Ensuring the long-term durability of glass fiber-reinforced polymer (GFRP) bars poses a significant challenge in practical applications, particularly in marine environments. Moisture absorption by GFRP bars causes hydrolysis and plasticization of the polymer matrix, resulting in a decline in both their stiffness and strength. Furthermore, the penetration of detrimental ions (e.g., OH-) with moisture accelerates the degradation of GFRP bars. Therefore, clarifying the moisture absorption behavior of GFRP bars is crucial for investigating their durability. Previous studies have indicated that the initial moisture absorption in GFRP bars conforms to the Fickian model. However, with prolonged exposure, anomalous diffusion behavior emerges, characterized as non-Fickian diffusion. This paper reviews existing models for non-Fickian diffusion, highlighting their shortcomings. Gravimetric experiments were then conducted on GFRP bars with diameters of 6, 10, and 14 mm, immersed in portable water at temperatures approximately 23, 40, and 60 °C. Based on the test results and underlying mechanisms, an improved model, named the Weibull relaxation (WR) model, was proposed and validated using the particle swarm optimization (PSO) algorithm for regression analysis. The new model not only exhibits better agreement with the test results but also incorporates fitting parameters with clear physical interpretations. Its distinct advantage over existing models is that it is able to more realistically capture the mechanisms governing the moisture absorption of GFRP bars.
Glass fiber-reinforced polymer (GFRP) bars serve as an ideal alternative to traditional steel reinforcement for marine concrete structures, primarily due to their good corrosion resistance and relatively low cost. However, despite being non-metallic, GFRP bars still exhibit degradation over extended service life [1,2]. Notably, research has shown that GFRP bars are particularly susceptible to moisture-related deterioration in real-world applications [3], [4], [5]. The moisture absorption by GFRP bars occurs via diffusion and significantly affects their properties due to multiple degradation mechanisms. Firstly, moisture causes matrix plasticization by breaking down intramolecular hydrogen bonds within the polymer and forming new polymer-water intermolecular bonds at hydrophilic sites [6,7]. Secondly, the absorbed moisture also triggers matrix hydrolysis, resulting in the chemical scission of ester groups [8]. Thirdly, the absorbed moisture exerts osmotic pressure that leads to the formation of micro-cracks in the surrounding materials [9,10], resulting in swelling of the matrix and leading to interfacial debonding between fibers and the matrix [11], thereby altering their internal stress state [12].
Therefore, a better understanding of the moisture absorption behavior of GFRP bars is paramount for accurate predictions of their long-term mechanical properties. Existing research suggests that moisture diffusion in fiber-reinforced polymer (FRP) composites adheres to the Fickian diffusion model [13], [14], [15]. However, contrasting findings reveal anomalous moisture diffusion behavior classified as non-Fickian diffusion [16], [17], [18]. The non-Fickian behavior is characterized by additional moisture uptake beyond the Fickian diffusion as exposure time increases. Manufacturing defects such as voids and un-impregnated fibers were thought to cause such non-Fickian diffusion [19]. However, these defects are relatively small. The advancements in manufacturing techniques, especially in pultrusion, have further improved the quality of FRP products, but the additional moisture uptake still occurs after extended exposure. This indicates that these defects are not the primary cause of the non-Fickian behavior. The primary mechanism driving the non-Fickian behavior appears to be matrix relaxation [20], [21], [22], [23], [24]. The absorbed moisture increases the free volume and enhances molecular chain mobility within the polymer matrix. This process can trigger matrix swelling, leading to fiber-matrix interfacial debonding and facilitating further moisture uptake [25,26].
The moisture absorption process occurs in two distinct phases. Initially, the changes in matrix microstructures are minimal owing to strong intermolecular forces and the long-chain polymer structure. During this period, Fickian diffusion predominantly governs moisture absorption, while relaxation has a minimal influence. As the exposure time increases, intermolecular and hydrogen bonds gradually break down, leading to rapid increases in moisture absorption, such as non-Fickian behavior.
While FRP composites in real applications may be exposed to complex environments, such as alkaline solutions in concrete pores that can cause chemical reactions and more complex mass changes, understanding moisture uptake in pure water is an essential first step. Over the past few decades, several models have been proposed to evaluate moisture absorption behavior related to relaxation [24],[27], [28], [29], [30]. This study conducted a comprehensive review and evaluation of the existing models. Gravimetric experiments were conducted on GFRP bars with diameters of 6, 10, and 14 mm, immersing in portable water at three different temperatures including room temperature of about 23, 40, and 60 °C. Based on the underlying mechanisms and test results, a new model was proposed to characterize moisture uptake due to relaxation. This model was also validated and compared with existing models.
2. Moisture absorption models
2.1. Fickian model
The Fickian model [31] has been widely adopted to describe moisture diffusion behavior in FRP [32], [33], [34]. This model is analogous to Fourier’s heat conduction theory and suggests that mass transport occurs due to the random motion of the penetrant, driven by the concentration gradient. In the polar coordinate system, the Fickian law can be expressed as follows:
where C is the penetrant concentration; r is the radial distance from the center; t is the time; and D is the diffusivity. The initial conditions for a circular FRP bar are assumed as follows: Before exposure to the solution, the internal of the FRP bar is totally dry (Eq. (2)). Upon exposure, the surface of the bar instantly becomes saturated (Eq. (3)).
$\begin{array}{c}C\left(0;r\right)=0\end{array}$
$C\left(t ; R_{0}\right)=C_{0}$
where R0 is the radius of the bar; C0 is the moisture concentration at saturation.
The solution for Eq. (1) subjected to Eqs. (2), (3) can be found in Crank [35]:
where αm is the mth positive root of the zero-order Bessel function; J0(x) and J1(x) are the zero- and first-order Bessel functions, respectively. The moisture absorption in the FRP bar at the time t, M(t), can be found by integrating the concentration over the cross-section:
where Md0 denotes the moisture absorption after infinite time due to diffusion (saturate moisture absorption).
2.2. Non-Fickian models
Several relaxation models [24],[27], [28], [29], [30] have been proposed to characterize the non-Fickian diffusion. They are summarised as follows.
2.2.1. Exponential relaxation model
Berens and Hopfenberg [24] proposed a model that considers relaxation as a first-order differential of moisture absorption. It is called the exponential relaxation model (ER model) and has been adopted in several studies [36], [37], [38]. The model is expressed as:
where Mrel(t) is the moisture uptake due to relaxation at time t, k is the relaxation rate, and Mr0 is the saturate moisture uptake due to relaxation. Integration of Eq. (6) yields:
In addition, the moisture absorption due to diffusion and relaxation was assumed to be a linear superposition. As a result, the total moisture absorption of an FRP bar with a radius of R0 can be found by adding Eq. (7) into Eq. (5):
Loh et al. [27] proposed that moisture absorption due to matrix relaxation follows the Fickian model. They used two parallel Fickian models to determine the moisture absorption due to diffusion and relaxation, respectively [28,37,39]. The moisture absorption behavior for an FRP bar with a radius of R0 can be expressed as:
Mubashar et al. [28] found that the PDF model was not able to adequately represent some test results. They introduced a sequential dual Fickian model that involves a Heaviside step function (ϕ) to delay the relaxation:
Compared with the PDF model, the SDF model assumes that relaxation only starts to occur at time td.
2.2.4. Time-dependent surface concentration model
Long and Richman [29] proposed a time-dependent modification to the surface saturation state of FRP, called the time-dependent surface concentration model (TDSC model). They assumed that the saturation concentration at the surface of the FRP bar is initially C0, and increases over time as a result of relaxation. They recommended an expression that incorporates this time-dependent surface condition as follows:
where C1 is the saturated moisture concentration after relaxation and R is the radius of the FRP bar.
The Duhamel’s theorem, originally applied to the heat conduction problem [40], was used to solve this problem (Eqs. (1), (2), and (12)). The theorem assumes that the concentration at (x, y, z) (where x, y, z are spatial coordinates) at time t is C = F(x, y, z, t) when the initial concentration is zero and the surface concentration is unity; if the surface concentration is replaced by f(t), the solution becomes:
They simplified the moisture absorption model due to diffusion and relaxation, which is called the simplified time-dependent surface concentration model (STDSC). For an FRP bar with a radius of R0, this model can be expressed:
where Md(t) is the moisture uptake due to diffusion at time t.
2.2.6. Langmuir model
Carter and Kibler [41] proposed the Langmuir (LAN) model for non-Fickian diffusion behavior, which assumes that diffusing moisture exists in both free and bound states:
where C and S represent the mass of mobile and bound water per unit volume, respectively; α and β are the probabilities per unit of time that mobile and bound molecules will change their respective states. The solution of the LAN model for an FRP plate was provided in Ref. [42]. However, the explicit solution for cylinders (i.e., FRP bars) is not available. For long-term exposure, an approximate solution of the LAN model can be expressed as:
where M0 is the saturate water uptake. This equation is in the exponential form, which is not capable of capturing the various types of non-Fickian behavior observed in FRP composites.
2.3. Assessment of existing models
Fig. 1 shows typical moisture uptake curves for the models reviewed above. It is seen that all but the LAN model follows the Fickian model initially but departs from it thereafter to simulate the non-Fickian diffusion behavior.
Various moisture uptake curves in FRP bars have been observed in experimental studies, as depicted in Fig. 2. These curves may be categorized into three types:
Type I: Relaxation begins after Fickian diffusion saturates (Fig. 2(a)) [28,43];
Type II: Relaxation begins just before Fickian diffusion saturates (Fig. 2(b)) [39,44];
Type III: Relaxation begins immediately at initial exposure (Fig. 2(c)) [27,37,45,46].
In addition, moisture uptake may show either a superlinear (a1, b1, and c1 in Fig. 2) [47,48] or a sublinear increase (a2, b2, and c2 in Fig. 2) [36,42,46,49,50] at the early stage of relaxation, depending on the material and environmental conditions.
The characteristics of the models reviewed are summarized in Table 1. Most relaxation models include two critical parameters: the relaxation rate and the saturate relaxation moisture uptake. The STDSC model only considers the relaxation rate, leading to infinite moisture uptake due to relaxation. Both PDF and SDF models assume that relaxation follows the Fickian diffusion form, which may not accurately represent the underlying mechanisms. Also, they are only suitable for modeling relaxation moisture uptake with sublinear increases. Similarly, the ER model is also appropriate only for modeling relaxation moisture uptake with sublinear increases. The LAN model is also unable to capture the different types of non-Fickian behavior observed in FRP composites since it primarily relies on an exponential form.
These assessments highlight the need for a model capable of better capturing the complexities of moisture uptake in FRP composites.
3. Experiment
A diverse range of materials has been used in different studies, making it difficult to compare and combine the test data from different sources. To gain a better understanding of moisture behavior, gravimetric experiments were conducted on GFRP bars with different diameters, immersing in portable water at different temperatures.
3.1. Materials
GFRP bars with diameters of 6, 10, and 14 mm were used. They were made of E-glass fibers and vinyl ester resin and purchased from the same manufacturer. Their surface was helically wrapped as shown in Fig. 3. Their densities were determined following ASTM D792 [51] to be 2.16, 2.13, and 2.07 g·cm−3, respectively, for the 6, 10, and 14 mm bars.
As the glass fiber volume content significantly influences the moisture absorption of GFRP bars [52], it was determined using both the density method and the ignition loss method. The density method calculates the glass fiber volume content based on the density of the fiber and resin:
where Vf is the volume content of glass fiber; ρf, ρr, and ρc are the densities of glass fiber, vinyl ester resin, and GFRP, respectively.
The ignition loss method was conducted following ASTM D7913 [53]. GFRP bars were dried and then placed in crucibles, which were subsequently heated in a furnace until only ash remained. After cooling the residue in a desiccator, its weight was measured. The glass fiber content was calculated as follows:
where mf and mc are the weight of the glass fiber and the GFRP bar, respectively.
The accuracy of the density method can be affected by voids or defects in the GFRP bars, leading to lower values. On the other hand, incomplete carbonization of the matrix can result in inaccurate results from the ignition loss method, leading to higher values. The average of the values from the two methods may be a closer representation of the actual fiber volume.
The test results are shown in Table 2. It can be seen that smaller bars had higher fiber content. This may partially explain the size effect of FRP bars.
3.2. Gravimetric test
GFRP bars were cut into 30 mm long pieces and immersed in water at room temperature (about 23, 40, and 60 °C, respectively). The elevated temperatures were maintained using a thermostatic water chamber. For each diameter, six identical GFRP bars were used and placed in a container as shown in Fig. 4(a). To prevent longitudinal water diffusion in order to simplify the problem, both ends of each specimen were sealed with epoxy (Fig. 4(b)) and cured before they were immersed in water.
Gravimetric tests were conducted to characterize the mass change of GFRP bars according to ASTM D5229-14 [54]. Prior to exposure, the bars were dried in an oven at 50 °C for 48 h and then cooled down to room temperature in a desiccator. The initial weight (W0) of each bar was measured. Subsequently, the weight of the bars was periodically measured using an electronic balance with a 0.1 mg accuracy. The surfaces of the specimens were dried using tissue paper before each weighing. The percentage of mass change for each bar was calculated as follows:
where Wt is the weight at time t, mg. Six identical bars for each condition were used.
3.3. Results and discussion
Fig. 5 presents the mass change of GFRP bars over time. Each data point represents the average of six identical bars, and the error bars indicate the standard deviation.
All bars exhibited an increasing trend in mass with time. At 23 °C, the mass increased quickly during the initial exposure and then slowed down gradually. At 40 and 60 °C, it followed a similar trend to those at 23 °C during the initial stage but exhibited an additional mass increase at the later stage, leading to stepped-shaped curves. The additional mass increase was due to matrix relaxation. The curves of bars at 23 °C were smooth, indicating that the contribution of relaxation was marginal. For GFRP bars at elevated temperatures, the relaxation process was accelerated, and its contribution became evident at the latter period of the test.
GFRP bars with smaller diameters displayed a higher mass increase, attributed to their larger specific surface area (i.e., the surface area-to-volume ratio). For instance, after 4500 h of exposure at 23 °C, the mass increases were approximately 1.0%, 0.7%, and 0.6%, respectively for GFRP bars with diameters of 6, 10, and 14 mm. The elevated temperature resulted in a quicker and larger mass increase. This is because the elevated temperature accelerated the molecular motion, leading to a quicker diffusion of water molecules. For instance, after 500 h, the mass increases of the 6 mm GFRP bars at 23, 40, and 60 °C were approximately 0.49%, 0.65%, and 1.33%, respectively.
4. New model
The limitations of existing models necessitate the development of a new model for moisture uptake that considers the relaxation more appropriately and reflects the underlying mechanisms. As discussed, the relaxation of FRP composites due to moisture ingress involves several mechanisms. Absorbed water molecules act as plasticizers, increasing the free volume and chain mobility within the polymer, which reduces the glass transition temperature (Tg) and weakens the strength and stiffness of the matrix. Water may also disrupt existing hydrogen bonds and can form new hydrogen bonds with polymer chains, causing the matrix to swell. In addition, moisture exposure can alter the cross-link density of the polymer through hydrolysis, further enhancing the matrix relaxation. Initially, because of the intermolecular force and long-chain structure, the matrix exhibits a slow microstructure change. Therefore, the contribution of relaxation is marginal during the early stage when the intermolecular and hydrogen bonds remain intact but increases rapidly after a certain period when these bonds start to break down. An appropriate model accounting for relaxation should incorporate three parameters: relaxation rate, maximum relaxation amount, and transition time. It must also satisfy the initial condition that the moisture uptake is zero at the start of exposure.
Taking these factors into account, a new model, using the Weibull function, thus termed the Weibull relaxation (WR) model, is proposed as follows:
The proposed model (Eq. (29)) involves two Fickian diffusion parameters and three relaxation parameters. The least-squares method (LSM) is a common regression technique but is sensitive to the initial values of the parameters, often leading to local optimization results. To overcome this drawback, the particle swarm optimization (PSO) algorithm proposed by Kennedy and Eberhart [55] is adopted. The PSO algorithm is based on swarm cooperation, starting with a random solution and iteratively searching for the optimal solution by updating generations. The updates follow local and global best values in each iteration.
where Pi(n) and Vi(n) are the position and velocity of the ith particle at the nth iteration; Pibest(n) and Pgbest(n) are the best positions of the ith particle (local optimum) and among all particles (global optimum), respectively; r1 and r2 are random numbers generated within the range of [0,1]; B1 and B2 control the maximum step length and are both set to two; ω is the inertia weight that influences the previous velocity on the current one. To prevent particles from exceeding the search space, the velocity (Vi) and position (Pi) are constrained to [υmin, υmax] (minimum and maximum velocities) and [pmin, pmax] (minimum and maximum positions), respectively. This process is repeated until the stopping criteria are met. The PSO algorithm is illustrated in Fig. 6.
5.2. Model comparison
This section compares existing models and the proposed one with the test data from this study. The parameters in each model were determined from regression analysis. All results are shown in Fig. 7.
Fig. 7(a) shows that all test results for GFRP bars at 23 °C fit well with the Fickian model. When the exposure temperature increased to 40 and 60 °C, deviations from the Fickian model are evident in the late stage of exposure. These deviations suggest that GFRP bars at elevated temperatures exhibited relaxation processes, which can not be accurately described by the Fickian model.
Relaxation models (ER, TDSC, and WR models) were used to regress the results at 40 and 60 °C. The results are shown in Figs. 7(b)-(d). The STDSC, LAN, PDF, and SDF models were excluded from consideration due to their less rigorous assumptions. Both the ER and TDSC models are unable to capture the stepped shape of test results, as shown in Figs. 7(b) and (c), highlighting their limitations in describing the relaxation behavior. In contrast, the proposed WR model shows a good agreement with the test results across both the diffusion-dominated and the relaxation-dominated stages (Fig. 7(d)).
The fitting parameters of these models are listed in Table 3. The temperature-dependent variation of diffusivity is expected to follow the Arrhenius relationship [26,56,57], which is representative of the activated transition state theory.
where D0 is the pre-exponential factor, which is a constant that represents the diffusion coefficient at infinite temperature; T is the temperature, K; Ea is the activation energy required for diffusion and Rg is the universal gas constant (≈ 8.3 J·mol−1·K−1). Take the natural logarithm of both sides of Eq. (32) gives:
Therefore, lnD follows a linear relationship with 1/T, with the slope value of −Ea/Rg.
Fig. 8 shows the Arrhenius plots of diffusivities for GFRP bars using different models. Fig. 8(a) shows that Arrhenius plots from the Fickian model at various temperatures are linear with R2 values greater than 0.97, indicating that elevated temperatures accelerate the diffusion without changing the mechanism. The diffusion activation energy can be determined from the slopes of the regressing lines (slope × gas constant), which are 31.8, 31.5, and 30.7 kJ·mol−1 for 6, 10, and 14 mm GFRP bars respectively. The activation energies between the bars with different sizes are very close as they were the same material. Larger bars have slightly larger diffusivities and smaller activation energies due to their lower glass fiber volume contents (Table 2), as glass fibers do not absorb moisture and may also act as barriers [58]. Larger bars may also contain more defects, contributing to this phenomenon.
Figs. 8(b) and (c) show the diffusivities for the ER and TDSC models. These plots do not follow a linear trajectory, which disobeys the Arrhenius theory. This suggests that these two models do not capture accurately the moisture uptake mechanism.
Fig. 8(d) shows that the Arrhenius plots of the diffusivity D in the proposed WR model exhibit a linear behavior at different temperatures, with R2 values greater than 0.98. This confirms that elevated temperatures accelerate diffusion without changing the underlying mechanism. The diffusion activation energies obtained for the WR model from data regression for the 6, 10, and 14 mm GFRP bars are 34.0, 32.9, and 36.5 kJ·mol−1, respectively. These values are slightly different from those obtained solely for the Fickian model, as relaxation occurs during the diffusion-dominated stage, albeit with a minor contribution.
The parameter δ in the WR model is related to the relaxation rate. Table 3 shows that δ increases when the temperature increases from 40 to 60 °C, indicating accelerated relaxation at higher temperatures. The parameter γ is related to the transition time when the relaxation contribution to moisture uptake becomes significant. Its value decreases slightly when the temperature increases from 40 to 60 °C, suggesting that the transition occurs earlier at higher temperatures, for example, higher temperatures accelerate water molecule movement while increasing the polymer chain’s flexibility and leading to easier relaxation. These indicate that the proposed WR model not only demonstrates a better mathematical agreement with test data but also more accurately reflects the moisture uptake mechanisms in GFRP bars compared to existing models.
5.3. Further model validation
To further validate the WR model, test data from two previous studies [44,59] were used. Four types of GFRP bars were tested in Ref. [59]: two using polyester as the matrix with diameters of 10 and 19.5 mm (labeled as P10 and P20), and two using vinyl ester as the matrix with the same two diameters (labeled as V10 and V20). These bars were immersed in an alkaline solution at 25 °C. In the other study [44], 9.53 mm diameter hybrid FRP bars made of carbon/glass fibers and epoxy matrix were used. They were immersed in water at three temperatures: 40, 60, and 90 °C.
Fig. 9 presents the fitting results of the WR model with the test data from Ref. [59]. Fig. 9(a) shows that smaller bars had higher mass increases, consistent with the findings from the present study. All curves exhibit stepped stages attributed to relaxation. The WR model is in good agreement with the test results during both diffusion-dominated and relaxation-dominated stages with R2 values greater than 0.99. Fig. 9(b) shows the parameters obtained for the WR model. The larger diameter bar had a higher diffusivity, also consistent with the observation in the present study. The values of δ and γ were smaller for the bars with smaller diameters, also consistent with the present study. Compared with vinyl ester matrix bars, polyester matrix bars had a smaller diffusivity but similar values of δ and γ. This suggests that vinyl ester matrix bars displayed faster moisture diffusion but similar relaxation behavior.
Fig. 10 presents the fitting results of the WR model in comparison with the test data from Ref. [44]. Fig. 10(a) shows that the WR model showed a good agreement with the test results in both diffusion-dominated and relaxation-dominated stages with R2 values greater than 0.99. FRP bars had higher mass increases at higher temperatures, and their mass increase curves also exhibited stepped stages. Fig. 10(b) presents the parameters fitted for the WR model. The diffusivity of the bars at higher temperatures is larger. The value of δ increases when the temperature is increased from 40 to 60 °C but decreases when the temperature is further increased to 90 °C, probably because the Tg of the matrix was exceeded, which altered the degradation mechanism. The value of γ is also considerably smaller at 90 °C, indicating a much earlier relaxation process.
6. Conclusions
This study has investigated the moisture absorption behavior of GFRP bars through experimental tests and theoretical analyses. The results show that both diffusion and relaxation contribute to moisture uptake. Matrix relaxation causes non-Fickian diffusion, resulting in deviations between the Fickian model and test results. Existing models accounting for matrix relaxation have been reviewed, and a new model has been proposed to better reflect relaxation-induced moisture uptake. The following conclusions can be drawn.
(1) The moisture absorption of GFRP bars is governed by the Fickian model during the initial stage. Larger diameter GFRP bars have larger diffusivities due to their smaller glass fiber volume content and probably more defects. Elevated temperatures accelerate the Fickian diffusion, and this acceleration effect conforms to the Arrhenius relationship within the test range.
(2) For GFRP bars at 23 °C, the contribution of relaxation to moisture absorption is marginal within the test time span. GFRP bars at higher temperatures (i.e., 40 and 60 °C) exhibit clearly additional increases due to matrix relaxation. Elevated temperatures not only enhance molecular movement, leading to faster diffusion but also increase the polymer chain’s flexibility, resulting in earlier relaxation.
(3) The proposed WR model takes into account the underlying mechanisms of relaxation effectively and performs better than other models. It adopts three relaxation parameters with clear physical meanings and accurately characterizes the moisture absorption behavior of GFRP bars during both diffusion-dominated and relaxation-dominated stages.
(4) The WR model is based on the moisture uptake mechanism of FRP composites within the test ranges. Its applicability beyond these ranges, such as very high temperatures, is not necessarily correct as materials may have different degradation mechanisms. Therefore, further research is needed to establish a suitable range of applicability.
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