Application of Fractional Calculus in Predicting the Temperature-Dependent Creep Behavior of Concrete

Abstract

Creep is an essential aspect of the durability and longevity of concrete structures. Based on fractional-order viscoelastic theory, this study investigated a creep model for predicting the temperature-dependent creep behavior of concrete. The order of the proposed fractional-order creep model can intuitively reflect the evolution of the material characteristics between solids and fluids, which provides a quantitative way to directly reveal the influence of loading conditions on the temperature-dependent mechanical properties of concrete during creep. The effectiveness of the model was verified using the experimental data of lightweight expansive shale concrete under various temperature and stress conditions, and the comparison of the results with those of the model in the literature showed that the proposed model has good accuracy while maintaining simplicity. Further analysis of the fractional order showed that temperature, not stress level, is the key factor affecting the creep process of concrete. At the same temperature, the fractional order is almost a fixed value and increases with the increase in temperature, reflecting the gradual softening of the mechanical properties of concrete at higher temperature. Finally, a novel prediction formula containing the average fractional-order value at each temperature was established, and the creep deformation of concrete can be predicted only by changing the applied stress, which provides a simple and practical method for predicting the temperature-dependent creep behavior of concrete.

1. Introduction

Concrete is widely used in the fields of construction engineering and road materials, and its durability and sustainable stability are essential for the safety of infrastructure [1]. However, creep deformation occurs in concrete with time under long-term loading, which results in pavement damage and foundational settlement [2,3]. Therefore, experimental research as well as theoretical modeling of the creep behavior of concrete have long been research concerns [4,5].

To date, a great deal of research has been devoted to establishing theoretical models to accurately predict the creep deformation of various concretes. Li et al. [6] developed a constitutive model for polypropylene concrete based on high-temperature dynamic experimental data to improve the predictability of mechanical behavior under varying thermal conditions. By including the transient creep explicitly in a uniaxial constitutive model, Gernay and Franssen [7] described the complex, unsteady temperature–stress relationship of concrete materials. Furthermore, Li et al. [8] proposed a thermodynamically coupled constitutive model for concrete derived from the Drucker–Prager strength criterion to predict creep behavior at various temperatures. Despite these advances, the existing concrete creep models are often too complex to be applied under different stresses and temperatures [9]. Therefore, developing a simple and effective model for predicting the creep behavior of concrete under various environmental conditions remains a key task, which is crucial for predicting the creep deformation of concrete in practical engineering applications.

In recent decades, more and more researchers have recognized fractal and fractional differential operators as alternative approaches for characterizing the properties of viscoelastic materials [10]. Among them, the fractal models are particularly suitable for predicting the effective thermal conductivity of porous materials because they have multiscale pores that are statistically self-similar [11]. In the field of concrete, Qu et al. [12] established a thermal resistance network model based on the Sierpinski carpet in parallel and series to model the effective thermal conductivity of novel autoclaved aerated concrete (AAC)-based composites. Jiang et al. [13] constructed a hybrid fractal model by mixing an n-stage Sierpinski carpet to reproduce the multiscale pore structure, which can be used for heat transfer as well as ion and gas transport simulations in porous cement-based materials. Fractal derivative constitutive models have also been applied to investigate the creep behavior of concrete [14]. In a recent study, Bouras and Vrcelj [15] developed linear creep models for reinforced and prestressed concrete by adopting fractal-derivative-based viscoelastic laws.

The key feature of fractional differential operators is the inherent long-term memory effect, which makes them particularly suitable for modeling phenomena with significant time dependence, damage progression, and microstructural changes [16,17]. The utility of fractional calculus in modeling viscoelastic materials is particularly notable due to its ability to reveal the evolution of the mechanical properties of materials between ideal solids and fluids [18]. For example, Bonfanti et al. [19] applied fractional viscoelastic models to power-law materials, effectively capturing the viscoelastic response of soft materials across a broad range of time scales and revealing the connection between the complex internal structure of soft materials and their macroscopic mechanical responses. Katicha et al. [20] developed a fractional viscoelastic model by simplifying the parameter settings of the generalized fractional Maxwell model, which can accurately predict the linear viscoelastic properties of asphalt concrete. Zatar et al. [21] applied fractional calculus to study the viscoelasticity of piles and soils, accurately identifying these properties in structural dynamics. The application of fractional-order calculus in these works significantly improved the practicability and adaptability of the models and more effectively reflected the complex viscoelastic behavior of materials with the change of fractional order.

As a typical viscoelastic behavior, creep is one of the domains where fractional calculus has been extensively applied [22,23]. For instance, Sapora et al. [24] proposed a method based on fractional calculus for simulating the creep recovery of modified asphalt binders, accurately describing their viscoelastic behaviors at different temperatures. Tang et al. [25] proposed a new geotechnical creep model based on variable-order fractional derivatives and continuum damage mechanics, which can simulate the creep behavior of rocks under long-term loading with greater accuracy, especially during the accelerated creep stage. Zhou et al. [26] applied fractional calculus to enhance the elasto-viscoplastic model to better describe and predict the creep behavior of soft soil and clay. Therefore, the present studies have proven that the use of fractional calculus can provide a rapid and reliable approach for describing creep behavior.

Fractional-order models have also been successfully applied to describe the creep behavior of concrete. Bouras et al. [27] developed a nonlinear thermoviscoelastic model using fractional derivatives to effectively simulate the creep behavior of concrete under different stresses and temperatures with very few parameters. Zhang et al. [28] developed a constitutive model for the nonlinear creep damage of concrete based on fractional calculus, improving the prediction of creep and fracture behavior at different stress levels. In addition, Ribeiro et al. [29] used fractional calculus to more accurately simulate the creep behavior of concrete and polymers with fewer rheological elements than traditional models, effectively characterizing the long-term creep in complex materials.

Although significant progress has been made in capturing the creep behavior of concrete with fractional calculus methods, existing models rarely consider the law of order variation, especially for the prediction of concrete creep, by establishing a quantitative relationship between order and environmental parameters. Therefore, the motivation of this study was not only to use fractional calculus to characterize the creep behavior of concrete but also to further analyze the relationship between the fractional order and the factors affecting the creep of concrete, including temperature and applied stress. Consequently, the innovation of this study is establishing a novel prediction formula considering the influence of environmental parameters on the value of the fractional order, providing a simple and practical method for predicting the temperature-dependent creep behavior of concrete.

In the following, Section 2 is dedicated to the derivation of a fractional creep model based on fractional viscoelastic theory. The model is capable of characterizing creep behavior under various conditions with simple parameters. In Section 3, the proposed model is fitted to the experimental data of lightweight expansive shale concrete to validate its effectiveness and obtain the model parameters. Section 4 provides a quantitative analysis of the relationship between order and different applied stress and temperature loading conditions to effectively predict the temperature-dependent creep behavior of concrete. Finally, the conclusions are drawn in Section 5.

2. Fractional Creep Model

In the conventional mechanical theories, the linear elastic behavior of ideal solids is typically characterized by Hooke’s law, $\sigma \left(t\right)=E\epsilon \left(t\right)=E{D}_t^0\epsilon \left(t\right)$, which illustrates that stress σ and strain ε are proportional, with the modulus of elasticity E serving as the constant of proportionality. The viscous behavior of ideal fluids is governed by Newton’s law, $\sigma \left(t\right)={\displaystyle \frac{\eta d\epsilon \left(t\right)}{dt}}=\eta D_t^1\epsilon \left(t\right)$, where stress is a linear function of the strain rate, with the viscosity coefficient η being the proportionality constant. This behavior is illustrated in the fractional “spring-pot” element depicted in Figure 1.

Therefore, by substituting the derivative order with the fractional order α, the stress–strain equation can express the intermediate mechanical behavior of viscoelastic materials between ideal solids and ideal fluids [30]:

$$\sigma \left(t\right)=E{\theta }^{\alpha }{D}_t^{\alpha }\epsilon \left(t\right)0<\alpha <1,$$

(1)

where E denotes the elastic modulus at α = 0, and θ is defined as η/E, representing the relaxation time parameter. On this basis, considering the creep behavior of concrete under constant stress, Equation (1) can be rewritten as follows:

$$\epsilon \left(t\right)={\displaystyle \frac{1}{E}}{\theta }^{-\alpha }{D}_t^{-\alpha }\sigma \left(t\right)0<\alpha <1.$$

(2)

Throughout the development of fractional calculus, numerous definitions have been proposed. For instance, the left-sided Riemann–Liouville fractional derivative of a function f(t) for order 0 < α < 1 is defined as follows [31]:

$$D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\alpha \right)}}{D}_t^1{\int }_0^t{\displaystyle \frac{f\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau ,$$

(3)

where Γ denotes the Gamma function:

$$\mathsf{\Gamma }\left(t\right)={\int }_0^t{e}^{-t}{\tau }^{t-1}d\tau .$$

(4)

The Gamma function has the following properties:

$$\mathsf{\Gamma }\left(x+1\right)=x\mathsf{\Gamma }\left(x\right).$$

(5)

The definition of the fractional integral operator [32] is expressed as

$$D_t^{-\alpha }f(t)={\int }_0^t{\displaystyle \frac{{(t-\tau )}^{\alpha -1}}{\mathsf{\Gamma }\left[\alpha \right]}}f(\tau )d\tau .$$

(6)

By substituting the fractional integral operator from Equation (6) and constant stress $\sigma \left(t\right)={\sigma }_0$ into Equation (2), the expression of the creep model is obtained:

$$\epsilon \left(t\right)={\displaystyle \frac{1}{E}}{\theta }^{-\alpha }{\int }_0^t{\displaystyle \frac{(t-\tau {)}^{\alpha -1}}{\mathsf{\Gamma }\left[\alpha \right]}}{\sigma }_{0}d\tau .$$

(7)

The integration of the right side of Equation (7), along with the use of the property of the Gamma function of Equation (5), allows for a further formulation to be deduced:

$$\epsilon \left(t\right)={\displaystyle \frac{{\sigma }_0}{E{\theta }^{\alpha }\mathsf{\Gamma }\left[\alpha \right]}}{\displaystyle \frac{-(t-\tau {)}^{\alpha }}{\alpha \left(t\right)}}{|}_0^t={\displaystyle \frac{{\sigma }_0}{E{\theta }^{\alpha }}}{\displaystyle \frac{t^{\alpha }}{\mathsf{\Gamma }\left[1+\alpha \right]}}.$$

(8)

Hence, it is evident that the fractional creep model represented by Equation (8) involves only three parameters: E, θ, and α, offering the advantage of easy data fitting. Theoretically, E and θ are material constants, while the fractional-order derivative α dictates the variation in the creep model’s response. Therefore, by simply adjusting the value of α, one can satisfy the creep curves of concrete under different stresses and temperatures.

In summary, the fractional creep model shows the ability to accurately characterize the creep behavior of concrete. In the following sections, the effectiveness of the fractional creep model is validated by fitting to the experimental data, and the law of the fractional order with experimental conditions is discussed in detail.

3. Results and Discussion

To evaluate the effectiveness of the proposed fractional creep model when applied to concrete, experimental data from the literature were used. Bouras et al. [27] conducted a series of creep loading tests on lightweight expansive shale concrete. In terms of temperature conditions, the experimental data under temperatures of 22 °C, 204 °C, and 316 °C were chosen for data fitting. The levels of applied stress ($\sigma /f_c^{\prime }$) contained 0.3, 0.45, and 0.6, where $f_c^{\prime }$ denotes the compressive strength, which was 24.0 MPa. Therefore, the selected applied stresses ${\sigma }_0$ corresponded to 7.2 MPa, 10.8 MPa, and 14.4 MPa, respectively.

In order to accurately describe the experimental data, the least squares fitting method was used to determine the parameter values of the fractional-order creep model. The calibration process was performed using MATLAB (MathWorks) to minimize the root mean square error (RMSE) and to maximize the coefficient of determination (R-squared), thus providing a quantitative measure of the goodness of fit. The RMSE and R-squared formulas used are as follows [33]:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n(y_i-{\widehat{y}}_i{)}^2}$$

(9)

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n(y_i-\widehat{y_i}{)}^2}{{\sum }_{i=1}^n(y_i-\overline{y}{)}^2}}$$

(10)

where n represents the number of observations, $y_i$ is the observed value, ${\widehat{y}}_i$ is the predicted value, and $\overline{y}$ is the mean of the observed values. The RMSE measures the average magnitude of prediction errors, providing an absolute measure of fit. A lower RMSE value indicates a better fit. The R-squared indicates the proportion of variance explained by the model, with a value closer to one indicating a better fit.

The RMSE is often more relevant for precise engineering predictions due to its direct reflection of prediction errors. Meanwhile, the R-squared is vital for understanding the explanatory power of a model [34]. MATLAB can produce a negative R-squared if the model fit is worse than a horizontal mean line, indicating poor model performance.

Throughout the process, the material constants E and θ were kept constant at the same temperature, while the values of the parameters were variably adjusted within a certain range to achieve an optimal fit for varying stress levels.

Based on the above methods, the model parameters for the expansive shale light concrete test data were obtained, as presented in

Table 1. Model parameters and fitting results for lightweight expansive shale concrete.

T (°C)		22			204			316
${\mathit{\sigma }}_0$ (MPa)	7.2	10.8	14.4	7.2	10.8	14.4	7.2	10.8	14.4
E (MPa)	20,841	20,345	18,570	11,720	11,458	11,298	4811	4390	4652
$\theta$ (s)	3.34 × 106	1.70 × 106	1.73 × 106	1.09 × 105	1.04 × 105	8.64 × 104	4.14 × 105	3.68 × 105	3.91 × 105
$\alpha$	0.3078	0.2993	0.2977	0.3683	0.3630	0.3629	0.4028	0.4059	0.3948

. It can be observed that the values of the material constants E and θ remain within a certain range at each temperature, regardless of the change in the applied stress. This shows that the model has the robustness to accurately represent material behavior under the influence of environmental factors without deviating from these effective parameter ranges. The model fitting results for lightweight expansive shale concrete at 22 °C, 204 °C, and 316 °C are demonstrated in Figure 2, Figure 3 and Figure 4. These figures show that the creep characteristics of concrete are significantly affected by the loading conditions and that the creep strain increases significantly at higher temperatures or greater stress levels. It is worth noting that under each loading condition, the fitting curve of the fractional order creep model has a strong correlation with the experimental data.

For the experimental data at temperatures of 22 °C, 204 °C, and 316 °C, comparisons of the model proposed in this paper with the model proposed by Bouras et al. [27] are depicted in Figure 5, Figure 6 and Figure 7, which prove that the proposed fractional creep model can more accurately match the experimental data. In the cases of low temperatures and low loading stresses, for example, when the temperature is 22 °C and the applied stresses are 7.2 MPa and 10.8 MPa, there is a large deviation between the results obtained by the model in the literature and the experimental data, but the proposed fractional order creep model accurately describes the experimental results.

Table 2. Comparison of fitting parameters between the fractional creep model and the nonlinear thermo-viscoelastic rheological model.

T (°C)			22			204			316
${\mathit{\sigma }}_0$ (MPa)		7.2	10.8	14.4	7.2	10.8	14.4	7.2	10.8	14.4
Bouras’s model	RMSE	3.20 × 10−5	1.81 × 10−5	2.54 × 10−5	3.29 × 10−5	2.68 × 10−5	3.57 × 10−5	3.96 × 10−5	5.01 × 10−5	2.10 × 10−5
	R2	−0.8034	0.8222	0.9395	0.9132	0.9787	0.9831	0.9235	0.9636	0.9951
Proposed Model	RMSE	5.80 × 10−6	5.19 × 10−6	1.44 × 10−5	2.03 × 10−5	2.14 × 10−5	2.31 × 10−5	2.56 × 10−5	3.63 × 10−5	4.25 × 10−5
	R2	0.9477	0.9889	0.9593	0.9724	0.9890	0.9927	0.9734	0.9841	0.9835

presents the RMSE and R-squared comparisons between the proposed fractional creep model and the previous model at these three temperatures, further validating that the model proposed in this paper has higher accuracy. Moreover, it can be observed that within the strain–time curves, the order of the fractional derivative lies between 0 and 1, covering the range from pure elasticity to Newtonian viscosity. In addition, it should be emphasized that during all fitting processes, the fractional order was strictly between 0 and 1 across various stress levels and temperatures, which conforms to the range from pure elasticity to Newtonian viscosity in the fractional viscoelastic spectrum.

In summary, according to the data fitting and model comparison procedure, it was proven that the proposed fractional model can effectively describe the creep behavior of concrete, and the fractional order can directly reflect the evolution of the mechanical properties of concrete. In the next section, the relationship between the value of fractional order α and the loading conditions is discussed in detail to finally achieve the prediction of concrete creep.

4. Prediction of Concrete Creep

The variation in the fractional order between 0 and 1 can reflect the hardness or softness of a material; therefore, the numerical value of the fractional order can intuitively reveal the influences of external environmental factors on the mechanical properties of a material.

During the data fitting process for lightweight expansive shale concrete, it is worth noting that the material constants E and θ changed little at a given temperature. The relationship between the fractional order α and applied stress levels is graphically depicted in Figure 8. It can be clearly seen that the numerical values of the fractional order α are basically consistent at the same temperature, regardless of the changes in the applied stress levels. Therefore, it is illustrated that the mechanical properties of lightweight expansive concrete and the law of the corresponding fractional order are principally governed by temperature.

The variation in the fractional order α with the experimental temperature is further plotted in Figure 9. It is shown that under the constant applied stress condition, the fractional order α demonstrates an upward trend with increasing temperature. This trend is consistent with the effect of temperature on creep behavior; that is, an increase in temperature tends to accelerate the creep phenomenon, thus making the material gradually softer. Therefore, the numerical variation in fractional-order values intuitively reflects the dominant impact of temperature on the creep behavior of lightweight expansive concrete. More importantly, Figure 8 and Figure 9 clearly show that the order curves are almost the same at all three temperatures, which provides a method of achieving the prediction of concrete creep.

At the same temperature, the average values of the material’s parameters and the fractional order under different stress levels are as presented in

Table 3. Average values of material parameters E, θ, and α at the same temperature.

T (°C)	22	204	316
E (MPa)	20,712	11,492	4618
$\theta$ (s)	1.76 × 106	9.98 × 104	3.91 × 105
$\alpha$	0.3034	0.3647	0.4012

. The model predictions employing the averaged parameters for the creep behavior of lightweight expansive shale concrete are depicted in Figure 10, Figure 11 and Figure 12. The results illustrate that the fractional-order creep model can effectively describe the creep behavior of concrete under different applied stress levels through the average parameter values at each temperature. This means that the temperature-dependent creep deformation of concrete can be directly predicted only by changing the applied stress values in the fitting formula. This is of great significance for the design and application of concrete in the fields of construction and road engineering.

In conclusion, the analysis of the rule of the fractional order shows that the value of the fractional order is mainly affected by temperature, and the creep deformation of concrete can be successfully predicted using the average fractional order value at different temperatures. In addition, the numerical value of the fractional order intuitively reflects the evolution of the mechanical properties of concrete with the variation in temperature.

5. Conclusions

In this work, a fractional creep model was derived from fractional viscoelastic theory to describe the temperature-dependent creep behavior of concrete, in which the fractional order has the ability to reveal the transition in mechanical properties from hard to soft. The effectiveness of the model was verified via data fitting and comparison with a lightweight expansive shale concrete model in the literature. The conclusions are summarized as follows:

(i)

The fractional creep model can accurately describe the experimental data and has good adaptability as well as robustness under different temperatures and applied stresses.

(ii)

The value of the fractional order tends to increase with the increase in temperature. This intuitively reflects that high temperatures accelerate the creep process of concrete, making its mechanical properties gradually softer.

(iii)

The prediction formula containing the average fractional-order value at each temperature was established, and the creep deformation of concrete was predicted only by changing the applied stress, which provides an innovative and practical method for predicting the temperature-dependent creep behavior of concrete.