Abstract
To investigate the evolution law of energy and the damage constitutive model of concrete with different rubber replacement ratios under uniaxial creep, this paper systematically studied the creep energy evolution law of rubber concrete using uniaxial creep-unloading tests. The study focused on the transformation and dissipation mechanisms of energy in rubber concrete during creep, alongside the establishment of damage creep constitutive models for concrete with varying rubber replacement ratios. The findings reveal that: (1) The elastic strain energy density (ue) of concrete with varying rubber replacement ratios demonstrates a linear decline over extended creep time, characterized by linear attenuation. (2) Based on the results of uniaxial creep-unloading tests, this study proposes a method for calculating creep energy in concrete based on the linear attenuation characteristics of ue. (3) The dissipated strain energy density (ud) and input strain energy density (u) of concrete with different rubber replacement ratios increase over time, and this growth can be divided into the decay growth stage, uniform growth stage, and accelerated growth stage. (4) During the creep process, as the creep time increases, the proportion of ud to u in concrete with different rubber replacement ratios gradually increases, reaching its maximum at the critical point of accelerated creep and post-peak, and the higher the rubber replacement ratio of the concrete, the smaller the proportion of ud to u at the critical point of accelerated creep and post-peak. (5) This research develops a viscoelastic-plastic-damage coupled constitutive model founded on energy dissipation, adept at delineating the deformation curves of concrete of varying rubber replacement ratios during creep. The research results not only have positive significance for improving the recycling of waste tires, but also provide new ideas for the establishment of the creep constitutive model of concrete.
Keywords: Rubber concrete, Creep, Strain energy evolution, Constitutive model
Subject terms: Civil engineering, Structural materials
Introduction
Waste tire rubber is recognized as a rapidly growing category of solid waste globally. It is estimated that more than 1 billion waste tires are generated worldwide each year, with a total weight exceeding approximately 20 million tons1. Rubber concrete is regarded as a novel composite material in which waste rubber particles are incorporated as aggregates or additives into concrete. This method not only achieves the resource utilization of solid waste but also effectively alleviates the over-exploitation pressure of natural aggregates, offering significant environmental benefits2. However, the long-term stability of concrete structures is considered a core issue in engineering safety assessments. Traditional research has predominantly focused on the macroscopic characterization of material mechanical properties, with an in-depth understanding of the creep damage mechanisms of rubber concrete under sustained loads being largely overlooked. Thermodynamics posits that the failure of solid materials is driven by energy, leading to state instability. Therefore, revealing the creep evolution laws of rubber concrete from the perspective of energy dissipation and constructing its damage constitutive model based on energy dissipation are regarded as having significant theoretical value and engineering importance.
In the research on rubber concrete, initial explorations primarily focused on its material properties, particularly its compressive strength3,4, shear performance5, fracture performance6,7, impact toughness8,9, and freeze-thaw characteristics10,11. As a widely used material in civil engineering, the long-term mechanical properties of concrete are directly related to the safety and durability of structures12. Among these, the creep effect (i.e., the slow deformation of materials over time under sustained load) is a critical factor that cannot be ignored in the design of concrete structures13,14. Therefore, the influence of the creep effect must be fully considered in the safety evaluation of concrete structures. Numerous experimental studies have shown that the creep deformation of concrete is significantly influenced by creep stress15–18. When the creep stress is low, the creep of concrete exhibits characteristics of decay creep and uniform creep. Meanwhile, various linear creep models have been proposed to predict the creep deformation of concrete, such as the CEB-FIP model19, ACI209 model20, and FIB 2010 Model21. However, with the deepening of research, under higher creep stress, the creep deformation of concrete not only exhibits significant decay creep and uniform creep but also a distinct nonlinear increase in the accelerated creep stage, which cannot be well characterized by traditional creep constitutive models. To address the nonlinear evolution characteristics of deformation in the accelerated creep stage, some researchers have constructed concrete creep constitutive models by introducing damage variables to modify the components of traditional creep models22,23.
In studies that introduce damage variables to modify model components, the methods for constructing damage variables are diverse. For instance, Bu et al.24 analyzed the long-term deformation characteristics of concrete and constructed the damage variable of concrete using statistical theory, deriving the concrete creep constitutive model based on the constructed damage variable. Alexander Dummer et al.25, based on continuum theory, used exponential softening laws to characterize concrete creep damage, further constructing a constitutive model that can represent material creep deformation, performance, and damage evolution. Zheng et al.26, considering water infiltration into concrete and high creep stress conditions, constructed a concrete creep damage variable by modifying the hard rock damage variable, and this model can well describe the creep curve of water-containing concrete. Liu et al.27 constructed a concrete creep damage variable using empirical methods, and this model can describe the concrete creep curve under different confining pressures and stresses.
Among the existing methods for constructing concrete creep damage variables, although they can relatively accurately describe the creep curve of concrete, their core mechanisms fail to fully capture the essence of creep failure in solid materials—namely, energy-driven state instability. In contrast, energy-based models can more profoundly reveal the energy evolution characteristics of concrete during the creep process. By constructing damage variables based on these characteristics and subsequently deriving the concrete creep constitutive model, this approach not only helps to uncover the fundamental features of concrete failure but also more precisely characterizes the creep behavior of concrete. By integrating energy evolution with damage variables, energy-based models can more comprehensively reflect the mechanical response of concrete under long-term loads, thereby providing a more reliable theoretical foundation for the long-term performance evaluation of concrete structures. Compared to traditional empirical methods, energy-based models exhibit significant advantages in theoretical depth and predictive accuracy, offering a more scientific framework for the study of concrete creep.
To construct a creep damage constitutive model based on energy evolution, this study designed a uniaxial creep-unloading experiment to analyze the energy evolution laws of rubber concrete during the creep process. Based on the energy dissipation evolution laws of rubber concrete, a viscoelastic-plastic-damage coupled constitutive model for rubber concrete was constructed, and the correctness of the model was verified using experimental data. The research findings can provide a theoretical basis for predicting the long-term mechanical properties of rubber concrete.
Uniaxial creep test of rubber concrete and its creep curve
Preparation of the sample
The concrete mix ratio was determined according to JG55-2011 “Specification for Mix Proportion Design of Ordinary Concrete”. The water, cement (ordinary Portland cement PO32.5), and coarse aggregate (crushed stone, particle size range of 4–16 mm) used in the concrete were all from the same origin; the fine aggregate used was river sand, with a fineness modulus of 2.5, an apparent density of 2640 kg/m3, and a bulk density of 1897 kg/m3; the rubber particles had a particle size range of 2–5 mm, an apparent density of 1260 kg/m3, and a bulk density of 840 kg/m3. The equal volume replacement method for fine aggregate was used to calculate the rubber content at replacement rates Q of 0%, 5%, 10%, and 15%, with the specific mix proportions for each specimen shown in Table 1; Fig. 1.
Table 1.
Proportions of concrete samples with different rubber substitution rates.
| Q/% | Cement | Water | Fine aggregate | Rubber particle | Coarse aggregate |
|---|---|---|---|---|---|
| 0% | 279 | 178 | 800 | 0 | 1012 |
| 5% | 279 | 178 | 760 | 17.71 | 1012 |
| 10% | 279 | 178 | 720 | 35.42 | 1012 |
| 15% | 279 | 178 | 680 | 53.13 | 1012 |
Fig. 1.
Preparation of the sample.
Sample size and test scheme
The testing machine used in this paper is the ZST-1500 microcomputer-controlled electro-hydraulic servo coal rock dynamic and static combined adaptive coupling test system, as shown in Fig. 2. Considering the characteristics of the testing machine, the rubber concrete specimens developed in this paper were processed into samples with a diameter of 50 mm and a height of 100 mm, ensuring that both ends of the samples are flat and the sides are smooth.
Fig. 2.
ZST-1500 microcomputer-controlled electro-hydraulic servo coal-rock dynamic and static combined adaptive coupling test system.
This paper mainly conducts uniaxial creep tests and uniaxial creep-unloading tests. For ease of analysis, the creep stress was uniformly set to 90% of the uniaxial compressive strength. The stresses for the uniaxial creep tests are shown in Table 2. The creep curves of rubber concrete are shown in Fig. 3.
Table 2.
Creep stress in uniaxial creep test.
| Number of uniaxial compression specimens | Q% | Peak strength | Number of uniaxial creep test | Q% | Creep stress |
|---|---|---|---|---|---|
| UC−0 | 0 | 32.12 | UCC−0 | 0 | 28.91 |
| UC−5 | 5% | 29.16 | UCC−5 | 5% | 26.24 |
| UC−10 | 10% | 27.77 | UCC−10 | 10% | 24.99 |
| UC−15 | 15% | 22.95 | UCC−15 | 15% | 20.66 |
Fig. 3.
Creep curves of concrete with different rubber substitution rates.
Based on the characteristics of the creep strain-time curve, the creep test of concrete specimens can be divided into three stages: decelerated creep, steady creep, and accelerated creep. By analyzing the creep curves of different rubber replacement rates in Fig. 3, the differences in creep behavior among different concretes can be further understood. It is shown by research that the rubber replacement rate is significantly correlated with the rate in the accelerated creep stage. Specifically, a faster increase in creep rate as it approaches failure is exhibited by concrete with a higher rubber replacement rate. This is because a faster rate of microcrack expansion and interconnection is possessed by concrete with a lower rubber replacement rate, making it more likely to reach the failure threshold under sustained load. In contrast, a slower increase in creep rate during the accelerated creep stage is shown by concrete with a higher rubber replacement rate, indicating better long-term stability.
Calculation method for uniaxial creep energy
Calculation principle of creep energy
During the loading process, the work done by the testing machine on the concrete is the u, with part of the energy stored inside the concrete and the other part dissipated during the loading process. As shown in Fig. 4. The energy stored inside the concrete is ue, and the dissipated energy is ud. The calculation formula is:
 |
1 |
Fig. 4.
Schematic diagram of strain energy density.
The area under the unloading curve represents the magnitude of the ue, while the area enclosed by the loading curve, unloading curve, and the horizontal axis represents the ud. The specific calculation formula is as follows:
 |
2 |
 |
3 |
In the formula: σ, ε represents stress and strain; 
and εʹ respectively represent loading of stress and strain; σʺ and εʺ respectively represent unloading of stress and strain.
Rubber concrete uniaxial creep-unloading test scheme
According to the above definition, the u of rubber concrete creep can be obtained by integrating the stress-strain curve of the uniaxial creep test. However, how to determine the ue and ud of creep is the key issue in creep energy analysis. In the method of calculating concrete energy, the ue during concrete creep requires integrating the concrete unloading curve. However, since creep tests do not have an unloading curve, it is necessary to design a new experimental scheme to analyze the evolution law of energy in rubber concrete creep tests, namely the uniaxial creep-unloading test. This study designed four groups of creep-unloading tests. The four groups of tests were unloaded after the completion of instantaneous deformation (uniaxial single loading-unloading test), unloaded during the decelerated creep stage, unloaded within the steady creep stage, and unloaded within the time range of the accelerated creep stage. To ensure the accuracy and repeatability of the tests, each specimen was not reused after completing the creep-unloading test to avoid the influence of previous tests on the material properties. The creep-unloading tests are as follows. The specific test parameters are shown in Table 3.
Table 3.
Test parameters of uniaxial creep-unloading test.
| Number of creep specimens | Creep unloading test specimen number | Q/% | Creep stress | Unloading time/h |
|---|---|---|---|---|
| UCC-0 | UCC-0-1 | 0 | 28.91 | 0.0000 |
| UCC-0-2 | 0 | 0.6324 | ||
| UCC-0-3 | 0 | 10.1123 | ||
| UCC-0-4 | 0 | 19.0123 | ||
| UCC-5 | UCC-5-1 | 5 | 26.24 | 0.0000 |
| UCC-5-2 | 5 | 0.7944 | ||
| UCC-5-3 | 5 | 15.7401 | ||
| UCC-5-4 | 5 | 24.8444 | ||
| UCC-10 | UCC-10-1 | 10 | 24.99 | 0.0000 |
| UCC-10-2 | 10 | 0.9166 | ||
| UCC-10-3 | 10 | 15.1667 | ||
| UCC-10-4 | 10 | 29.7166 | ||
| UCC-15 | UCC-15-1 | 15 | 20.66 | 0.0000 |
| UCC-15-2 | 15 | 0.9778 | ||
| UCC-15-3 | 15 | 22.5778 | ||
| UCC-15-4 | 15 | 31.6978 |
The linear Attenuation characteristic of elastic strain energy density in concrete creep
Table 4 shows the energy at different stages of the uniaxial creep test for different rubber replacement rates. Analysis of the ue in the creep test reveals that as the creep time continues to increase, the ue stored inside the concrete gradually decreases. According to the law of energy conservation, the gradual decrease in ue indicates that the ue stored inside the concrete is gradually converted into ud over time during the creep process.
Table 4.
Evolution of strain energy density under different stages of uniaxial creep experiment of rock.
| Number of creep specimens | Completion of instantaneous deformation | Decelerated creep stage | Steady creep stage | Accelerated creep stage | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Time (h) | u | ue | Time (h) | u | ue | Time (h) | u | ue | Time (h) | u | ue | |
| UCC-0 | 0 | 0.1264 | 0.0681 | 0.6324 | 0.1935 | 0.0278 | 10.1123 | 0.2437 | 0.0256 | 19.0123 | 0.3317 | 0.0243 |
| UCC-5 | 0 | 0.1455 | 0.0843 | 0.7944 | 0.2124 | 0.0435 | 15.7401 | 0.2748 | 0.0389 | 24.8444 | 0.3504 | 0.0325 |
| UCC-10 | 0 | 0.1517 | 0.1021 | 0.9166 | 0.2203 | 0.0509 | 15.1667 | 0.3025 | 0.0446 | 29.7166 | 0.4092 | 0.0403 |
| UCC-15 | 0 | 0.1616 | 0.1152 | 0.9778 | 0.2385 | 0.0897 | 22.5778 | 0.3187 | 0.0787 | 31.6978 | 0.3880 | 0.0729 |
In summary, it can be seen that during the creep process of concrete, there is not only energy input but also energy transformation. To further explore the evolution law of the transformation of ue into ud during the creep process, the ue over time was fitted. Table 5 shows the fitting results of uniaxial creep elastic energy density versus time.
Table 5.
Fitting results of uniaxial creep elastic energy density reduction and time.
| Number of creep specimens | a | b | Fitting formula | Adj. R-square |
|---|---|---|---|---|
| CUU-0 | − 0.00124 | 0.068042 | ue = − 0.00124t + 0.068042 | 0.99904 |
| CUU-5 | − 0.00104 | 0.08435 | ue = − 0.00104t + 0.08435 | 0.99909 |
| CUU-10 | − 0.00091 | 0.10191 | ue = − 0.00091t + 0.10191 | 0.99837 |
| CUU-15 | − 0.00086 | 0.11509 | ue = − 0.00086t + 0.11509 | 0.99983 |
Based on the fitting results, it can be known that the ue of concrete storage gradually decreases with the increase of creep time, and the ue has a linear relationship with the creep time, indicating the existence of a linear attenuation law.
Analysis of evolution of creep energy in concrete
By making use of the uniaxial creep-unloading test and the linear attenuation characteristic of ue, the energy within any time period of the uniaxial creep test can be quantified.
From Fig. 5, it can be observed that as the creep time increases, the ue stored in the concrete gradually decreases. For ease of analysis, the proportion of ue converted into ud during the creep process is referred to as the attenuation ratio. The calculation formula for the attenuation ratio is as follows:
 |
4 |
Fig. 5.
Evolution law of elastic strain energy density during creep.
In the formula, φ represents the attenuation ratio, ∆ue indicating the conversion value of elastic strain during creep.
The maximum conversion ratios for concrete specimens with rubber replacement rates of 0%, 5%, 10%, and 15% are 38.07%, 33.48%, 27.36%, and 25%, respectively. The higher the rubber replacement rate, the smaller the ultimate conversion ratio of ue. Based on the linear attenuation characteristics of ue in concrete creep, the evolution law of energy density in concrete creep tests can be obtained. Figure 6 shows the energy evolution of uniaxial creep tests for concrete specimens with rubber replacement rates of 0%, 5%, 10%, and 15%, as well as the evolution curves of the proportion of ue and ud to the u.
Fig. 6.
Evolution law of energy in uniaxial creep test.
From Fig. 6, it can be seen that the ud and u of concrete specimens with rubber replacement rates of 0%, 5%, 10%, and 15% exhibit similar evolutionary characteristics, i.e., they increase with increasing creep time, while the ue gradually decreases. The growth rates of ud and u decrease with time during the decelerated creep stage, exhibit approximately linear growth during the steady creep stage, and increase with time during the accelerated creep stage. The ratio of ud to u for concrete specimens with rubber replacement rates of 0%, 5%, 10%, and 15% increases with time, showing the same evolutionary trend as the ud, i.e., there are decelerated growth, steady growth, and accelerated growth stages. The higher the rubber replacement rate in concrete specimens with rubber replacement rates of 0%, 5%, 10%, and 15%, the smaller the ratio of ud to u, with ratios approximately ranging from 0.46 to 0.89, 0.42–0.86, 0.32–0.82, and 0.28–0.80, indicating that rubber can effectively reduce energy dissipation during the creep process. To conclude, concrete creep entails the ongoing dissipation of strain energy density and the progressive conversion of ue into ud.
A viscoelastic-plastic-damage coupled constitutive model based on strain energy dissipation
Establishment of the constitutive model
When the time required to load concrete to the creep stress level is relatively small compared to the creep time of concrete, the elastic strain can be considered as instantaneous strain and represented by a linear elastic body. When concrete is in the decelerated creep, the Kelvin model is used to describe the creep curve of concrete. The combination of Hooke body and Kelvin body in series constitutes the generalized Kelvin body. Figure 7 shows the generalized Kelvin model.
Fig. 7.
Generalized Kelvin body.
Thus, the following equation of generalized Kelvin model can be obtained:
 |
5 |
After differentiating both sides of the equation, we can obtain the following result:
 |
6 |
It can be obtained through Laplace transformation that:
 |
7 |
In the formula: ε1 represents the instantaneous strain of concrete, ε2 represents the strain generated during the attenuation creep stage of concrete; σ represents the creep stress of concrete; E1 represents the elastic modulus of concrete; E2 and η1 represent the elastic modulus and viscosity coefficient of the viscoelastic body; 
is the first derivative of creep stress with respect to time.
Figure 8 shows the schematic diagram of viscous body.
Fig. 8.
Viscous body.
In the process of identifying rheological model parameters, the distinction between the decelerated creep stage and the steady creep stage represented by each component in the combined model is frequently neglected, resulting in certain differences in the constitutive model parameters and unclear physical significance. Additionally, the Burgers model is illustrated in Fig. 9, where the generalized Kelvin model and the viscous body are connected in series to describe the strain-time curve of concrete during the decelerated and steady creep stages. This approach ensures consistency with the Burgers model and highlights the linear characteristics exhibited by the strain of concrete during the steady creep stage, which are effectively captured by the viscous body.
Fig. 9.
Burgers model.
Based on the Buegers model, the following Eqs. (8) and (9) can be obtained:
 |
8 |
 |
9 |
In the formula, η2 represents the viscous body; 
are the first derivatives of ε1, ε2, ε3 with respect to time;
are the second derivatives of ε2, ε3 with respect to time.
By substituting and transforming the equation, the constitutive equation of the model can be obtained:
 |
10 |
In the formula, ε represents the creep deformation of concrete; 
is the second derivative of creep stress with respect to time; 
and 
are the first and second derivatives of ε with respect to time, respectively.
In the study of the time-dependent characteristics of concrete damage degradation, it is difficult to adequately describe and characterize accelerated creep using elastic bodies, viscous elements, and plastic elements. Therefore, many scholars have introduced damage mechanics to describe its nonlinear deformation using damage variables. Xie Heping28, starting from thermodynamic principles, first proposed a damage variable that considers the energy dissipation and large deformation of cracked concrete. According to the definition of damage, it can be described as:
 |
11 |
In the formula: Di represents the three-dimensional damage variable, where i = 1, 2, 3, respectively indicating the maximum principal stress, the intermediate principal stress and the minimum principal stress; Yi represents the damage strain energy dissipation; Y0i is the initial strain energy dissipation; B, n are parameters related to strain energy dissipation.
However, in the application of this damage variable, because the ud at each stage of the creep process is difficult to determine, this damage variable exists only with theoretical rationality. Yang [29] simplified this damage variable and constructed a damage constitutive model for rock salt. The simplified damage variable was used by Xu et al. [30] to establish a creep damage constitutive model for green schist, and the accelerated creep stage of green schist was fitted, achieving good fitting results. In the above studies, the damage variable was simplified to one dimension, which, although convenient for application and with good fitting results, did not reflect the advantages of the damage constitutive model based on strain energy dissipation, merely describing the accelerated creep stage curve. It is suggested by thermodynamics that concrete failure instability is a state instability phenomenon driven by strain energy. If the evolution law of strain energy in concrete deformation and failure can be analyzed in detail, and a creep damage constitutive relationship with strain energy change as the damage factor can be established, it is expected that the essential characteristics of concrete failure will be better reflected and engineering practice will be better served. The above discussion on the rheological element models for each stage of creep, combined with the uniaxial creep-unloading tests in this study, resulted in the evolution law of ud during the concrete creep process being obtained. Based on the creep energy evolution law, a viscoelastic-plastic-damage coupled constitutive model was established. The viscoelastic-plastic-damage coupled constitutive model is shown in Fig. 10.
Fig. 10.
Viscoelastic-plastic damage coupling constitutive model.
It is commonly accepted that damage does not arise during the decelerated and steady creep phases of the creep process. Damage initiates as concrete transitions into the accelerated creep phase, meaning the damage dissipation strain energy in concrete occurs during the accelerated creep phase. In accordance with the effective stress principle, the effective stress is:
 |
12 |
Then, according to the superposition principle, the creep equation of concrete is:
 |
13 |
In the formula: Yi is the total strain energy dissipation; Y0i represents the initial strain energy dissipation, it represents the energy dissipation that occurs before the rock enters the accelerated creep stage. The acceleration stage is determined when the second derivative of strain with respect to time is greater than 0, indicating that the rock has entered the accelerated creep state.
Parameter identification of creep constitutive model
To verify the accuracy and practicality of the model proposed in this paper, we conducted detailed creep tests on concrete and identified the model parameters based on the experimental data. The parameter identification in this paper employs the rheological curve decomposition method. Through parameter identification, the parameters for the uniaxial creep tests of the four types of concrete are shown in Table 6, and the comparison between the theoretical curves and the experimental curves is shown in Fig. 11. During the attenuated creep stage, the authors used fractional-order elements to describe this stage. Through the fitting, it was found that the Adj. R-Square values for all four rocks were above 0.997. In the uniform creep stage, the strain-time curve of the rocks showed a linear evolution trend, and the Adj. R-Square values for all four rocks were above 0.999. Based on the fact that the Adj. R-Square values for both stages were above 0.99, it can be concluded that the fitting effect was good.
Table 6.
Parameter identification of constitutive model.
| Specimen number | Parameter | Fitting result | ||||||
|---|---|---|---|---|---|---|---|---|
| E1 (GPa) | E2 (GPa) | η1 (GPa h) | η2 (GPa h) | η3 (GPa h) | n | B | Adj. R-square | |
| CUU-0 | 2.70 | 1.42 | 3.14 | 98.33 | 17.43 | 1.01 | 13.91 | 0.98972 |
| CUU-5 | 2.23 | 1.73 | 3.76 | 201.84 | 25.44 | 1.17 | 10.32 | 0.99127 |
| CUU-10 | 1.73 | 1.91 | 4.79 | 233.99 | 113.49 | 2.13 | 8.69 | 0.99887 |
| CUU-15 | 1.38 | 1.35 | 3.12 | 155.34 | 145.36 | 2.49 | 9.18 | 0.99258 |
Fig. 11.
Theoretical curve and experimental curve.
Comparing the theoretical curves with the experimental curves, it can be seen from Fig. 11 that the viscoelastic-plastic-damage coupled concrete damage constitutive model based on strain energy dissipation constructed in this paper can well describe and characterize the various stages of concrete with different rubber replacement rates during the creep process.
The constitutive model we constructed offers significant advantages over the Burgers model in describing the creep behavior of rocks. The model incorporates a damage variable based on energy dissipation, which effectively describes the accelerated creep stage of rocks. This damage variable accounts for the energy loss during creep, providing a more accurate representation of the nonlinear strain acceleration observed in the tertiary creep phase. In contrast, the Burgers model, while effective for describing primary and secondary creep, lacks the capability to adequately represent the accelerated creep stage due to its linear viscoelastic framework. Therefore, the proposed model provides a more comprehensive and accurate description of the full creep process.
Analysis of concrete damage evolution
Based on the parameter range of concrete with four rubber replacement rates, by setting parameter n to 0.5, 0.8, 1, 2, and 4, and parameter B to 1, 5, 10, 15, and 20.
The results from Fig. 12 show that as parameters n and B increase, the damage generated in concrete within the same time period becomes more significant, indicating that the damage sensitivity and failure rate of concrete increase with the increase of these two parameters. This phenomenon reveals a close correlation between the damage growth rate and material parameters, meaning that with larger values of parameters n and B, concrete can reach the critical damage state more quickly.
Fig. 12.
Influence of parameter changes on rock damage evolution.
Conclusion
This paper systematically studied the creep energy evolution law of rubber concrete using uniaxial creep-unloading tests. A constitutive model of concrete creep based on energy dissipation was further constructed. The main research conclusions are as follows:
The ue of rubber concrete samples with different replacement rates shows a linear decay trend with the increase of creep time, that is, there is a linear decay characteristic. The creep process of concrete is the process of ue gradually transforming into ud, and the higher the rubber replacement rate, the smaller the proportional limit of ue transformation.
Based on uniaxial creep-unloading tests and the linear decay characteristics of ue, a new method for calculating creep strain energy of rubber concrete is proposed, providing a new approach for studying the evolution law of energy during the creep process.
During the creep process of four different rubber replacement rate concretes, the ud and the u increase with time, and their growth can be divided into the decay stage, the steady growth stage, and the accelerated growth stage.
The higher the rubber replacement rate of concrete samples with rubber replacement rates of 0%, 5%, 10%, and 15%, the smaller the ratio of ud to u, with ratios approximately between 0.46 and 0.89, 0.42–0.86, 0.32–0.82, and 0.28–0.80, respectively, indicating that rubber can effectively reduce energy dissipation during the creep process. As creep time increases, the proportion of ud to u in concrete gradually increases and reaches its maximum at the end of accelerated creep.
A viscoelastic-plastic-damage coupled constitutive model based on strain energy dissipation was constructed. This model effectively characterizes the mechanical behavior of concrete with different rubber replacement rates during the creep process. Sensitivity analysis of the damage variable parameters found that when parameters n and B are higher, concrete is more likely to reach the critical damage state.
Acknowledgements
This research was funded by the Key Project of the Scientific Research Program for Higher Education Institutions in Anhui Province(2024AH051854), Tongling University High-level Talent Research Start-up Fund Project (2024tlxyrc040), National Planning Office of Philosophy and Social(22BSH044). General Project of the School Level of Tongling University.
Abbreviations
- ue
Elastic strain energy density
- ud
Dissipated strain energy density
- u
Input strain energy density
Author contributions
Author Contributions: Liu Xiaodie: Conceptualization, methodology, data curation, writing-original draft. Cui Lei: Supervision. Liu Zhixi: Supervision.
Funding
This research was funded by the Key Project of the Scientific Research Program for Higher Education Institutions in Anhui Province(2024AH051854), Tongling University High-level Talent Research Start-up Fund Project (2024tlxyrc040).
Data availability
Data sets generated and/or analyzed during the current study period may have an impact on the subsequent development of the study due to the disclosure of preliminary data but may be obtained from the corresponding author upon reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Author contributions
Author Contributions: Liu Xiaodie: Conceptualization, methodology, data curation, writing-original draft. Cui Lei: Supervision. Liu Zhixi: Supervision.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
Data sets generated and/or analyzed during the current study period may have an impact on the subsequent development of the study due to the disclosure of preliminary data but may be obtained from the corresponding author upon reasonable request.