35], Gao model [41], and proposed model are 0.9038, 0.9154, and 0.9632, respectively. Furthermore, compared with the experimental results reported by Lagerblad [52], the R-squared values of the three models are 0.6960, 0.77757, and 0.8770, respectively. These results suggest that the proposed model is the most accurate. The fundamental reason for this is that the other two models do not consider the effect of expansion stress caused by chemical products on the expansion deformation. In addition, the impact of continuous cement hydration on enhancing the porosity of concrete is disregarded. Moreover, the proposed model can be used to obtain the internal expansion stress, time-space evolution of the expansion deformation, and porosity evolution of the sample during the corrosion process. The calculation results for the expansion stress, expansion deformation, and porosity are given based on the experiments reported by Ferraris et al. [51].

Fig.4 and Fig.5 show the calculated expansion stress, P, and expansion deformation, respectively, of each composite concentric spherical element along the radial corrosion direction as a function of time. The triggering positions of the expansion deformation and expansion stress gradually expand inward with increasing corrosion time. At the same spatial point, the expansion deformation increases with the corrosion time, whereas the expansion stress initially increases and then decreases. In accordance with the process of microstructural change, the expansion stress calculation results suggest three stages of concrete degradation caused by ettringite crystals. Scanning electron microscopy (SEM) images of the cement paste with a w/c ratio of 0.485 at different corrosion stages are shown in Fig.6. The first stage involves the formation and filling of chemical products. During this stage, the sulfate ions and hydration products react to generate expansion products that fill the original pores. There is no mutual extrusion between the generated expansion products and the pore wall, as shown in Fig.6(a). Owing to the refinement of the mesopore structure, the strength of concrete typically increases during the initial corrosion stage [37]. The second stage is the chemical–mechanical conversion stage. The expansion products generated in this stage gradually grow and are confined by the concrete to yield a progressively increasing local internal expansion stress [22,23]. Finally, in the corrosion damage stage, a gradual increase in the expansion stress leads to the initiation and propagation of corrosion damage. Furthermore, the SEM images reveal that the volume of ettringite crystals increases, resulting in microcracks around the pores, as shown in Fig.6(b).

Fig.7 shows the spatiotemporal evolution of the porosity of cement mortar under ESA. It is clear that the porosity of cement mortar is not constant throughout the corrosion period, and it is affected by the continuous hydration of cement and corrosion damage. Therefore, the porosity evolution of concrete is a dynamic time-space evolution process that is affected by the continuous hydration and damage. Fig.8 clearly shows the porosity evolution at different corrosion times along the depth direction of the cement mortar.

According to the calculated spatial evolution of the porosity, the concrete can be divided into three zones along the corrosion direction. Taking the pink curve in Fig.8(a) as an example, the concrete can be divided into a deterioration zone, a filling zone, and an undamaged zone from the outside surface to the center at the cross-section, as shown in Fig.8(b). In the continuous hydration zone, the external sulfate ions do not penetrate the concrete. The slight decrease in porosity is mainly due to the continuous hydration of concrete, which leads to a slight enhancement in the mechanical performance of concrete [43]. In the chemical product filling zone, the decrease in porosity relative to the initial porosity is greater than that in the undamaged zone. There are two main reasons for the reduction in porosity: the continuous hydration of cement and filling by the generated chemical products. The porosity primarily decreases from the inside to the outside because the expansion products decrease along the corrosion direction. In the deterioration zone, expansion stresses are formed and cause loads on the pore walls after the pores have been partially filled with chemical products. Once the tensile stress induced by the internal expansion stress exceeds the strength limit, damage starts [7], thereby forming the deterioration zone. By dividing the concrete into the damage zone and undamaged zone along the diffusion direction, the damage-triggering position is shown to spread gradually inside the concrete with time, indicating that the corrosion damage zone of concrete extends inside. In addition to identifying the damaged area, the evolution of the damage-trigger position will improve the life prediction of concrete under ESA.

To summarize, the proposed model can characterize the deterioration behavior of the mechanical properties of concrete during the entire degradation process. Compared with the experimental results and other calculation models, the proposed model is more reliable, simple, and effective for comprehensively analyzing the deterioration behavior of concrete under ESA.

5.2 Effect of cement composition on the sulfate resistance

The tricalcium aluminate content is varied in the calculation model to determine the effect of cement on the sulfate attack resistance of the concrete; the other concrete parameters are selected from the study by Ferraris et al. [51]. The linear expansion strain of concrete with different tricalcium aluminate contents is shown in Fig.9.

Fig.9 shows the macroscopic average expansion strain for different initial molar concentrations of tricalcium aluminate. The tricalcium aluminate content has a significant effect on the macroscopic average deformation. The initial concentration of tricalcium aluminate increases from 197.9 to 272.9 mol/m³, a percentage increase of 37.9%. After 118 d of corrosion, the average strain of the sample increases from 3.42 × 10⁻⁴ to 5.15 × 10⁻³ (a 14-fold increase). The tricalcium aluminate content in concrete plays a crucial role in its resistance to ESA [53,54]. The proposed model provides a quantitative relationship between the expansion deformation and tricalcium aluminate content of concrete under ESA. Corrosion damage can be induced by expansion deformation, which reduces the service life of concrete [40]. Therefore, the proposed model can evaluate the deterioration of concrete containing different amounts of tricalcium aluminate.

5.3 Effect of the elastic modulus of cement mortar

Fig.10 shows the expansion deformation of cement mortars with different elastic moduli under ESA.

It is clear that when the initial elastic modulus of cement mortar increases from 15.6 to 36 GPa, the linear expansion strain of concrete decreases from 0.00498 to 0.00457 after 113 d of corrosion, representing a decrease of 8.23%. Therefore, an increase in the initial elastic modulus of materials can reduce the expansion deformation of concrete under ESA, which improves the corrosion resistance and slows the deterioration of concrete. These results further support recent studies indicating that improving the properties of cement paste can help enhance the sulfate corrosion resistance of concrete [11–14].

5.4 Effect of damage and continuous hydration on sulfate attack

Using the proposed simulation models, the evolution of the average expansion deformation of concrete samples is obtained without considering the corrosion damage or continued hydration; only the effects of corrosion damage and hydration on the concrete porosity and sulfate ion diffusion coefficient are considered. The results are shown in Fig.11.

As illustrated in Fig.11, the influence of different factors on the expansion deformation in decreasing order is as follows, continuous hydration and damage, damage alone, continuous hydration alone, and no continuous hydration or damage. Therefore, continuous hydration and damage are crucial factors for the deformation under ESA. The corrosion damage of concrete under ESA can increase the penetration rate of external sulfate ions and significantly increase the content of chemical expansion products [36]. Hence, the calculated expansion deformation is lower when the damage is not considered. The continuous hydration of concrete improves the pore structure and reduces sulfate ion infiltration, thus improving the resistance of concrete to sulfate attack [23,55]. However, according to Fig.11, continuous hydration of the cement causes increased expansion deformation, which exacerbates the corrosion damage. We examined the sulfate ion concentration by ignoring and considering continuous hydration for a more in-depth understanding of how continuous hydration influences concrete durability. The results are shown in Fig.12.

As shown in Fig.12, the concentration of sulfate ions calculated without considering continuous hydration is higher than that calculated with continuous hydration after 16 or 30 d of corrosion. However, after 60 d of corrosion, the calculated sulfate ion concentration without continuous hydration is far lower than that with continuous hydration. The main reason for this is that the continuous hydration of cement reduces the porosity of the concrete, which has two effects. First, the ion diffusion coefficient decreases owing to the improvement in the pore structure, which enhances the sulfate resistance of the materials [56]. Second, continuous hydration reduces the porosity, and thus the internal porosity for free volume expansion of ettringite crystals decreases, resulting in a larger expansion stress and strain [10]. However, a larger internal expansion stress will cause more corrosion damage, which is equivalent to greater porosity. Whenever the increase in porosity caused by the continuous hydration of cement exceeds the decrease in porosity, the continuous hydration will accelerate concrete deterioration. Therefore, continuous hydration can help reduce the porosity and improve the strength and stiffness of concrete, both of which can delay sulfate corrosion. Thus, the strength of concrete under sulfate attack is improved in the early stages in most previous studies. Nevertheless, subsequent hydration also increases the degree of concrete damage and accelerates sulfate corrosion by reducing the pore filling space of ettringite. The results are similar to those reported by Naik et al. [55], who noted that refinement of the pore structure and pore size makes concrete more prone to damage.

Therefore, using methods to increase the hydration reaction rate, such as reducing the particle size of the cement powder to increase the particle surface area, increasing the water−cement ratio within a certain range, and appropriately increasing the curing temperature, will allow the initial cement reaction degree of concrete to be increased. This improves the initial pore structure of the cement mortar, increases the initial elastic modulus of the cement mortar, reduces continuous hydration, and improves the ESA resistance of concrete.

6 Conclusions

In this study, an effective model describing the entire process of concrete deterioration was established based on three stages: the formation of expansion products under sulfate corrosion, the filling of pores by expansion products, and the corrosion damage induced by expansion stress.

A time−space evolution model of the porosity and expansion deformation of concrete under ESA was established. Continuous hydration of cement, filling of expansion products, and local damage caused by internal expansion stress were the main reasons for pore changes.

The amount of tricalcium aluminate in cement could significantly influence the corrosion expansion deformation of the concrete. In addition, a quantitative relationship between the expansion deformation and tricalcium aluminate content of concrete under ESA was obtained.

The larger the initial elastic modulus of the cement mortar, the less expansion deformation of concrete would occur under ESA. Increasing the initial elastic modulus of the materials could help improve the corrosion resistance of concrete.

Damage and continuous hydration played an important role in corrosion deterioration. The continuous hydration of cement could enhance the corrosion resistance of concrete by reducing the porosity. At the same time, the continuous hydration of cement reduced the accommodation space for expansion products in pores and accelerated the deterioration of concrete. Furthermore, continuous hydration of cement accelerated the concrete degradation during the later stages of corrosion.

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