A Novel Simulation Method for the Spatiotemporal Variation in Relative Humidity in Early Age of Polypropylene Fibers Reinforced Concrete

Abstract

Early-age cracking remains a major durability challenge for concrete. It is primarily caused by internal restraint stresses induced by humidity and temperature gradients during hydration. Conventional approaches often fail to capture the coupled and non-uniform nature of heat and moisture transport, limiting their ability to predict cracking risk and evaluate mitigation strategies. To address this limitation, we characterize the spatiotemporal evolution of internal humidity and temperature using a spatial coefficient of variation. From a numerical standpoint, the influence of polypropylene fibers (PPFs) on internal relative humidity is elucidated by adopting an unconditionally stable backward-Euler finite-difference scheme to resolve multiple coupled physicochemical processes—hydration, heat release, self-desiccation, heat and moisture diffusion to the environment—and their mutual interactions. Furthermore, a one-dimensional homogeneous random-field model is proposed to quantify the spatial non-uniformity of humidity in PPF concrete. On this basis, the effects of polypropylene fibers (PPFs) in mitigating internal humidity is quantitatively revealed. Good agreement is achieved between simulations and tests, with standard deviations of 0.0119 for normal concrete and 0.0041 for PPF concrete, thereby validating the model’s predictive capability for the spatiotemporal distribution of internal relative humidity (RH) in PPF concrete. According to the numerical analysis, owing to the moisture-sorption characteristics of PPFs, at a depth of 25 mm, the internal RH in PPF concrete has decreased by 16% at 28 days, whereas normal concrete exhibits a 28% decrease. With increasing depth, the RH reduction at 28 days is approximately 13% for both PPF concrete and plain concrete, and the time-dependent evolution of RH in PPF concrete is broadly similar to that of normal concrete. Furthermore, the mitigating influence of PPFs decreases with hydration age and distance from the surface, reflecting the gradual decline of diffusion heterogeneity over time and depth. These findings provide new numerical evidence for the effectiveness of PPFs in reducing the early-age cracking risk in concrete.

1. Introduction

In recent years, fiber-reinforced cement-based concrete has been widely used due to its ability to enhance crack resistance and improve concrete durability [1,2]. Early-age cracking in concrete has become critical for both service life and structural safety [3]. The main causes can be classified into two aspects. First, the heat released during cement hydration generates temperature gradients inside the concrete. Due to the heterogeneous nature of this multiphase material, non-uniform thermal strains occur, producing differential volumetric changes and residual stresses that lead to cracking [4,5,6]. Second, internal humidity gradients cause the migration of pore water toward the surface, which induces shrinkage stresses under capillary tension and further promotes cracking [7,8]. Therefore, investigating the coupled evolution of temperature and humidity fields in early-age concrete is essential for understanding its cracking mechanisms.

Recent studies have increasingly focused on the spatiotemporal variation in the internal relative humidity in concrete, often referred to as the relative humidity field. Ding et al. [9,10] demonstrated that humidity variation in cementitious materials can be described by diffusion equations, while Shen et al. [11] further incorporated nonlinear diffusion coefficients to account for the time dependence of humidity transport. Since relative humidity is strongly governed by hydration, the coupling between moisture diffusion and hydration heat generation has been recognized as a key mechanism. Zhao et al. [12] proposed a coupled mass–heat transfer model to capture this interaction in porous media. Similarly, Bouziadi and Puatatsananon et al. applied finite element methods to study concrete under multi-field coupling, providing theoretical insights into creep and durability [13,14]. Extending this framework, Brugger et al. [15] incorporated the combined influence of temperature, humidity, and load to investigate deformation mechanisms. These studies highlight that the pore structure of concrete provides pathways for diffusion, permeation, and migration, and that hydration-induced pore evolution further affects the coupled transport of heat and moisture [16].

Among various fibers, polypropylene fibers (PPFs) are widely used due to their low cost and ability to suppress plastic shrinkage cracks [17,18]. Researchers have demonstrated that PPF-reinforced concrete exhibits improved mechanical properties and durability compared with ordinary concrete [19]. Mechanically, better tensile strength, shear resistance, and impact toughness, etc., are possessed through the incorporation of PPFs with the enhancement of ductility [20,21]. However, when considering the durability, PPFs contribute to better crack resistance [22], freeze–thaw resistance [23], high-temperature stability [24], and shrinkage control [25,26]. Upon incorporating PPFs, a spatially distributed three-dimensional fiber network forms while the concrete is in its plastic state. Water can adsorb onto PPFs’ surfaces, retarding moisture migration and mitigating plastic-shrinkage effects; this reduces the likelihood of capillary-crack initiation and propagation and markedly improves early-age crack resistance [27,28,29,30,31]. In parallel, using fine-diameter and surface-modified PPFs further enhances interfacial bonding with the cementitious matrix [32,33,34,35,36]. Together with frictional bridging, this stress-redistribution effect increases the material’s energy-absorption capacity, yielding clear advantages in impact resistance [37] and resistance to water penetration [38]. Ma et al. [39] reported that even a low PPF dosage (0.1%) suppresses moisture loss in 3D-printed concrete (3DPC) layers, providing an internal curing effect and reducing drying deformation. Xu et al. [40] showed that adding PPFs to an alkali-activated slag (AAS) binder not only lowers the drying shrinkage of AAS mortars but also decreases their average pore size, thereby increasing both their flexural and compressive strengths. Abusogi et al. [41] observed that, owing to the bridging and moisture-retention effects of PPFs, PPF-containing mortars exhibit delayed crack occurrence and reduced crack width. Lu et al. [42] investigated hybrid PPF–basalt–fiber (BF)-reinforced concrete, leveraging the early-age crack-mitigation of PPFs and the stiffness enhancement of BF; both the workability and mechanical properties improved. Cao et al. [43] synthesized functional PPFs bearing hydrophilic groups, which strengthened bonding to cement paste and reduced drying shrinkage by 25%, clearly outperforming conventional PPFs. These findings indicate that PPFs not only improve the strength and durability performance of concrete, but also significantly impact the evolution of its internal temperature and humidity fields at an early age. The limited literature indicates that the experimental methods for the spatiotemporal distribution of internal relative humidity in early-age concrete are relatively difficult. Therefore, numerical simulation methods provide an effective way to solve this problem [44].

Nowadays, most research efforts on early-age cracking rely on qualitative assessments, which are beneficial for comparing the cracking sensitivity of different mixtures and additives but remain insufficient for quantifying the fibers’ effects on the spatiotemporal evolution of temperature and relative humidity in concrete. However, this quantitative insight is crucial for developing effective techniques and materials to mitigate early-age cracking [45]. It is found from the literature review that although numerical analyses are employed to characterize the coupling between hydration and heat–moisture diffusion, only the temporal evolution of this coupling is considered, whereas spatial variability within the concrete is not addressed. This ignorance is critical, as the heterogeneous pore structure of concrete induces random, non-uniform distributions of heat and moisture transport. Considering the influence of PPFs on the internal moisture retention, it is therefore necessary to study their effect on the spatial distribution of relative humidity in early-age concrete [46,47].

In this paper, the spatiotemporal evolution of relative humidity in PPF-modified concrete was investigated using a numerical simulation framework. To solve the coupled heat–moisture diffusion equations, an unconditionally stable backward difference scheme was applied, with the diffusion coefficient expressed as a time-dependent parameter linked to hydration. In addition, the coupling effect of the hydration and heat–humidity diffusion process was managed by modifying the diffusion factor within the hydration process. In the simulation, several models were used to account for the hydration process and self-drying effects, as well as the temperature and humidity diffusion processes of concrete at an early age. To assess the effects of spatial variability of the diffusion factor on relative humidity distribution, a one-dimensional homogenous random field model was proposed to simulate the spatial variability of diffusion. Subsequently, the method was validated against experimental data, demonstrating its accuracy and potential as a reliable tool for predicting early-age humidity distribution and guiding crack control in fiber-reinforced concrete. Finally, the remainder of this paper is organized as follows. Section 2 (Materials and Methods) presents the mix design and experimental reference, the hydration and self-desiccation models, the coupled heat–moisture diffusion equations, the unconditionally stable backward-difference solution scheme, and the lognormal random-field representation for the spatial variability of diffusion. Section 3 (Results and Discussion) reports model-experiment comparison for internal RH, analyzes the spatiotemporal RH distribution with depth, quantifies the effect of spatial variability, and discusses the underlying mechanisms and practical implications. Section 4 (Conclusions) summarizes key findings and outlines directions for future work.

2. Materials and Methods

2.1. Mixing Ratio and Production of Concrete in the Literature

Numerical simulations were conducted based on the mixed design and experimental data of the humidity field in the PPF concrete reported in Ref. [1]. The relevant parameters are shown in

Table 1. C30 normal concrete mix ratio (kg/m³).

Label	Strength	Cement	Water	Sand	Coarse Aggregate	Fly Ash	Superplasticizer	Polypropylene
C30-0	C30	288 (P.O.32.5)	180	750	1100	72	0.65%	0
C30-C-0.9	C30	288 (P.O.32.5)	180	750	1100	72	0.65%	0.9

. Concrete specimens were cast in a 200 × 200 × 400 mm wooden mold covered with plastic film, leaving only the top surface exposed to air to allow for one-dimensional moisture diffusion. The environmental temperature was maintained at 20 °C and the RH of 60%. Ordinary Portland cement (P.O 32.5, Jinyu brand) was used (density = 3.1 g/cm³). The coarse aggregate was crushed limestone with a particle size of 5–25 mm. the fine aggregate was natural sand with a fineness modulus of 2.64. A high-range water-reducing admixture (FDN-A) and a naphthalene-based water reducer (SPA) were employed. Class I low-calcium fly ash produced by the Yuanbaoshan Power Plant (Inner Mongolia, China) was used, and silica fume from the Guizhou Hongfeng Ferroalloy Plant was adopted, with an amorphous SiO₂ content of 89.22% and a specific surface area of 200 × 103 cm²/g. PP fibers with a length of 12 mm, a diameter of 56 μm, and a fiber content of 0.9 kg/m³ were used in the test and marked as C30-C-0.9 in Table 1. The particle size of the aggregates in the mix ratio of the two was 5~25 mm. After concrete casting, sensors were embedded. Each sensor was housed in a plastic tube (inner diameter 15 mm, length 75 mm), the tube’s bottom was sealed, and a slit was reserved in the lower sidewall. Two O-rings (2 mm thickness) were fitted around the upper portion of the sensing probe to create a 3 cm sealed cavity at the tube’s bottom. After insertion, the gap between the sensor and the tube at the top end was sealed with a polymer liquid sealant. The device was a capacitive digital temperature–humidity sensor with an RH measurement range of 0–100% and an accuracy of ±3% RH. Data were automatically recorded every 10 min, as shown in Figure 1a. Based on the experimental data in Ref. [1] (all extracted from Figure 2), together with our numerical results, we analyzed relative humidity at different times and spatial locations.

2.2. Physical and Chemical Models of the Hydration and Diffusion Processes

Figure 1b,c present the flowchart of the research methods in this paper and interaction between temperature and relative humidity in early-age concrete. The physical and chemical processes of early-age concrete, including hydration, heat diffusion, chemical drying, and drying, can cause moisture changes inside the concrete. Moreover, the heat and moisture diffusion are interconnected. These comprehensive processes could be expressed as a function of the degree of hydration, which depends, to some extent, on the water content within the concrete, which is typically quantified by relative humidity (RH). To simulate the spatiotemporal variations in internal humidity, several existing models for the heat of hydration, self-drying, and heat and moisture diffusion were integrated, as follows.

2.2.1. Concrete Hydration and Self-Drying Model

The hydration process of cementitious concrete is associated with the ingredients, water–cement ratio, water content, and environmental conditions such as temperature and humidity. However, for simplicity, the heat of hydration is often expressed as a function of the degree of hydration, α, as described in Equation (1) [1]. Such a degree of hydration is also influenced by the internal relative humidity of the cement concrete, caused by water loss during hydration. Following a large number of experiments, Zhao et al. [49] proposed an empirical expression between α and RH, as shown in Equations (2)–(4). Accordingly, the degree of hydration, α, can be determined by integrating with the equivalent age of hydration, as shown in Equation (5). According to the Arrhenius equation, Bažant et al. [50] proposed the formulation of the equivalent age, $t_e$, corresponding to the reference temperature of 20 °C, as shown in Equation (6).

$$Q=\alpha \cdot Q_{total}$$

(1)

$${\displaystyle \frac{d\alpha }{d{t}_e}}=\kappa (RH{)}^n+p$$

(2)

$$\kappa ={\alpha }_c{\displaystyle \frac{B}{A}}{\left[\mathit{ln}{\displaystyle \frac{{\alpha }_u}{{\alpha }_c}}\right]}^{(B+1)/B}-p$$

(3)

$${\alpha }_u={\displaystyle \frac{1.031w/c}{0.194+w/c}}<1$$

(4)

$$\alpha =\int \left\{\left[{\alpha }_c•{\displaystyle \frac{B}{A}}{\left(\mathit{ln}\left({\displaystyle \frac{{\alpha }_u}{{\alpha }_c}}\right)\right)}^{\left(B+1\right)/B}-p\right](RH{)}^n+p\right\}d{t}_e$$

(5)

$$t_e={\int }_0^t\mathit{exp}\left({\displaystyle \frac{1}{R}}\left({\displaystyle \frac{U_{ar}}{273+20}}-{\displaystyle \frac{U_{aT}}{273+T}}\right)\right)dt$$

(6)

where $Q_{total}$ is the total heat of complete hydration of the concrete, $n$ and $p$ are the empirical constants that can be obtained according to the experimental data, $\kappa$ is the comprehensive hydration rate constant, R is the constant ($8.314 \mathrm{J}/\mathrm{mol}\cdot \mathrm{K}$) of an ideal gas, ${\alpha }_c$ and ${\alpha }_u$ correspond to the real-time degree of hydration and ultimate degree of hydration, respectively, when the humidity inside the concrete starts to decrease. According to the experimental studies, the hydration of concrete cannot be fully performed in most cases, and the ultimate degree of hydration will depend on the initial water–cement ratio: ${\alpha }_u={\displaystyle \frac{1.031w/c}{0.194+w/c}}<1$, where $U_{ar}$ and $U_{aT}$ represent the activation energy of cement hydration at a reference temperature of 20 °C and the actual temperature, respectively [51,52]. In addition, at a temperature of 20 °C < T < 80 °C, $U_{ar}=U_{aT}=\mathrm{33,500} \mathrm{J}/\mathrm{mol}$. In Equations (3) and (5), A and B are empirical fitting constants, which can be determined through isothermal experiments [50].

According to the heat conduction theory, the temperature increment per unit volume of concrete is proportional to the hydration heat and can be represented by the following expression:

$$\Delta T={\displaystyle \frac{\Delta Q}{c\rho }}={\displaystyle \frac{\Delta \alpha }{c\rho }}·Q_{total}=\int \left\{\left\{{\alpha }_c·{\displaystyle \frac{B}{A}}{\left[ln\left({\displaystyle \frac{{\alpha }_u}{{\alpha }_c}}\right)\right]}^{(B+1)/B}-p\right\}{\left(RH\right)}^n+p\right\}d{t}_e·{\displaystyle \frac{Q_{total}}{c\rho }}$$

(7)

where $c$ and $\rho$ are the specific heat capacity and density of concrete, respectively.

The hydration process will accelerate in the early stage and slow down with the loss of water. However, as suggested by Zhang et al. [53], the relative humidity of concrete at an early age can be expressed as a function of the degree of hydration:

$${\displaystyle \frac{1-R{H}_{s,\alpha }}{1-R{H}_{s,{\alpha }_u}}}={\left({\displaystyle \frac{\alpha -{\alpha }_c}{{\alpha }_u-{\alpha }_c}}\right)}^{\beta }$$

(8)

where $R{H}_{s,\alpha }$ and $R{H}_{s,{\alpha }_u}$ denote the relative humidity caused by self-drying with the degree of hydration $\alpha$ and ${\alpha }_u$, respectively, and $\beta$ is an empirical constant.

2.2.2. Temperature and Moisture Diffusion Models

In addition to the hydration effects, the diffusion process under environmental conditions also influences the temperature and relative humidity inside concrete. According to the heat conduction theory of solid materials, the temperature change can be predicted by Equation (9). Equation (10) shows the thermal boundary condition along the x-direction, where Equation (10a) represents the adiabatic state, and Equation (10b) applies to one-dimensional diffusion situations, which assumes that heat that passes through the concrete surface is proportional to the difference between the concrete surface temperature (T_S) and the environment temperature (T_C):

(a) Temperature change prediction:

$${\displaystyle \frac{\partial T}{\partial t}}=D_T\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}\right)$$

(9)

(b) Adiabatic boundary conditions:

$$-D_T{\displaystyle \frac{\partial T}{\partial x}}=0$$

(10a)

(c) One-dimensional heat diffusion condition:

$$-D_T{\displaystyle \frac{\partial T}{\partial x}}={\beta }_T\left(T_s-T_c\right)$$

(10b)

where $D_T=k_h/c\rho$ is the diffusion factor of temperature, $k_h$ and $c$ denote the heat-transfer coefficient and specific heat coefficient of concrete, respectively, $\rho$ is the concrete density, and ${\beta }_T$ signifies the convective heat transfer coefficient between concrete and air, generally taking the value of $\mathrm{17,100} \mathrm{W}/({\mathrm{m}}^2\cdot °\mathrm{C})$.

The moisture diffusion equation shown in Equation (11) was similar to the temperature diffusion equation, with the T for temperature in Equation (9) replaced with RH for relative humidity. Similarly, by replacing the diffusion factor of temperature $D_T$ in Equations (9) and (10) with $D_H$, the diffusion factor of relative humidity, the isolated humidity boundary condition could be determined:

$${\displaystyle \frac{\partial \left(RH\right)}{\partial t}}=D_H\left({\displaystyle \frac{{\partial }^2\left(RH\right)}{\partial x^2}}+{\displaystyle \frac{{\partial }^2\left(RH\right)}{\partial y^2}}+{\displaystyle \frac{{\partial }^2\left(RH\right)}{\partial z^2}}\right)$$

(11)

The one-dimensional diffusion boundary condition of relative humidity is given by Equation (12):

$$-D_H{\displaystyle \frac{\partial \left(RH\right)}{\partial x}}=f\left(R{H}_s-R{H}_e\right)$$

(12a)

$$f=C\left(0.253+0.06v_a\right)\left(RH-R{H}_e\right)$$

(12b)

where $f$ is the surface humidity exchange coefficient of the concrete, which depends on the difference between the relative humidity of the concrete surface (RH_S) and the environment humidity ($R{H}_e$), as well as the average wind speed $v_a$, and $C$ is an empirical coefficient.

2.2.3. Correction of Diffusion Factors Based on the Coupling Effect

Moreover, the diffusion factors for heat and humidity in early-age concrete change as the hydration process progresses. According to Schindler et al. [54], the temperature diffusion factor decreases linearly with an increasing degree of hydration, as described by Equation (13):

$$D_T\left(\alpha \right)=D_u\left(1.33 - 0.33\mathsf{\alpha }\right)$$

(13)

where $D_u$ is the final thermal conductivity of concrete.

The 2010CEB-FIP model code for concrete structures establishes an empirical relationship with the coefficient of RH diffusion [32], $D_H\left(RH\right),$ with respect to early age concrete, as shown in Equation (14):

$$D_H(RH)={D_{max}\left[m+{\displaystyle \frac{1-m}{1+{\left[\left(1+RH\right)/\left(1-{RH}_c\right)\right]}^v}}\right]}_{max}$$

(14)

where $D_{max}$ is the coefficient of humidity diffusion with 100% RH, ${RH}_c$ is the critical value of relative humidity, and $m$ and v are empirical coefficients.

2.3. Numerical Simulation Method for Temperature and Relative Humidity

2.3.1. Backward Difference Method for Diffusion Simulation

The difference method, employing difference quotients instead of derivatives, approximates functions at finite discrete points to satisfy the control equations. Known for its efficiency and accuracy, it is widely applied to diffusion problems like heat conduction and seepage. In this study, we selected the backward difference method for its stable and accurate calculation of previous values from current ones. This method was used to solve heat and humidity diffusion equations and simulate the evolution of temperature and humidity in early-age concrete under one-dimensional conditions.

During the difference analysis simulation, the analyzed region is discretized into a grid model, and the diffusion equations, such as Equations (9) and (11), are transformed into second-order difference equations. Below is the second-order difference equation corresponding to Equation (11):

$$\begin{array}{c}-r{\left(RH\right)}_{i+1}^k+\left(1+2r\right){\left(RH\right)}_i^k-r{\left(RH\right)}_{i-1}^k\\ ={\left(RH\right)}_i^{k-1},i=1,2,\dots ,M; k=1,2,\dots ,N\end{array}$$

(15a)

where

$$r=D_H{\displaystyle \frac{\Delta t}{{\left(l\right)}^2}}$$

(15b)

where i and k represent the number and iteration time steps of the nodes, respectively, t signifies time, $t\in \left[0,\Gamma \right],$ $\Gamma$ is the time period of concern, $\Delta t$ is the time step, l is the mesh size, $l\in \left[0,L\right],$ $L$ is the length of the concrete specimen, and $r$ is a parameter related to the difference in step size and the grid scale. The grid model and the computational schema are presented in Figure 2.

2.3.2. Analysis Process Considering Interaction Between Temperature and Humidity

The interaction between temperature and humidity in early-age concrete can be manifested in two aspects. Firstly, as the hydration process develops, it consumes water, leading to a reduction in relative humidity, which in turn slows down the hydration rate [55]. Secondly, the diffusion of heat and humidity with environmental conditions alters the distribution of temperature and relative humidity, subsequently resulting in changes in the temperature and humidity diffusion factors.

To simulate the spatiotemporal distribution of relative humidity in early-age concrete while considering the temperature–humidity interaction, a semi-decoupled approach was adopted. The process of temperature and relative humidity was divided into multiple time steps. Within each time step, two sub-stages were considered. In the first sub-stage, it is focused on the internal processes within the concrete. The changes in temperature and relative humidity due to hydration and self-drying, excluding diffusion effects, were considered. In the second sub-stage, the influence of environmental diffusion was incorporated to revise the temperature and relative humidity. For simplicity while maintaining accuracy, it was assumed that the diffusion factors for temperature and relative humidity remained constant within each time step, provided the step size was sufficiently small. At the end of each step, the diffusion factors were updated based on the current temperature and relative humidity values, ensuring an accurate representation of their dynamic interaction in early-age concrete.

The simulation process is outlined below and illustrated in the flow chart shown in Figure 3.

(1)

Firstly, the analyzed concrete specimens are divided into grid models. Set up time step $\Delta t$ and boundary conditions. Initialize the temperature, $T_0$, relative humidity, $R{H}_0$, temperature diffusion factor, DT₀, and relative humidity diffusion factor, DH₀.

(2)

At time step $t=t+\Delta t$, update the temperature diffusion factor, $D_T^{t+\Delta t}$, and humidity diffusion factor, $D_H^{t+\Delta t}$, using Equations (13) and (14), respectively, based on the temperature T_t and the relative humidity RH_t at the end of the previous time step, t. Then, simulate the current temperature and relative humidity through the following two sub-stages.

(i).

Sub-stage one: Internal processes with hydration and self-drying. Based on the current hydration degree, ${\alpha }^{t+\Delta t}$ (using Equation (5)), calculate the temperature increment, $\Delta T_1^{t+\Delta t}$, due to hydration (using Equation (6)), and the relative humidity increment, $\Delta R{H}_1^{t+\Delta t}$, caused by self-drying (using Equations (7) and (8)). Subsequently, update the current temperature to $T^t+\Delta T_1^{t+\Delta t}$ and the relative humidity to $R{H}^t-\Delta R{H}_1^{t+\Delta t}$.

(ii).

Sub-stage two: Diffusion process with environment. Using the updated temperature and relative humidity from sub-stage one, predict the temperature increment, $\Delta T_2^{t+\Delta t}$, and the relative humidity increment, $\Delta R{H}_2^{t+\Delta t}$, due to diffusion by the application of the backward difference method, as shown in Equation (15).

(iii).

Final revision: Superimpose the increments of temperature and relative humidity from both stages to obtain the final temperature, $T^{t+\Delta t}=T^t+\Delta T_1^{t+\Delta t}-\Delta T_2^{t+\Delta t}$, and the final relative humidity, $R{H}^{t+\Delta t}=R{H}^t+\Delta R{H}_1^{t+\Delta t}-R{H}_2^{t+\Delta t}$, at the end of the current time step.

(3)

Iterate Step 2 to obtain the spatiotemporal distribution of the temperature and relative humidity within the early-age concrete until the desired time point is achieved.

2.4. Random Field Model of Diffusion Factor Considering Spatial Uncertainty

As discussed before, many previous studies treated diffusion factors as constants and overlooked the potential impact of spatial variations in these factors on heat and humidity diffusion. Concrete, as a composite material, possesses non-uniform internal structures. The spatial heterogeneity of these multiphase mediums is a key contributor to the anisotropic nature of concrete’s physical and mechanical properties, including the diffusion factors for heat and humidity. Neglecting the spatial variability of diffusion factors can lead to inaccurate predictions concerning the temperature and humidity evolution inside early-age concrete, as well as the resulting shrinkage. Therefore, in this section, we further explored the impact of non-uniform spatial distribution of diffusion factors on the humidity levels within concrete.

The homogeneous random field model has been widely used for the spatial variability of parameters. We could assume that the diffusion factor of concrete consisted of a homogenous random field, where the mean and standard deviation of the diffusion factor were constant in space, and the correlation of the diffusion factor would change with distance between the two measured points, rather than with specific locations. In a homogenous random field, the larger the intervals between the two consideration points, the weaker the correlation, as shown in Figure 4, where b denotes the correlation length [15]. If the interval between the two consideration points was less than b, they were considered correlated. Otherwise, these two points would not be correlated. Thus, the homogenous random field model of parameters could be obtained by giving the mean value, standard deviation, and correlation function.

Several discretization methods can be used for the random field model. Among these methods, the spectrum representation method proposed by Kakooei et al. [48] is popular due to its applicability and robustness. The basic principle of the spectrum representation method involves generating sampling functions using the power spectral density function corresponding to the correlation function. Because the diffusion factor cannot be negative, it can be further assumed to be a lognormal random field [16]. A one-dimensional log-normal random field for the diffusion factor of concrete can be expressed by Equation (16a).

$$f(x)=\mathit{exp}\left(\sqrt{2}\sum _{i=0}^{N-1}{A}_i\mathit{cos}({\omega }_{i}x+{\varphi }_i)+{\mu }_{ln}\right)$$

(16a)

$${\omega }_i=i\Delta \omega$$

(16b)

$$\Delta \omega ={\omega }_u/N$$

(16c)

$$A_i=\left\{\begin{array}{cc}0,& i=0\\ \sqrt{2{\sigma }_{ln}^2{S}_{ff}\left({\omega }_i\right)\Delta \mathsf{\omega }} ,& i=1,2,\mathrm{L},N-1\end{array}\right.$$

(16d)

where $A_i$ and ${\omega }_i$ are the standard deviation and the frequency of item i, (i = 1~N), respectively, $\Delta \omega$ is the frequency increment, $S_{ff}\left(\omega \right)$ and ${\omega }_u$ denote the power spectral density function corresponding to the autocorrelation function $R_{ff}\left(\xi \right)$ and the related upper cut-off frequency, respectively, $x_i$ and ${\varphi }_i$ represent the coordinate and an independent random phase angle that follows a uniform distribution of $[0,2\pi ]$, respectively, and ${\mu }_{ln}$ and ${\sigma }_{ln}$ denote the logarithmic mean value and standard deviation of the diffusion factor, respectively, ${\mu }_{ln}=\mathit{ln}(\mu )-{\displaystyle \frac{1}{2}}\mathit{ln}(1+{\mathit{cov}}^2)$ and ${\sigma }_{ln}=\sqrt{ln(1+{cov}^2)}$.

The upper cut-off frequency, ${\omega }_u$, was calculated by Equation (17), and ε was 0.001 in this study:

$${\int }_0^{{\omega }_u}{S}_{ff}\left(\omega \right)d\omega =(1-\epsilon ){\int }_0^{\mathrm{\infty }}{S}_{ff}\left(\omega \right)d\omega$$

(17)

According to Xu et al. [30], the correlation function has few effects on the correlation length. Therefore, we selected the square exponential (Gaussian) function shown in Equation (18a) as the correlation function for the proposed random field model, following the power spectral density function given by Equation (18b):

$$R_{ff}(\xi )=\mathit{exp}\left[-{\left({\displaystyle \frac{\xi }{b}}\right)}^2\right]$$

(18a)

$$S_{ff}(\omega )={\displaystyle \frac{b}{2\sqrt{\pi }}}\mathit{exp}\left(-{\displaystyle \frac{b^2{\omega }^2}{4}}\right)$$

(18b)

where $R_{ff}$ is the autocorrelation coefficient of the two consideration points in the logarithmic random field, $\xi$ signifies the interval between the points, and $b$ denotes the correlation length.

Figure 5 presents the flow chart of the spectral representation method for random field discretization of the one-dimensional diffusion factor of concrete.

However, until now, no specific data have been obtained to determine the spatial variability of the water diffusion factor in concrete. Considering the water diffusion mainly occurs in the cement in concrete, it is reasonable to assume the correlation length of the water diffusion factor approaches the size of the aggregate particles, e.g., 20 to 40 mm for C30 concrete. Figure 6 shows typical random samples of the water diffusion factor with correlation lengths of 20 mm and 40 mm, using a variation coefficient of 0.5, based on the concrete strength variation. The remaining parameters were the same as those listed in Table 1 and

Table 2. Parameters for relative humidity simulation [53,57].

	C30-0	C30-C-0.9		C30-0	C30-C-0.9
D1	18	1.1	n	82.08	82.08
Du	8.49	8.49	P	0.00005	0.00005
f	34.976	21.763	${\alpha }_c$	0.5202	0.5202
βT	120	120	${\alpha }_u$	0.8246	0.8246
$\kappa$	0.00958	0.00958	$T_c$	20	20
$H_{s,u}$	0.835	0.835	$H_c$	0.6	0.6

. Notably, the smaller correlation lengths lead to more pronounced spatial variations in diffusion factors within the concrete.

3. Results and Discussion

3.1. Numerical Simulations

Using the proposed method, the internal relative humidity of early-age concrete was simulated, as shown in Figure 7, and compared with the experiment results of Hou et al. [1]. The simulation parameters are listed in Table 2. According to Figure 7, the simulated RH of early-age normal concrete and PPF-reinforced concrete were similar to the experimental data. For normal concrete, the average error and standard deviation were 0.0104 and 0.0119, respectively, while for PPF concrete, they were even smaller, at 0.0043 and 0.0041, respectively. These results confirm that the proposed numerical analysis method is effective for simulating the early-age relative humidity field in both normal and PPF-reinforced concrete.

As shown in Figure 7, the RH started to decrease at the end of Day 4, 25 mm from the surface of the normal concrete specimen, with a decrease of 28% on Day 28. For the concrete mixed with 0.9 kg/m³ PPF, the RH started to decrease on Day 9 and decreased by 16% after 28 days. We found that the PPFs slowed down the RH decrease near the surface of the concrete and had a good water-retaining effect on the concrete.

3.2. The Spatiotemporal Distribution of the Humidity of the PPF Concrete

Further, Figure 7 shows the simulated temporal evolution RH at different depths. By Day 28, RH at a depth of 250 mm decreased by 28% in normal concrete, whereas it only decreased by 16% in PPF-modified concrete. At 100 mm, the decreases were 16% and 13% for normal and PPF-modified concrete, respectively. However, at 180 mm, the water retention effects of both mixtures decreased by 12%, indicating that the water retention effect of PPFs weakens as the distance from the concrete surface increases.

Figure 8 illustrates the spatial distribution of RH over time. Assuming uniform water loss due to hydration, the spatial variation in RH was mainly due to water’s diffusion with the environment. In general, PPF-modified concrete had higher RH, and less spatial gradients than normal concrete. Beyond 180 mm, the influence of PPF was sufficiently reduced, resulting in a similar value of RH for both concretes. On Day 7, the PPF-modified concrete was saturated (100% RH), while the normal concrete reached 75.4% saturation on Day 14. By Day 28, the RH on the surface of the normal concrete was equal with the environment (62%), while PPF-modified concrete remained at a higher level (81%).

These simulated results suggested that PPFs significantly affected humidity diffusion and distribution, especially near the surface. This impact can be attributed to fiber-induced modification of the micropore structure, which reduced the humidity diffusion factor (e.g., 1.1 for PPF-modified concrete vs. 18 for normal concrete in Table 2). In addition, the fibers can also retain a certain amount of water, preventing its migration and loss. The higher humidity and smoother gradients in PPF-modified concrete help mitigate plastic shrinkage and associated tensile stress, therefore lowering the risk of early-age cracking in concrete.

3.3. The Influence on the Humidity of Spatial Variability of Diffusion Factor

In this section, we further discussed the influence of spatial variability on the distribution of RH of PPF concrete and normal concrete. Two characteristics of spatial variability of the humidity diffusion factor, namely, the correlation length and the variation coefficient, were considered.

Based on the results presented in the (i) of Figure 9a–c, it was observed that the mean value of the RH of normal concrete remained consistent across various correlation lengths of the humidity diffusion factor. However, notable variations were seen in the coefficients of variance for RH, particularly in proximity to the concrete surface, as illustrated in the right column of Figure 9. These variations intensified closer to the surface and diminished rapidly with increasing depth, reaching a threshold at approximately 25 mm. At greater depths (150 mm), the spatial variability of the diffusion factor had a negligible impact on RH. Furthermore, the influence of this spatial variability weakened progressively over time. Specifically, the maximum coefficient of humidity variation near the surface was approximately 0.35 on Day 14, whereas it decreased to 0.25 by Day 28.

The impact of diffusion factor variations on RH in PPF-modified concrete is illustrated in Figure 10. In this simulation, the correlation length and variation coefficient of the diffusion factor were set to 20 mm and 0.5, respectively, while other parameters followed Table 1 and Table 2. A comparison with normal concrete indicates that the RH variation in PPF-modified concrete, caused by the spatial variations in the diffusion factor, was much lower. Specifically, at a depth of 20 mm, the RH variation coefficient was 0.02 for PPF-modified concrete, compared to 0.18 for normal concrete. As depth increased, such as at 100 mm and 180 mm, the variation coefficient in PPF-modified concrete decreased rapidly to zero. These results suggested that the RH distribution in PPF-modified concrete is less sensitive to the spatial variability of the diffusion factor than in normal concrete. The RH field in PPF-modified concrete was nearly uniform, except near the surface, thereby reducing the risk of shrinkage and early-age cracking.

3.4. Discussion

The moisture-retention effect of PPFs is the result of multiple complementary mechanisms: (i) Fibers act as obstacles within the paste, decreasing the connectivity of capillary channels and increasing transport-path tortuosity, thereby markedly lowering the effective moisture diffusivity and moderating near-surface RH decline. (ii) PPFs suppress bleeding pathways and early-age microcracks during the plastic stage and provide crack-bridging in the hardened stage, weakening preferential routes along cracks and through-going pores. (iii) The hydrophobic fiber–matrix interface reduces capillary wicking along the fibers and induces local water holdup at the interface, further delaying outward evaporation and diffusion. The moisture-retention benefit diminishes with depth, primarily because humidity gradients and the exchange with the environment are the greatest near the exposed surface, where the PPF-induced reduction in pathway connectivity is most consequential; deeper in the matrix, the external drying drive is substantially weaker and RH evolution is governed more by the relatively uniform hydration process, leading to a gradual convergence of RH between PPFs and plain concretes.

4. Conclusions

In this study, we numerically assessed the spatiotemporal variation in the relative humidity in early-age PPF concrete under one-dimensional diffusion conditions, considering comprehensive models for hydration heat, heat and humidity diffusion, and their interactions. The spatial variability of diffusion factors was considered in the simulation process. The proposed simulation method for relative humidity evaluation in early-age PPF concrete was validated against experimental results.

(1) The simulation results of the relative humidity of ordinary concrete had an average error and a standard deviation of errors of 0.0104 and 0.0119, respectively, compared to the experimental results. For PPF concrete, the average error and standard deviation of errors were even lower, at 0.0043 and 0.0041, respectively. These results demonstrate the applicability of this numerical method for simulating the relative humidity field within PPF concrete.

(2) Early-age shrinkage was delayed in PPF concrete because higher internal humidity was maintained. At 28 days, the surface’s relative humidity decreased by about 16% in PPF concrete and by 28% in normal concrete. The benefit was concentrated near the surface and diminished with depth. Beyond approximately 180 mm, the humidity evolution in PPF concrete was similar to that in normal concrete. The spatial variability of the diffusion factor mainly affected the near-surface region. Stronger spatial heterogeneity and larger humidity fluctuations were observed for smaller correlation length, whereas larger correlation length was associated with smaller variability. Its influence became negligible beyond about 150 mm.

(3) Early-age cracking risk is closely associated with the magnitude and gradient of internal RH. Within the validated coupled humidity–temperature framework, the dosage of PPFs and the water–cement ratio are adjusted in design to establish a controllable RH field, whereby shrinkage-induced cracking, rework, and waste in prefabrication are reduced.