Uncertainty-Aware Virtual Physics-Based Chloride Resistance Analysis of Metakaolin-Blended Concrete

Abstract

Metakaolin (MK) obtained from calcined kaolinitic clay is a highly reactive pozzolanic ingredient for use as an emerging supplementary cementitious material (SCM) in modern sustainable binder productions. It provides elevated alumina to promote formations of Alumina Ferrite Monosulfate (AFm) and Calcium-Aluminium-Silicate-Hydrate (C-A-S-H) phases, enhancing the chloride binding capacity. However, due to inherent material uncertainty and lack of approach in quantifying hydration kinetics and chloride binding capacity across varied mixes, robustly assessing the chloride resistance of metakaolin-blended concrete remains challenging. In light of this, a machine learning-aided framework that encompasses physics-based material characterisation and ageing modelling is developed to bridge the knowledge gap. Through applying to laboratory experiments, the impacts of uncertainty on the phase assemblage of hydrated system and chloride penetration are quantified. Moreover, the novel Extended Support Vector Regression (XSVR) method is incorporated and verified against a crude Monte Carlo Simulation (MCS) to demonstrate the capability of achieving effective and efficient uncertainty-aware chloride resistance analyses. With the surrogate model established using XSVR, quality control of metakaolin towards durable design optimisation against chloride-laden environments is discussed. It is found that the fineness and purity of adopted metakaolin play important roles.

1. Introduction

Over the past few decades, the advances of sustainable concrete technologies have given rise to a variety of supplementary cementitious materials (SCMs) [1], with an emerging one being metakaolin (MK), i.e., Al₂O₃⋅2SiO₂ (AS₂) [2]. Metakaolin is a major pozzolanic ingredient, normally obtained from calcining kaolinitic clay, which has gained increasing popularity due to its high reactivity and availability amid an undersupply of fly ash under energy structure reforms [2,3,4]. According to the literature [5], concrete prepared with the addition of SCMs (up to 50% of cement replacement, in practice), involving metakaolin, exhibits greater electrical resistivity and lower permeability, without compromising mechanical performance in comparison to pure cement-based concretes.

Thus, a huge volume of recent studies have been dedicated to exploiting the benefits of metakaolin as an SCM, especially for achieving durable and sustainable designs [6].

As mentioned above, greater electrical resistivity and lower permeability of the concrete can generally be interpreted as being indicative of better durability potential in the material. Indeed, such characteristics would normally lead to reduced ionic diffusivity, and thus an enhanced capacity to withstand material degradation and corrosion in reinforced concrete under various conditions [7]. Among them, the resistance of metakaolin-blended concretes against chloride penetration is widely investigated, considering its importance and close correlation with structural service life [4,8]. Specifically, a notable contributing factor to chloride resistance is the chloride binding capacity of the material, which can be categorised into physical and chemical binding due to their distinct mechanisms [9]. Physical binding traps chloride ions in the Calcium-Aluminium-Silicate-Hydrate (C-A-S-H) gel, and chemical binding indicates the formations of chloride-bearing solids, such as Friedel’s and Kuzel’s salts, from Alumina Ferrite Monosulfate (AFm) phases. It is thus evident that the contents and characteristics of C-A-S-H and AFm phases govern the chloride binding capacity. In this regard, metakaolin provides additional alumina to promote the formation of the AFm phase, while increasing the surface area on C-A-S-H [10,11], potentially enhancing the chloride binding capacity. Nevertheless, robustly quantifying binding capacity through experiments remains challenging, and the overall chloride resistance cannot be judged solely by the available alumina in the mix, where the detailed phase assemblage is considered influential [12]. As a result, attempts at numerical innovations have been put forward to advance relevant knowledge.

To model chloride ingress in cementitious materials, assessment of chloride ion transportation and binding effect at the same time is required. Most studies employed Fick’s law, accompanied with empirical chloride binding isotherms, for modelling [13,14,15]. The main drawbacks of these reported methods are two-fold. Firstly, chloride ingress in concrete materials never proceeds as the diffusion of a single ion. Instead, it is a multi-ionic transport problem involving simultaneous reactions with the hydrated cement [16]. In chemical reactions, there is not only chemical chloride binding, but also other reactions due to the disturbance of the system equilibrium by ionic exchanges with the external environment. A notable effect is the simultaneous leaching, which may result in decalcification of calcium-bearing hydrates, including the AFm and C-A-S-H phases. This leads to the second main drawback regarding the uses of empirical chloride binding isotherms [13,14,15], where the binding capacity is often assumed to be constant, overestimating material chloride resistance for long-term exposure. Moreover, concrete materials, especially for those prepared with SCMs, are of significant inherent uncertainty, which may further complicate the effective assessments of chloride resistance in metakaolin-blended concretes.

In light of the aforementioned research gaps, this study proposes a novel numerical framework to achieve an uncertainty-aware chloride resistance analysis of metakaolin-blended concrete. The core method involves physics-based material characterisation and ageing modelling, assisted by machine learning to realise virtual probabilistic modelling with enhanced efficiency. The detailed model development is presented in Section 2, and then carefully examined in terms of performance in Section 3 through applying to real-life experimentation. Various machine learning techniques are compared, including Support Vector Regression (SVR), Gaussian Process Regression (GPR), Kriging, and novel Extended Support Vector Regression (XSVR) [17,18,19], to showcase the general applicability of the proposed framework. A series of surrogate models for evaluating chloride ingress depth in metakaolin-blended concrete is constructed and further exploited to investigate the specific influence of material uncertainty in metakaolin resources on the chloride ingress process in Section 4. By doing so, new insights on achieving durable design optimisation based on the uncertainty-aware physics-based analyses are obtained and discussed, with key findings and implications for future studies drawn in Section 5.

2. Methodology

2.1. Approach Overview and Computational Framework

To achieve a comprehensive uncertainty propagation analysis on chloride resistance of metakaolin-blended concrete, it is crucial to capture the complex binder chemistry of a material. To do so requires physics-based modelling for both material characterisation and ageing degradation analyses. Specifically, the coupled kinetic–thermodynamic model, as recently developed by Yu et al. [20], is adopted and modified in this study to assess the initial material characteristics before chloride exposure. This includes analysing binder hydration, phase assemblage, and ionic diffusivity based on the porosity of the hydrated system. Following the obtained characterisation, multi-ionic reactive transportation analyses are then performed for chloride resistance assessments, where solving a complex system of partial differential equations (PDEs) is required. Overall, it is difficult to retrieve an analytical solution to the aforementioned physics-based problem, where probabilistic modelling may further lead to intractable computational burden. Thus, to boost numerical efficiency, a virtual framework assisted with machine learning is considered promising and is hereby proposed using the illustration in Figure 1 below.

As shown in Figure 1, the proposed framework is designed to be versatile in terms of considering both material and model uncertainties as system inputs, where the detailed learning algorithm can be interchangeable. Specifically, within the enclosed physics-based model domain, i.e., blue box, material characterisation modelling proceeds first to process the potential uncertain input variables with kinetics–thermodynamic hydration analyses, which is followed by reactive transport modelling. To do so, the sequential non-iterative solver is applied, where the Poisson–Nernst–Planck (PNP) model is adopted for multi-ionic transportation analyses, including physical chloride binding evaluation. In terms of the simultaneous reactions, it is computed standalone within each time step using the Gibbs Energy Minimisation solution (GEMS) [21,22], i.e., the same thermodynamic method as that used in characterisation modelling. By doing so, changes in system phase assemblage and microstructure can then be evaluated and updated in the damage module for the next time step, including ionic diffusivity as well as chloride binding capacity. This would generate time-dependent chloride profiles along the exposure surface into the concrete core, providing quantitative data as the system outputs, such as penetration depth, for further chloride resistance assessments.

In order to expedite the numerical modelling, the uncertainty propagation process, i.e., the physics-based model domain, is replaced with surrogate models constructed using machine learning, i.e., green box, using the aforementioned uncertain inputs and outputs. Crude Monte Carlo Simulation (MCS) is implemented to verify the validity of the proposed framework. Moreover, as mentioned above, the framework is designed to be generally applicable. Traditional SVR, GPR, and Kriging are thus incorporated and compared with the novel XSVR [17,18,19] to highlight its advanced performance in the uncertain chloride resistance quantification of metakaolin-blended concrete, which involves solving multiphysical modelling problems. In the following sections, the key mathematical models and relevant modifications are derived to complete the methodology.

2.2. Material Characterisation Modelling

Since the to-be-employed thermodynamic modelling approach, i.e., GEMS, has been well established in the literature [21,22], the model derivation and modification for material characterisation centres around kinetics modelling for hydration degree estimations. This is considered crucial, as binder hydration degree governs the effective amounts of source materials participating in the actual reactions to produce time-dependent evolvements of phase assemblages and pore structures. Focusing on the scope of this study, the binder contains cement and metakaolin as the SCM, where kinetics modelling needs to be performed for these two ingredients accordingly. In general, cement hydration proceeds ahead of the SCM to produce portlandite to trigger the following pozzolanic reactions [3]. Herein, the governing equation for cement hydration kinetics, developed by unit-cell shrinking-core theory [23], is adopted and given in Equation (1).

$${\displaystyle \frac{\mathrm{d}{\alpha }_i}{\mathrm{d}{t}_{\mathrm{h}}}}={\displaystyle \frac{3(S_w/S_0){\rho }_w{C}_{w-\mathrm{free}}}{0.4r_0{\rho }_c}}\times 1/\left[({\displaystyle \frac{1}{k^d}}-{\displaystyle \frac{r_0}{D_i^e}})+{\displaystyle \frac{r_0}{D_i^e}}{(1-{\alpha }_i)}^{-1/3}+{\displaystyle \frac{1}{k_i^r}}{(1-{\alpha }_i)}^{-2/3}\right]$$

(1)

$$C_{w-\mathrm{free}}={\left({\displaystyle \frac{w_c-0.4\alpha }{w_c}}\right)}^{\tilde{r}} \mathrm{where} \left\{\begin{array}{l}\tilde{r}=1\\ \tilde{r}=2.6-4w_b\end{array}\right. s.t. \begin{array}{c}{w}_b\ge 0.4\\ w_b<0.4\end{array}$$

(2)

$$\alpha ={\displaystyle {\sum }_{i=1}^4{\alpha }_i{g}_i}/{\displaystyle {\sum }_{i=1}^4{g}_i}$$

(3)

where ${\alpha }_i$ is hydration degree (-) of ith mineral, including Alite, Belite, Aluminate, and Ferrite; $t_{\mathrm{h}}$ denotes time (h); ${\rho }_w$ and ${\rho }_c$ are water and cement density ($\mathrm{g}/{\mathrm{cm}}^3$); $k^d$ and $k_i^r$ are mineral reactivity (cm/h) at dormant and phase-boundary periods; $D_i^e$ is water diffusivity (${\mathrm{cm}}^2/\mathrm{h}$) in the hydration cell in the diffusion period, recently modified by Yu et al. [20]. In addition, $r_0$ and $S_w/S_0$ are essential parameters related to the hydration cell [20]; and $C_{w-\mathrm{free}}$ estimates water content (-) exterior to the C-A-S-H gel as a function of water-to-binder and cement ratios, i.e., $w_b$ (-) and $w_c$ (-), as well as the overall hydration degree $\alpha$ (-), calculated through weighted average over each cement mineral $g_i$ (%).

From Equation (1), in addition to mineral reactivity, the parameters that control cell dimensions also govern the cement hydration kinetics. Following the recent development from the authors [20], these parameters are defined below and consider the pore size refinement effect due to the addition of metakaolin.

$$r_0=3/(S_c\cdot {\rho }_c)$$

(4)

$$l=\sqrt[3]{4\pi ({\rho }_c\cdot w_c+1)/3}\cdot r_0$$

(5)

$$R=f\left({\overline{p}}_{\mathrm{CC}}, {\mu }_{\mathrm{MK}}\right)\cdot \sqrt[3]{1+(\xi -1)\alpha }\cdot r_0$$

(6)

$$f\left({\overline{p}}_{\mathrm{CC}}, {\mu }_{\mathrm{MK}}\right)=1+{\lambda }_1\cdot {\overline{p}}_{\mathrm{CC}}\cdot {{\mu }_{\mathrm{MK}}}^{{\lambda }_2}$$

(7)

where $r_0$ indicates the radius (cm) of an assumed spherical cement particle, further defining length $l$ (cm) of the cubic hydration cell as a function of ${\rho }_c$, $w_c$, and Blaine fineness of cement $S_c$ (${\mathrm{cm}}^2/\mathrm{g}$). Hydrates are considered to form on the cement particle surface, which would increase the radius $R$ (cm) of the hydrating particle with a growth rate of $\xi =2$ [23]. Furthermore, $f\left({\overline{p}}_{\mathrm{CC}}, {\mu }_{\mathrm{MK}}\right)>1$ is to consider the pore size refinement effect as a function of normalised metakaolin content ${\overline{p}}_{\mathrm{CC}}$ (-) and purity ${\mu }_{\mathrm{MK}}$ (-). According to Yu et al. [20], the fitting parameter in Equation (7) has been calibrated based on the experiments by Avet and Scrivener [5], yielding ${\lambda }_1=8$ and ${\lambda }_2=0.333$, respectively.

With the multiplier as expressed in Equation (7), the size growth of hydrating cement particles is enhanced within the confined hydration cell, accelerating reduction in water accessibility $S_w/S_0$ for continuing hydration. The detailed evaluation of $S_w/S_0$ is well documented in the literature [23], hence not repeated herein. Following the progressive hydration of clinkers, pozzolanic reactions from SCM, i.e., metakaolin, in this study, may proceed. Though no consensus on pozzolanic reaction formula of metakaolin is drawn, it is well known to be triggered by the available portlandite in the system. Thus, a simplified formula is proposed in Equation (8) for estimating the stoichiometry of the reaction.

$$6\mathrm{CH}+2{\mathrm{AS}}_2+(7+x)\mathrm{H}\to {\mathrm{CASH}}_x+2{\mathrm{C}}_{1.5}{\mathrm{SH}}_{2.5}+{\mathrm{C}}_2{\mathrm{ASH}}_8$$

(8)

where standard cement chemistry notation is adopted. For detailed explanations, one may refer to Appendix A.

Based on Equation (8) and the unit-cell shrinking-core theory [23,24], the governing equation for metakaolin reaction kinetics is written below as

$${\displaystyle \frac{\mathrm{d}{\alpha }_{\mathrm{MK}}}{\mathrm{d}{t}_{\mathrm{h}}}}=m_{\mathrm{CH}}(t_{\mathrm{h}})\times {\displaystyle \frac{3{\rho }_w}{v_{\mathrm{MK}}^{\mathrm{CH}}{r}_{\mathrm{CC}}{\rho }_{\mathrm{CC}}{\overline{p}}_{\mathrm{CC}}}}\times {\displaystyle \frac{1}{(\frac{1}{k_{\mathrm{MK}}^d}-\frac{r_{\mathrm{CC}}}{D_{\mathrm{MK}}^e})+\frac{r_{\mathrm{CC}}}{D_{\mathrm{MK}}^e}{(1-{\alpha }_{\mathrm{MK}})}^{-1/3}+\frac{1}{k_{\mathrm{MK}}^r}{(1-{\alpha }_{\mathrm{MK}})}^{-2/3}}}$$

(9)

$$m_{\mathrm{CH}}(t_{\mathrm{h}})={\overline{R}}_{\mathrm{OPC}}^{\mathrm{CH}}\cdot {\overline{p}}_c\cdot \alpha -v_{\mathrm{MK}}^{\mathrm{CH}}\cdot {\alpha }_{\mathrm{MK}}\cdot {\overline{p}}_{\mathrm{MK}}$$

(10)

where parameters in Equation (9) are differentiated from those in Equation (1) through subscripts CC and MK to indicate metakaolin resource and pure metakaolin component based on resource purity ${\mu }_{\mathrm{MK}}$. Equation (10) is the portlandite availability function, with the first term used to estimate portlandite content from clinker hydration by ${\overline{R}}_{\mathrm{OPC}}^{\mathrm{CH}}=0.583\cdot {\rho }_{\mathrm{CH}}/{\rho }_c$ [23], and the second term to calculate its consumption under pozzolanic reaction using $v_{\mathrm{MK}}^{\mathrm{CH}}=1.001$ following Equation (8). For the reactivity-related model parameters, they are determined based on the calibrations reported by Yu et al. [20].

As expressed in the above governing equations, cement hydration kinetics and metakaolin pozzolanic reaction kinetics are related, causing computational difficulty in solving the implicit PDE system. To address this, this study adopts a model framework proposed by Yu et al. [20], which features a sequential iterative solver for robust computing. Integrating this with the GEMS thermodynamic model, time-dependent evolvements of phase assemblage and porosity can be evaluated. Specifically, porosity is crucial for determining the intrinsic diffusivity of various ions upon curing completion, where the analytic model, proposed by Oh and Jang [25], is implemented for estimations.

2.3. Reactive Transport Modelling for Chloride Ingress

To achieve a comprehensive chloride penetration analysis, it is essential not only to distinguish the physical chloride binding from the chemical counterpart in modelling, but also to consider other reactions under given exposure conditions. With the detailed phase assemblage obtained from material characterisation modelling, general chemical reactions and chemical chloride binding can be evaluated by the GEMS thermodynamic model [21,22]. Thus, the method for analysing ionic transport while considering physical chloride binding is introduced. Focusing on chloride immersion under an isothermal condition, the influences of convection and heat conduction are ignored. Following the computational framework of Figure 1, the Poisson–Nernst–Planck (PNP) model is applied for multi-ionic transport analyses, where the adopted governing function forms are given below as

$$w_{\mathrm{L}}{\displaystyle \frac{\partial c_i}{\partial t}}-\mathrm{div}\left[D_i{w}_{\mathrm{L}}\mathrm{grad}(c_i)+{\displaystyle \frac{D_i{\nu }_i\mathrm{F}}{\mathrm{R}T}}{c}_i\mathrm{grad}(\psi )+D_i{w}_{\mathrm{L}}{c}_i\mathrm{grad}(\ln {\gamma }_i)\right]=0$$

(11)

$$({c^{\prime }}_{\mathrm{phy}}+w_{\mathrm{L}}){\displaystyle \frac{\partial c_{\mathrm{f}}}{\partial t}}-\mathrm{div}\left[D_{\mathrm{Cl}}{w}_{\mathrm{L}}\mathrm{grad}(c_{\mathrm{f}})+{\displaystyle \frac{D_{\mathrm{Cl}}{\nu }_{\mathrm{Cl}}\mathrm{F}}{\mathrm{R}T}}{c}_{\mathrm{f}}\mathrm{grad}(\psi )+D_{\mathrm{Cl}}{w}_{\mathrm{L}}{c}_{\mathrm{f}}\mathrm{grad}(\ln {\gamma }_{\mathrm{Cl}})\right]=0$$

(12)

$$\mathrm{div}\left(\tau w_{\mathrm{L}}\mathrm{grad}(\psi )\right)+{\displaystyle \frac{\mathrm{F}}{\epsilon }}{w}_{\mathrm{L}}{\displaystyle \sum _{i=1}^N{\nu }_i{c}_i}=0$$

(13)

where Equation (11) is for generic ions using subscript $i$ to indicate the ith ion, $w_{\mathrm{L}}$ is the volume fraction (-) of pore solution, the same as porosity $\varphi$ if saturated; $c_i$, $D_i$, and ${\nu }_i$ are ion concentration ($\mathrm{mmol}/\mathrm{L}$), diffusivity (${\mathrm{m}}^2/\mathrm{s}$), and valence (-). In addition, $\mathrm{F}$ is the Faraday constant ($\mathrm{96,488.46} \mathrm{C}/\mathrm{mol}$), $\mathrm{R}$ is the ideal gas constant ($8.3143 \mathrm{J}/\mathrm{mol}/\mathrm{K}$), $\psi$ is system potential (V), and ${\gamma }_i$ is chemical activity (-) estimated by modified Davies’ law [26]. Equation (12) is for chloride transport with subscript Cl to denote distinctions, $c_{\mathrm{f}}$ is free chloride ion concentration ($\mathrm{mmol}/\mathrm{L}$), and ${c^{\prime }}_{\mathrm{phy}}=\partial c_{\mathrm{phy}}/\partial c_{\mathrm{f}}$ is derivative term (-) to account for the physical binding effect. Equation (13) is Poisson’s equation, where $\tau$ is tortuosity (-) [26], $\epsilon$ is medium permittivity ($7.08\times {10}^{-10} {\mathrm{C}}^2/\mathrm{N}/{\mathrm{m}}^2$), and $N$ counts the number of ions as simulated in the multi-ionic transport problem, including chloride.

To complete the governing equation for chloride transportation, i.e., Equation (12), it is essential to define the physical binding term. Considering that physical chloride binding is triggered by the C-A-S-H gel to trap drifting chloride ions, a time-dependent physical binding term is proposed in this study. Specifically, advanced from the general Langmuir isotherm, as widely used in the literature [27], the following function form is adopted.

$$c_{\mathrm{phy}}={\displaystyle \frac{\kappa \cdot c_{\mathrm{f}}}{1+\kappa \cdot c_{\mathrm{f}}}}\cdot N_{\mathrm{C}\text{-}\mathrm{A}\text{-}\mathrm{S}\text{-}\mathrm{H}}$$

(14)

$$\kappa =1.27\times {10}^{-4}\times {\left({\displaystyle \frac{\mathrm{Al}/\mathrm{Si}}{0.05}}\right)}^{2.4}$$

(15)

where $c_{\mathrm{phy}}$ is bound chloride by the C-A-S-H gel ($\mathrm{mmol}/\mathrm{L}$ of the material), $\kappa$ is the affinity parameter of a unit as reciprocal to free chloride concentration, and $N_{\mathrm{C}\text{-}\mathrm{A}\text{-}\mathrm{S}\text{-}\mathrm{H}}$ is C-A-S-H gel content ($\mathrm{mmol}/\mathrm{L}$ of the material). In Equation (15), $\kappa$ is defined as a function of the Al/Si ratio of C-A-S-H gel obtained from characterisation modelling, where the model constant is calibrated against the case study as introduced in Section 3.1.

By using Equation (14), it is noted that the proposed physical chloride binding term is able to account for the changes in C-A-S-H content due to simultaneous decalcification. Solved with the GEMS thermodynamic model for chemical reactions [21,22], Equation (14) is thus capable of considering the potential time-dependency under a long-term exposure. Furthermore, for the affinity parameter as defined in Equation (15), it is clear that the enhanced physical binding capacity brought by the elevated alumina content from the addition of metakaolin can also be simulated. By doing so, the physically bound chloride content and its impact on chloride transport are evaluated. Combining with the chemically bound chloride content calculated according to the formations of Friedel’s and Kuzel’s salts, the total chloride profile along the ingress path is obtained with addition of $c_{\mathrm{f}}$. What can also be computed from thermodynamic modelling is the volumetric changes in solid products due to potential reactions. It leads to porosity change, calculated by Equation (16), which may further impact the overall ionic diffusivity. Such an impact is estimated through the modified Kozeny–Carman relation [26]; see Equation (17).

$$\varphi ={\varphi }_0-\mathrm{\Delta }{V}_{\mathrm{S}}$$

(16)

$$H_D^{\varphi }(\varphi )={\left({\displaystyle \frac{\varphi }{{\varphi }_0}}\right)}^3{\left({\displaystyle \frac{1-{\varphi }_0}{1-\varphi }}\right)}^2$$

(17)

where ${\varphi }_0$ is initial porosity (-) obtained from material characterisation modelling, $\mathrm{\Delta }{V}_{\mathrm{S}}$ is total volumetric variation (-) calculated from all heterogeneous reactions by GEMS, and $H_D^{\varphi }$ is the ionic diffusivity multiplier (-) as a function of initial porosity and porosity $\varphi$ at the current time step.

With Equation (17), the influence of the calculated multiplier should be included to update the ionic diffusivity in each time step. According to Figure 1, since the sequential non-iterative solver is adopted to solve the reactive transportation problem for numerical stability, a proper time stepping is required to ensure accuracy. Based on the reported convergence studies [16,26], a $\mathrm{\Delta }t=1800 \mathrm{s}$ is demonstrated to balance well between efficiency and accuracy, hence adopted here. To further reduce computational burden, numerical tests are conducted to identify key ionic species for effectively representing chloride ingress into metakaolin-blended cementitious materials. Following a chemistry handbook [28], the ionic radius and intrinsic diffusivity in the free water of these key species are gathered in

Table 1. Radius and intrinsic diffusivity of key ionic species [28].

Ionic Species	Radius (×10−10 m)	Intrinsic D (×10−9 m2/s)	Ionic Species	Radius (×10−10 m)	Intrinsic D (×10−9 m2/s)
${\mathrm{AlO}}_2^{-}$	1.89	1.10	${\mathrm{SiO}}_3^{2-}$	2.20	0.90
${\mathrm{Ca}}^{2+}$	1.00	0.79	${\mathrm{CO}}_3^{2-}$	1.78	0.92
${\mathrm{FeO}}_2^{-}$	1.90	1.00	${\mathrm{Cl}}^{-}$	1.81	2.03
${\mathrm{K}}^{+}$	1.38	1.96	${\mathrm{SO}}_4^{2-}$	2.40	1.07
${\mathrm{Mg}}^{2+}$	0.72	0.70	${\mathrm{OH}}^{-}$	1.40	5.27
${\mathrm{Na}}^{+}$	1.02	1.33	${\mathrm{H}}^{+}$	0.9	9.31

. In addition, the essential hydration products and chloride bearing solids, together with their standard thermodynamic data, are identified and collated in Appendix A with references to Cemdata 18 [29].

2.4. Extended Support Vector Regression (XSVR)

With the comprehensive material characterisation modelling and reactive transport modelling, detailed chloride profiles, such as time-dependent free, physically bound, and chemically bound chloride, can be obtained. In this regard, the potential uncertainties associated with the source materials and employed model can be reflected in the obtained chloride profiles. Nevertheless, as elaborated above, the physics-based chloride resistance analysis is complex and computationally intensive. Numerical solutions, such as the finite element method, are required to solve the governing PDEs, leading to the uses of crude MCS and intractable numerical burden in probabilistic modelling. To enhance efficiency, virtual surrogate models constructed through machine learning to replace time-consuming physics-based models have been widely explored. Among various well-established models, such as SVR, GPR, and Kriging, the novel XSVR, recently developed by the authors [17,18,19], has demonstrated advanced performance. Thus, considering the highly nonlinear multiphysical problem as investigated herein, XSVR is adopted, where the model fundamentals are briefly introduced below.

Distinct from traditional SVR [30], XSVR is extended from the doubly regularised support vector machine [31], and incorporates a quadratic ε-insensitive loss function to enhance numerical stability. For addressing complex multiphysical problems, empirical kernel mapping is utilised [32]. In model training, the input variable dataset is defined as $x_{train}=[x_1,x_2,\dots ,x_i]\in {\Re }^{m\times n} (x_i\in {\Re }^n, i=1,2,\dots ,m)$, with m and n indicating the number of training samples and input variables, containing all potential randomness. The output is defined as $y_{train}\in {\Re }^m$, which includes the stochastic response of time-dependent chloride profiles for assessing the chloride resistance of metakaolin-blended concrete. Leveraging the empirical kernel mapping [32], the kernelisation process is formulated as follows.

$$x_i={[x_{i,1},x_{i,2},\dots ,x_{i,n}]}^T\mapsto \widehat{k}(x_i)=\left[\begin{array}{c}\Phi {(x_1)}^T\Phi (x_i)\\ \Phi {(x_2)}^T\Phi (x_i)\\ ⋮\\ \Phi {(x_m)}^T\Phi (x_i)\end{array}\right]=\left[\begin{array}{c}K(x_1,x_i)\\ K(x_2,x_i)\\ ⋮\\ K(x_m,x_i)\end{array}\right]$$

(18)

$$K_{\mathrm{train}}=\left[\begin{array}{cccc}K(x_1,x_1)& K(x_1,x_2)& \cdots & K(x_1,x_m)\\ K(x_2,x_1)& K(x_2,x_2)& \cdots & K(x_2,x_m)\\ ⋮& ⋮& \cdots & ⋮\\ K(x_m,x_1)& K(x_m,x_2)& \cdots & K(x_m,x_m)\end{array}\right]$$

(19)

where $K(x_i,x_j)$ is the kernel constructed with mapping function $\Phi (x)$, and the generalised Gegenbauer kernel (GGK), as reported in Yu et al. [33], is implemented herein. Moreover, $\widehat{k}(x_i)$ is a feature vector comprising random input variable information, and $K_{\mathrm{train}}\in {\Re }^{m\times m}$ is a kernel matrix used to establish the XSVR in Equation (20).

$$\underset{p_{\mathrm{k}},q_{\mathrm{k}},\gamma ,\xi ,\widehat{\xi }}{\min }:{\displaystyle \frac{{\lambda }_1}{2}}({‖p_{\mathrm{k}}‖}_2^2+{‖q_{\mathrm{k}}‖}_2^2)+{\lambda }_2{e}_m^T(p_{\mathrm{k}}+q_{\mathrm{k}})+{\displaystyle \frac{c}{2}}({\xi }^T\xi +{\widehat{\xi }}^T\widehat{\xi })$$

(20)

$$s.t.\left\{\begin{array}{l}{K}_{\mathrm{train}}(p_{\mathrm{k}}-q_{\mathrm{k}})-\gamma e_m-y_{train}\le \epsilon e_m+\xi \\ y_{\mathrm{train}}-(K_{\mathrm{train}}(p_{\mathrm{k}}-q_{\mathrm{k}})-\gamma e_m)\le \epsilon e_m+\widehat{\xi }\\ p_{\mathrm{k}}, q_{\mathrm{k}}, \xi , \widehat{\xi }\ge 0_m\end{array}\right.$$

(21)

where ${\lambda }_1, {\lambda }_2>0$ and ${‖•‖}_2$ are regularisation parameters and $l_2$ norm, $\xi$ and $\widehat{\xi }$ are non-negative constraints of the slack variable, $e_m$ is defined as ${[1,1,\dots ,1]}^T\in {\Re }^m$, $p_{\mathrm{k}}, q_{\mathrm{k}}\in {\Re }^m$ are non-negative variables determined by Feng et al. [34], $\gamma \in \Re$ represents bias, $c>0$ is constant to denote penalty, and $\epsilon$ is tolerance in output predictions.

Based on the optimisation as presented in Yu et al.’s theory [33], Equation (20) is re-written into Equation (22) and solved as a quadratic programming problem by adopting a non-negative Lagrange multiplier $u_k\in {\Re }^{4m}$ given in Equation (24).

$$\underset{z_{\mathrm{k}},\gamma }{\min }:{\displaystyle \frac{1}{2}}(z_{\mathrm{k}}^T{\widehat{C}}_{\mathrm{k}}{z}_{\mathrm{k}}+{\gamma }^2)+{\lambda }_2{b}_{\mathrm{k}}^T{z}_{\mathrm{k}}$$

(22)

$$s.t. ({\widehat{A}}_{\mathrm{k}}+I_{4m\times 4m})z_{\mathrm{k}}+(\epsilon I_{4m\times 4m}+\gamma {\widehat{G}}_{\mathrm{k}}){\widehat{e}}_{\mathrm{k}}+{\widehat{d}}_{\mathrm{k}}\ge 0_{4m}$$

(23)

$$\underset{u_{\mathrm{k}}}{\min }:{\displaystyle \frac{1}{2}}{u}_{\mathrm{k}}^T{Q}_{\mathrm{k}}{u}_{\mathrm{k}}-m_{\mathrm{k}}^T{u}_{\mathrm{k}}; s.t. u_{\mathrm{k}}\ge 0_{4m}$$

(24)

where

$$Q_{\mathrm{k}}=({\widehat{A}}_{\mathrm{k}}+I_{4m\times 4m}){\widehat{C}}_{\mathrm{k}}^{-1}{({\widehat{A}}_{\mathrm{k}}+I_{4m\times 4m})}^T+{\widehat{G}}_{\mathrm{k}}{\widehat{e}}_{\mathrm{k}}{\widehat{e}}_{\mathrm{k}}^T{\widehat{G}}_{\mathrm{k}}, Q_{\mathrm{k}}\in {\Re }^{4m\times 4m}$$

(25)

$$m_{\mathrm{k}}={\lambda }_2({\widehat{A}}_{\mathrm{k}}+I_{4m\times 4m}){\widehat{C}}_{\mathrm{k}}^{-1}{b}_{\mathrm{k}}-\epsilon {\widehat{e}}_{\mathrm{k}}-{\widehat{d}}_{\mathrm{k}}$$

(26)

where for the definition of ${\widehat{C}}_{\mathrm{k}}, {\widehat{G}}_{\mathrm{k}}, {\widehat{A}}_{\mathrm{k}}, b_{\mathrm{k}}, {\widehat{e}}_{\mathrm{k}}, {\widehat{d}}_{\mathrm{k}}, \mathrm{and} z_{\mathrm{k}}$, one may refer to Yu et al. [33].

Considering $u_{\mathrm{k}}^{\ast }\in {\Re }^{4m}$ as a potential solution to the quadratic programming problem of Equation (24), the XSVR-aided surrogate model is defined in Equation (27) to bypass the physics-based models presented in Section 2.2 and Section 2.3 for enhanced efficiency.

$$\widehat{f}(x)={(p_{\mathrm{k}}-q_{\mathrm{k}})}^T\widehat{k}(x)-{\widehat{e}}_{\mathrm{k}}^T{\widehat{G}}_{\mathrm{k}}{u}_{\mathrm{k}}^{\ast }$$

(27)

3. Model Examination and Analysis

3.1. Problem Description

To examine the advanced performance of the proposed method and computational framework, chloride ponding tests on both pure cement and metakaolin-blended concrete, reported by Ferreira et al. [35], are numerically investigated. The reported mix proportions are listed in

Table 2. Concrete mix proportion information from Ferreira et al. [35].

Mix Identification	Mix Proportions (kg/m3)
	Cement	Metakaolin	Aggregate	Water
PC set	440	-	1840	200
MK set	352	88

, labelled with PC set and MK set for identification. According to the literature [35], concrete materials were prepared and cured under 20 °C for 28 days before testing. The sample was properly sealed to ensure a one-dimensional exposure in the chloride environment sustained using 3% of NaCl solution, and the chloride profiles were measured after 90-day immersed ponding from exposure surface to concrete core. For the source material properties, i.e., cement and used metakaolin, one may refer to Ferreira et al. [35]. In terms of uncertainties inherent to source materials and/or associated with a physics-based model, five key parameters are identified, such as $S_c$, ${\mu }_{\mathrm{MK}}$, Blaine fineness of metakaolin $S_{\mathrm{MK}}$, $k_{\mathrm{MK}}^r$, and $D_{\mathrm{MK}}^e$. These are selected based on their known influence on material durability, and to highlight the advanced performance of the developed method, varied random distributions are assumed, with details given in

Table 3. Design of random variables for physics-based chloride ingress analysis.

Random Variables	Distribution	Mean	Coefficient of Variation (CoV) or Range
$S_c$$\left({\mathrm{cm}}^2/\mathrm{g}\right)$	Lognormal	4100	0.05
$S_{\mathrm{MK}}$$\left({\mathrm{cm}}^2/\mathrm{g}\right)$	Lognormal	15,000	0.05
${\mu }_{\mathrm{MK}}$ (-)	Uniform	0.9	[0.85, 0.95]
$k_{\mathrm{MK}}^r$$\left(\mathrm{cm}/\mathrm{h}\right)$	Gumbel	$5\times {10}^{-8}$	0.1
$D_{\mathrm{MK}}^e$$\left({\mathrm{cm}}^2/\mathrm{h}\right)$	Gumbel	$9.5\times {10}^{-13}$	0.1

.

In the following sections, considering the listed random variables, analyses are conducted first through using the MCS approach. These include material characterisation modelling and subsequent reactive transportation modelling, to gauge the impact of uncertainty on binder hydration as well as chloride penetration. The stochastic outputs in terms of chloride penetration depth ${\tilde{d}}_{\mathrm{crit}}$ (mm), determined by critical chloride content ${\tilde{c}}_{\mathrm{crit}}$ (% by binder mass), are then retrieved from MCS outcomes for machine learning-aided chloride resistance analysis. Surrogate models that correlate random variable inputs with stochastic ${\tilde{d}}_{\mathrm{crit}}$ measured under certain ${\tilde{c}}_{\mathrm{crit}}$ are examined in detail for both computational efficiency and accuracy. In this regard, MCS obtained by ${10}^4$ of physics-based model evaluations ($N_{\mathrm{fun}}$) is adopted as a reference, where various well-established models, such as SVR, GPR, and Kriging, are implemented and compared against XSVR. All simulations are performed using MATLAB R2022b on a desktop workstation (Dell XPS Desktop 8960 sourced in Melbourne, Australia) of 24 Intel Core i9-13900 CPU@3.0 GHz, lasting around 40 min for each physics-based model run. The accuracy of surrogate modelling is assessed using the below error metrics, i.e., coefficient of determination ($R^2$), root mean square error (RMSE), relative error for entire data sets (RE), and $R{E}_i$ for relative error of cumulative distribution function (CDF) curve.

$$R^2=1-{\displaystyle {\sum }_{N_{\mathrm{fun}}}{\left[y-\widehat{f}(\mathrm{x})\right]}^2}/{\displaystyle {\sum }_{N_{\mathrm{fun}}}{\left[y-\overline{y}\right]}^2}$$

(28)

$$RMSE=\sqrt{{\displaystyle {\sum }_{N_{\mathrm{fun}}}{\left[y-\widehat{f}(\mathrm{x})\right]}^2}/N_{\mathrm{fun}}}$$

(29)

$$RE=\sqrt{{\displaystyle {\sum }_{N_{\mathrm{fun}}}{\left[y-\widehat{f}(\mathrm{x})\right]}^2}/{\displaystyle {\sum }_{N_{\mathrm{fun}}}{y}^2}} \mathrm{and}/\mathrm{or} R{E}_i=\left[y_i-{\widehat{f}}_i(\mathrm{x})\right]/y_i$$

(30)

where $\overline{y}$ indicates the mean of stochastic chloride resistance response $\mathrm{y}$ from MCS.

3.2. Influence of Uncertainty

In this section, the impact of material and model uncertainties on concrete hydration and chloride penetration is examined through MCS. Numerical results are showcased in colour-shaded range graphs and compared with reported experiments where applicable.

3.2.1. Impact on Hydrated Metakaolin-Blended Concrete

Generally speaking, the chemical composition of ordinary Portland cement as used in concrete production is stable with minor variations. Thus, as listed in Table 3, only the Blaine fineness of cement $S_c$ is considered as a random variable, given its known impact on hydration kinetics. The remaining four random variables are all associated with the metakaolin, which may lead to more complex uncertainty propagation for metakaolin-blended concrete. Herein, by performing the MCS, the time-dependent progressions of the hydration/reaction degree of binder materials for both PC and MK sets are obtained, which are showcased in Figure 2.

From Figure 2a, it is noted that the influence of randomness in $S_c$ on cement hydration kinetics propagates with time, which eventually leads to around 5% variation in the degree of hydration after 28 days of curing. For the set with the metakaolin blend shown in Figure 2b, the variation in cement hydration kinetics stays roughly the same as that from the PC set. On the other hand, the uncertainty propagates more evidently in the reaction kinetics of metakaolin, as it is governed by multiple random factors associated with both material resources and reactivity. Indeed, roughly 20% of variation is observed in the reaction degree of metakaolin after the prescribed curing period. Combined with the uncertainty in cement hydration, it creates complex stochastic responses in hydrate phase assemblage. Focusing on the present topic, the MCS-guided analyses of key hydration products and properties, in relation to chloride resistance assessments, are performed through GEMS thermodynamic modelling. Specifically, the time-dependent stochastic results of system porosity as well as formations of portlandite, C-A-S-H, and AFm are obtained and shown in Figure 3 and Figure 4 for PC and MK sets, respectively.

In Figure 3 and Figure 4, the developments of hydration products are presented as the mass percentage of the hydrated binder at paste level, so as system porosity. Comparing the two figures, several distinctive features are observed. First, due to enhanced pozzolanic reactions provided by metakaolin addition, portlandite is gradually consumed over time to produce more C-A-S-H phase in the MK set. It compensates for the reduction in cement use, where the C-A-S-H contents are comparable for both PC and MK sets after 28 days of curing. Second, with the enhanced alumina availability provided by metakaolin, the AFm phase production is higher in the MK set than that in the PC set, potentially improving chemical chloride binding capacity. Third, due to the additional randomness with metakaolin, the uncertainty propagation in stochastic responses from characterisation modelling appears to be much greater for the MK set. Overall, the impact of uncertainty on hydrated systems is evident, where the stochastic responses in phase assemblage and porosity are to create varied chloride binding capacity and ionic diffusivity in material, which may further lead to probabilistic system chloride resistance.

3.2.2. Impact on Chloride Penetration

Following the experimental setups as reported by Ferreira et al. [35], the stochastic phase assemblage at Day 28 is regarded as the initial material state to facilitate MCS-guided stochastic chloride penetration analyses. Considering that the total chloride profile was measured once, as the mass percentage to binder content, after 90 days of exposure [35], numerical results are obtained accordingly to compare against the tests. Furthermore, as a comprehensive physics-based reactive transportation model is implemented, this study is able to distinguish the two different chloride binding mechanisms. Thus, in addition to total chloride profile, the stochastic responses in terms of porosity change, as well as free, physically bound, and chemically bound chloride contents along the penetration path, can also be computed. Herein, focusing on the PC set first, the detailed chloride profiles at Day 90 and time-dependent porosity profiles obtained at Day 7, 14, 28, and 90 are showcased in Figure 5 and Figure 6, respectively. The same set of stochastic responses for the MK set are then also obtained and displayed in Figure 7 and Figure 8.

As shown in Figure 5a–c, the chloride profiles along the penetration path after 90 days of exposure are broken down into details as $c_{\mathrm{f}}$, $c_{\mathrm{phy}}$, and $c_{\mathrm{che}}$, respectively. Their summation yields the total chloride content $c_{\mathrm{total}}$ in Figure 5d, which is found to agree very well with the trend as observed in the experimental measurements. Indeed, since no error bar is given by Ferreira et al. [35], the reported measurement is regarded as the mean curve of total the chloride profile along the ingress path, and is covered by the colour contour in Figure 5d. It demonstrates the effectiveness of the physics-based model as well as the impact of uncertainty. Furthermore, it is also noted that there is a drop in $c_{\mathrm{phy}}$ and $c_{\mathrm{che}}$ at the region close to the exposure surface due to simultaneous leaching of calcium-bearing hydrates. In particular, the decalcification of C-A-S-H and AFm reduces the physical and chemical chloride binding capacity [36,37], hence the observed drops in Figure 5b,c. As a result, the dissolution of a collective of solid products would lead to an increase in porosity at the corresponding region. As exhibited in Figure 6, such a stochastic process is also time-dependent, where a rise in variation range is observed close to the surface as continuous exposure towards 90 days, indicating further uncertainty propagations.

Comparing Figure 5 and Figure 7, two distinctive features are observed for the MK set, where the chloride penetration depth is much lower and the peak chloride content is much greater than those found in the PC set. According to Figure 7a–c, the peak free chloride content is found to be comparable between two sets, where the peak physically and chemically bound chloride contents are much higher in the MK set. In terms of $c_{\mathrm{phy}}$, the addition of metakaolin increases the potential Al/Si ratio of the C-A-S-H phase, enhancing the physical binding capacity based on Equation (15). Furthermore, as shown in Figure 3c and Figure 4c, a much higher AFm production after 28-day hydration is observed in the MK set, also due to the additional alumina provided by metakaolin for hydration. As a result, a significant higher peak in $c_{\mathrm{che}}$ is observed for the MK set, compared to that of the PC set. With the enhanced physical and chemical binding capacity, a slower chloride ingress for the MK set is found, reflected in the total chloride content $c_{\mathrm{total}}$, which agrees very well with the experimental measurement. The slowed chloride penetration process due to enhanced binding is also reflected in the stochastic response of porosity changes exhibited in Figure 8. There occurs a slight drop in porosity in the region of the $c_{\mathrm{total}}$ peak, before rising closer to the exposure surface due to simultaneous leaching.

3.3. Machine Learning-Aided Chloride Resistance Analysis

In general, to assess the chloride resistance based on bulk ponding is to examine the chloride ingress depth (${\tilde{d}}_{\mathrm{crit}}$) according to the prescribed critical chloride content threshold (${\tilde{c}}_{\mathrm{crit}}$) [38]. Such a threshold is often adopted to determine the depassivation and onset of reinforcement corrosion. In this study, four ${\tilde{c}}_{\mathrm{crit}}$ values in terms of total chloride content $c_{\mathrm{total}}$, i.e., 0.001%, 0.1%, 0.2%, and 0.4% of binder mass, are considered to determine the stochastic responses of ${\tilde{d}}_{\mathrm{crit}}$. XSVR and several other methods, such as SVR, GPR, and Kriging, are employed to build surrogate relations between ${\tilde{d}}_{\mathrm{crit}}$ and random variables as listed in Table 3, where computational accuracy and efficiency are examined.

3.3.1. On Concrete Without Metakaolin Blend

In this section, XSVR-aided chloride resistance analysis is first applied to concrete without the metakaolin blend, i.e., the PC set. To examine the effectiveness of XSVR, ${\tilde{d}}_{\mathrm{crit}}$ measured at ${\tilde{c}}_{\mathrm{crit}}$ of 0.4% binder mass, it is studied using various $N_{\mathrm{fun}}$ from 10 to 100. The crude MCS outcome obtained with $N_{\mathrm{fun}}={10}^4$ is taken as a reference; see Figure 9.

According to Figure 9, the effectiveness of the XSVR model is fully demonstrated, which achieves over 0.99 of $R^2$ value with only 10 actual physics-based model evaluations. It shows significant improvement in terms of computational efficiency, compared to using the crude MCS. Especially for $N_{\mathrm{fun}}=100$, an outstanding prediction is observed with the highest $R^2$ as well as the lowest RMSE and RE, enhancing the efficiency by 100 times. Considering that only the Blaine fineness of cement is regarded as a random variable for the PC set, the machine learning-aided probabilistic modelling may become more challenging for the MK set. Thus, $N_{\mathrm{fun}}=100$ is to be used throughout the following analyses for consistency. Herein, based on the four selected ${\tilde{c}}_{\mathrm{crit}}$ values, the stochastic responses of the corresponding ${\tilde{d}}_{\mathrm{crit}}$ are calculated with surrogate relationships established using the XSVR, where the estimated probability density function (PDF) and cumulative distribution function (CDF) are compared against MCS in Figure 10.

As highlighted in Figure 10, with an increase in ${\tilde{c}}_{\mathrm{crit}}$, a significant drop in average chloride penetration depth is observed. Moreover, by considering a lower ${\tilde{c}}_{\mathrm{crit}}$, such as 0.001%, a higher variance generally occurs. Overall, regardless of the selected ${\tilde{c}}_{\mathrm{crit}}$ values, outstanding agreements between XSVR-aided analyses and crude MCS are achieved. By carefully examining the $R{E}_i$ from the estimated CDFs, a very low relative error rate is found. Generally, the observed $R{E}_i$ is within around 0.2%, with the largest point being less than around 1%, as found in the case of ${\tilde{c}}_{\mathrm{crit}}=0.4\%$ binder mass. For this particular case, the slightly larger $R{E}_i$ is potentially caused by the choice of ${\tilde{c}}_{\mathrm{crit}}$ value, which corresponds to a region where notable $c_{\mathrm{f}}$, $c_{\mathrm{phy}}$, and $c_{\mathrm{che}}$ are presented simultaneously; see Figure 5. As a result, the complex chloride profile introduces a challenge to surrogate model fidelity. Overall, the proposed computational framework, featuring the XSVR-aided physics-based model, is demonstrated to be effective and efficient for stochastic chloride ingress depth analysis of the PC set.

3.3.2. On Concrete with Metakaolin Blend

Following the previous section, XSVR-aided chloride resistance analysis is extended to concrete with metakaolin blend, i.e., the MK set. According to the convergence study in Figure 9 and subsequent discussions, the same $N_{\mathrm{fun}}=100$ is used in surrogate model building for consistency. Adopting the same series of ${\tilde{c}}_{\mathrm{crit}}$ as those for the PC set, the stochastic responses of ${\tilde{d}}_{\mathrm{crit}}$ are calculated, where the XSVR-aided estimations of PDF and CDF are compared against MCS in Figure 11.

A similar trend is observed in Figure 11, where the average chloride penetration depth decreases as the selected ${\tilde{c}}_{\mathrm{crit}}$ rises. In contrast to the PC set from Figure 10, the variances of stochastic responses remain stable, regardless of the ${\tilde{c}}_{\mathrm{crit}}$ values. This is potentially due to a higher chloride resistance of metakaolin-blended concrete, where the general chloride penetration rate is reduced with enhanced binding capacity, leading to a more stabilised uncertainty propagation. In terms of predictive accuracy, a $R^2$ value of over 0.99 in comparison to crude MCS has been achieved for all cases in Figure 11, showcasing the advanced ability of the proposed framework, featuring XSVR. By further examining the $R{E}_i$ from estimated CDFs, a slightly higher rate of error is observed in the MK set, especially for the case obtained when considering higher ${\tilde{c}}_{\mathrm{crit}}$. It is considered reasonable, as all five random variables, specified in Table 3, contribute to the stochastic responses of chloride penetration depth. Overall, the proposed method is shown to be capable of uncertainty-aware physics-based chloride resistance analysis of metakaolin-blended concrete. Herein, to further highlight the advanced capability of XSVR, performance comparisons against other established methods, such as SVR, GPR, and Kriging, are conducted. Taking ${\tilde{c}}_{\mathrm{crit}}$ of 0.4% binder mass as an example and using $N_{\mathrm{fun}}$ of 100 as the baseline for consistency, model comparisons for ${\tilde{d}}_{\mathrm{crit}}$ assessment of the MK set are illustrated in Figure 12.

As shown in Figure 12, the combination of various random input variables associated with cement and metakaolin challenges the effectiveness of different machine learning models in building surrogate relations for critical chloride penetration depth assessments. Indeed, it appears that, with a small $N_{\mathrm{fun}}$ of 100, the conventional SVR and GPR fail to generate meaningful relationships, where undesirable $R^2$ values are obtained. The Kriging model demonstrates much better performance in comparison to SVR and GPR, reaching a $R^2$ value of around 0.97. Among all four machine learning methods, XSVR is found to be of the best performance in correlating stochastic chloride ingress responses with random material properties, showing a $R^2$ value of over 0.99. Overall, it demonstrates the significant challenges rooted within uncertainty-aware physics-based chloride ingress analyses, involving binder chemistry and reactive transport modelling. More importantly, it highlights that the proposed framework, featuring the XSVR-aided physics-based model, is capable of probabilistic material modelling on chloride resistance of metakaolin-blended concrete with enhanced computational efficiency.

4. Implications for Durable Design Optimisation

In this section, leveraging the verified XSVR-aided surrogate models for chloride ingress depth assessment, virtual tests are conducted. This is with the intention to generate new insights into effectively adjusting binder resource characteristics for optimising chloride resistance towards enhanced durability. In this regard, given that the property of ordinary Portland cement often remains stable and comparable, focus is placed on the potential variations in metakaolin resources, where the influences of purity and the Blaine fineness of metakaolin, i.e., ${\mu }_{\mathrm{MK}}$ and $S_{\mathrm{MK}}$, are investigated. With reference to the reported data and literature [39], high-grade metakaolin resources with a ${\mu }_{\mathrm{MK}}$ between 0.85 and 0.95 are studied, where a wide range of $S_{\mathrm{MK}}$ between 1.2 ${\mathrm{m}}^2/\mathrm{g}$ and 1.8 ${\mathrm{m}}^2/\mathrm{g}$ is considered. These would create a series of combinations regarding the metakaolin characteristics for examining their corresponding impact on chloride penetration. Herein, similar to Section 3.3, four ${\tilde{c}}_{\mathrm{crit}}$ values measured at 0.001%, 0.1%, 0.2%, and 0.4% of binder mass are used to determine 90-day ingress depth, where numerical outcomes are shown in Figure 13.

A distinctive trend is observed in Figure 13, where ${\tilde{d}}_{\mathrm{crit}}$ drops, regardless of the prospective ${\tilde{c}}_{\mathrm{crit}}$ values used for measurements, with the rises of ${\mu }_{\mathrm{MK}}$ and $S_{\mathrm{MK}}$. It indicates the importance of pozzolanic reactivity towards enhancing the chloride resistance of metakaolin-blended concrete. Indeed, both the increase in ${\mu }_{\mathrm{MK}}$ and $S_{\mathrm{MK}}$ promotes the reaction kinetics of metakaolin, according to Section 2.2. The elevated Al/Si ratio for the C-A-S-H gel gives rise to better physical chloride binding, where more AFm produced through hydration also promotes a greater chemical binding capacity of chloride. Thus, based on Figure 13, it is concluded that, with the metakaolin resource of certain ${\mu }_{\mathrm{MK}}$, the additional grinding for a higher particle fineness is generally beneficial to enhancing the chloride binding capacity of the metakaolin-blended concrete, leading to much improved material chloride resistance and overall product durability.

5. Conclusions

Metakaolin obtained from calcined kaolinitic clays is an emerging pozzolanic SCM for sustainable binder production due to its vast availability. As reported in the literature, metakaolin also offers enhanced chloride binding capacity to concrete, desirable for long-term durability. Nevertheless, due to resource variety and production discrepancy, the reactivity of different metakaolin sources varies significantly, which may challenge the consistency of product performance. To better facilitate scientifically robust binder design, this study presents a novel numerical framework to achieve uncertainty-aware chloride resistance analysis of metakaolin-blended concrete. The proposed methodology features comprehensive physics-based models for material characterisation and reactive transport analyses, assisted with XSVR for efficient probabilistic modelling. A variety of random inputs associated with cement and metakaolin resources, such as $S_c$, ${\mu }_{\mathrm{MK}}$, $S_{\mathrm{MK}}$, $k_{\mathrm{MK}}^r$, and $D_{\mathrm{MK}}^e$, are considered, where the investigation reveals the following findings.

(1)

The influence of material uncertainty on binder hydration propagates as curing time. The stochastic response in terms of phase assemblage suggests notable impact on the amount of alumina-bearing hydrate, i.e., AFm phase, for the metakaolin-blended mix.

(2)

The MCS-guided physics-based stochastic chloride ingress analysis is capable of distinguishing different chloride profiles, i.e., $c_{\mathrm{f}}$, $c_{\mathrm{phy}}$, $c_{\mathrm{che}}$, and $c_{\mathrm{total}}$. The metakaolin blend exhibits considerably higher binding capacity, hence slower chloride ingress process.

(3)

The proposed framework, featuring the XSVR, is demonstrated to be effective for correlating stochastic responses of chloride penetration depth with material randomness, significantly improving the efficiency of probabilistic modelling.

(4)

In the case of dealing with multiple sources of uncertainty, the XSVR showcases advanced performance compared to conventional SVR, GPR, and Kriging methods.

(5)

Regardless of the purity of metakaolin resources, it is concluded that grinding for finer particle size is considered beneficial for achieving greater chloride resistance of the metakaolin-blended concrete.

(6)

Future studies should focus on generalisation of the affinity parameter $\kappa$ for the proposed approach to be more versatile and accurate in reflecting the varied physical binding capacities of metakaolin-blended concrete, considering uncertainty in resources.