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Article

Fractional Order Analysis of Creep Characteristics of Sandstone with Multiscale Damage

College of Energy and Mining Engineering, Xi’an University of Science and Technology, Xi’an 710054, China
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Author to whom correspondence should be addressed.
Mathematics 2025, 13(16), 2551; https://doi.org/10.3390/math13162551
Submission received: 30 June 2025 / Revised: 31 July 2025 / Accepted: 7 August 2025 / Published: 9 August 2025

Abstract

Deep mining is often accompanied by complex geological conditions, which can cause damage to the coal seam roof surrounding rock, thereby reducing its safety and stability. Therefore, analyzing the long-term mechanical behavior of multiscale damaged sandstone under deep mining conditions is of great significance. To describe the long-term deformation and damage evolution of multiscale damaged sandstone under deep mining conditions, this work establishes a fractional-order multiscale damage creep model by incorporating fractional calculus and damage mechanics theory into the Nishihara model. The model parameters were determined by fitting the creep data of damaged sandstone using the least squares method. The results demonstrate that the proposed model can accurately simulate the complete creep process, including the decelerated, steady-state, and accelerated stages. Compared with the classical integer-order multiscale damage creep model, the fractional-order model can better capture the time-dependent behavior of materials and thus shows superior performance in characterizing the nonlinear features of the accelerated creep stage. Furthermore, through sensitivity analysis of the parameters reveals the influence of key parameters on different creep stages, thereby validating the model’s effectiveness and reliability. This model provides a solid theoretical foundation for evaluating the long-term stability of coal mine roof strata in deep mining environments.
MSC:
26A33; 44A10; 74S40; 74D10

1. Introduction

The coal seam roof, as a critical structure bearing the overburden pressure during coal mining, plays a vital role in ensuring the safety and efficiency of mining operations. As deep mining increasingly becomes the dominant method of coal extraction, greater attention has been paid to the safety and stability of the coal seam roof [1]. Compared to shallow mining, deep mining is subjected to more complex geological and stress conditions, such as high in situ stress, elevated geothermal gradients, and high pore pressure [2,3,4]. These factors collectively increase the possibility of roof collapse, fracturing, and other related accidents. Moreover, the uneven distribution of in situ stress fields and the disturbances caused by mining activities can lead to the initiation of micro damage or even significant damage within the roof strata [5], thereby reducing its load-bearing capacity. The deformation and failure process under such conditions becomes more complex and less predictable [6]. Therefore, investigating the stability of coal seam roofs under complex geological environments in deep mining and understanding their long-term deformation and failure characteristics are critical to enhancing mine disaster prevention and control capabilities, as well as ensuring the safe and sustainable extraction of coal resources.
Sandstone is a common and critical rock in coal seam roof, and its mechanical behavior directly influences the structural stability of the roof. Typically exhibiting moderate strength and pronounced brittleness, sandstone can provide initial load-bearing capacity, but it is prone to creep deformation under prolonged high-stress conditions, potentially leading to accelerated failure [7]. In deep mining environments with complex stress fields, the development of micro damage and degradation of mechanical properties in sandstone become more pronounced [8], further compromising its long-term stability [9]. Therefore, it is essential to establish creep models to describe and predict the deformation and failure processes of sandstone under sustained loading, providing theoretical support for roof control and disaster prevention in deep mining operations.
As a crucial bridge between experimental observations and engineering design, creep models are often closely integrated with practical engineering applications. In deep mining, creep analysis helps predict the long-term deformation of roadway surrounding rock, enabling the optimization of support design and parameters to ensure stability [10,11]. The surrounding rock in empty areas also undergoes creep under sustained stress, affecting overall safety. Creep models can be used to predict deformation and failure in these zones, guiding their management and utilization [12,13]. Additionally, for mining-induced surface subsidence, creep models help estimate its extent and severity, supporting the protection of surface structures and infrastructure [14]. However, with the increasing complexity of engineering environments, these models often fall short in capturing the full range of time-dependent responses. Therefore, in recent years, more advanced mechanical theories such as fractional-order models have been introduced into the long-term stability analysis of surrounding rocks to improve prediction accuracy and engineering reliability.
The traditional creep models such as the Nishihara model, composed of spring, viscoelastic body and viscoplastic body, can only capture basic characteristics and struggle to represent the nonlinear deformation features observed throughout all stages of rock creep. To overcome these limitations, researchers have introduced fractional-order into the Nishihara model, thereby enhancing their ability to simulate complex creep responses more comprehensively. For instance, Zhou et al. [15] improved the classical Nishihara model by replacing its Newtonian dashpot with an Abel dashpot based on a fractional-order derivative, and derived its analytical solution. This enhancement enabled a more accurate description of the time-dependent creep behavior of rock using the fractional derivative approach. Compared with traditional integer-order models, fractional-order models can more accurately capture the viscoelastic properties [16] and memory effects of rock [17], offer a better representation of the time-dependent behavior during the nonlinear creep process. Fractional-order models are capable of simulating the entire creep evolution from initial loading to long-term deformation, make them particularly suitable for analyzing the long-term deformation and damage accumulation of rock masses under complex stress conditions [18].
In further creep modeling research, scholars have integrated fractional-order theory with damage mechanics to construct fractional-order damage creep models [19,20,21]. The introduction of a damage variable allows the model to dynamically reflect the degradation of rock micro structure and strength during the creep process [22,23], helping to more comprehensively reveal the creep failure mechanisms of fractured rock [24,25]. Zhou et al. [26] based on the Weibull statistical distribution principle, coupled fractional-order derivatives with the strength distribution of rock mass micro elements, established a statistical damage fractional-order model that more realistically simulates the heterogeneity and suddenness of damage accumulation during the creep process. Furthermore, researchers have developed a unified creep damage coupled model that integrates fractional calculus, nonlinear damage mechanics, and statistical heterogeneity. This model enables a more systematic and accurate characterization of the time-dependent deformation and progressive damage evolution of rocks under long-term loading conditions [27].
However, to better reflect actual mining conditions, it is necessary to simulate the multiscale damage in rock induced by factors such as joint development [28], initial defects [29], stress variations [30], and temperature effects [31,32]. This work introduces fractional derivative operators and time-varying damage factors into the classic Nishihara structure, so that it can reflect both the viscoelastic response of the rock and the whole process of damage evolution. The core of the model is to construct a set of fractional constitutive equations, in which the relationship between stress and strain is jointly regulated by the fractional-order, viscoelastic modulus and damage factor. Moreover, the effectiveness and applicability of the model are confirmed by the fitting analysis of damage creep data.
The rest of the paper is organized as follows. Section 2 introduces the theories of fractional calculus and multiscale damage mechanics and provides a detailed derivation of the proposed fractional-order multiscale damage creep model. Section 3 presents the parameters identification and models validation, followed by a sensitivity analysis of key parameters. Section 4 summarizes the main conclusions of this study.

2. Mathematical Modeling

2.1. Definition of Fractional Derivative

The Riemann–Liouville and Grünwald–Letnikov formulations are commonly used fractional derivative operators [33]. However, in the context of constitutive modeling of creep behavior, these formulations require the fractional derivative of the function at the initial time [34], which complicates both the numerical implementation and the assignment of physically meaningful initial conditions [35]. In contrast, the Caputo fractional derivative computes the integer-order derivative of the function first and then performs fractional integration. This allows for the direct use of measurable physical quantities, such as initial stress or strain, which are typically defined in terms of integer-order derivatives [36]. Therefore, the Caputo definition is not only more consistent with the physical characteristics of creep processes but also simplifies the incorporation of initial and boundary conditions. In this work, the Caputo fractional derivative is invoked, which is defined as
d γ f t d t γ = 1 Γ 1 γ 0 t f τ t τ γ d τ ,
where 0 < γ ≤ 1 is a fractional-order, f′(τ) is the first order derivative of function f(τ), Γ(·) is the Gamma function defined as Γ w = 0 e t t w 1 d t .

2.2. Multiscale Damage Mechanics

During the exploitation of underground projects, the redistribution of principal stress within the rock mass promotes the expansion and coalescence of initial existing fractures [37], pores, and other initial defects, thereby triggering the onset of macro damage in the rock mass [38]. Meanwhile, the inherent micro damage within the rock continues to evolve over time, leading to the accumulation of micro damage [39,40]. This process is further intensified by the development of macro damage, eventually resulting in a multiscale damage structure spanning from the micro to the macro level [41]. Previous studies have shown that the coupling effect between damage at different scales significantly alters the mechanical response and creep evolution behavior of the rock [42,43]. Therefore, introducing multiscale damage theory is not only rational but also essential for accurately analyzing rock mechanical behavior. Traditional single-scale damage models often fail to capture the full progression from micro structural defects to macroscopic mechanical degradation, leading to inadequate predictions of rock behavior under long-term and multi-physical conditions.
In this background, multiscale damage theory, by simultaneously accounting for micro damage and macro damage, enables a more comprehensive and realistic representation of the damage evolution process in rock masses. This theoretical perspective substantially enhances our understanding of long-term rock stability and improves the predictive capacity for failure-related phenomena [44]. Building on this foundation, several researchers have developed new constitutive models that incorporate multiscale damage mechanisms to investigate the long-term mechanical behavior of damaged rocks [45,46,47], and conducted simulation experiments to study the damage to rocks under various conditions such as pore pressure [48], temperature [49,50], and mining disturbances [51]. The results indicated that models incorporating damage variables can more effectively describe the complete creep stages.
In essence, multiscale damage refers to the coupled process whereby micro-defects evolve under stress and progressively deteriorate the macro-scale mechanical properties of the rock [52]. Micro damage reflects internal structural deterioration, while macro damage represents overall performance degradation, they characterize the complete damage behavior of the rock together. As such, developing multiscale damage models that bridge micro structural mechanisms with macroscopic responses is both theoretically grounded and practically necessary.
Hou et al. [53] proposed a new formula to describe the macro damage using strain and elastic modulus, as follows
D T = 1 ε ε ε ε u E E ,
where DT denotes the macro damage variable, ε, ε′, and εu represent the total strain, residual strain after unloading, and crack compaction strain, and E′ and E refer to the unloading modulus and initial elastic modulus, respectively.
For the micro damage (DC), an exponent-negative function is available for defining the degree of damage evolution induced by creep constant load [54]
D C = 1 e α t ,
where the velocity of damage variable change with time t can be controlled by the decay parameter α. The larger the parameter α, the faster the multiscale damage evolution of fractured rock under external loading.
Based on the Lemaitre equivalent strain principle, the multiscale damage of fractured rock can be written as [55]
D = 1 1 D T 1 D C 1 D T D C .

2.3. Fractional Multiscale Damage Creep Model

In Figure 1, Eve and ηve are the elastic modulus and viscosity coefficient of the viscoelastic elements, ηvp is the viscosity coefficient of the viscoplastic elements, σ represents the constant stress on the model, σs is the long-term strength of sandstone, α and β are different forms of the decay parameter in Equation (3), used to distinguish between different elements.
The creep model is composed of a Hooke element, a damaged viscoelastic body and a damaged viscoplastic body, with strains εe, εve and εvp respectively. According to the principle of component combination, the overall strain of the model can be expressed as
ε = ε e + ε v e + ε v p .
The constitutive equation of the Hooke element
ε e = σ E e ,
where Ee represents the elastic modulus.
In the proposed model, the nonlinear increase in creep strain is governed by the damaged viscoelastic and viscoplastic elements, whose elastic modulus and viscosity coefficients are no longer treated as constants throughout the creep process, but are instead considered as variables that evolve with the deterioration of rock properties
E v e D = 1 D E v e η v e D = 1 D η v e η v p D = 1 D η v p ,
where Eve(D), ηve(D) and ηvp(D) are the elastic modulus and viscosity coefficients under multiscale damage conditions.

2.3.1. Multiscale Damage Viscoelastic Body

The damaged viscoelastic body is composed of a damaged Newtonian dashpot and a damaged spring (Figure 2). Its constitutive relationship is given as follows
ε v e = ε H = ε A σ = E v e D ε v e + η v e D d γ ε v e d t γ ,
where εH and εA represent the strains of the damaged Hooke element and the damaged Abel element. Eve(D) and ηve(D) are the damaged elastic modulus and the damaged viscosity coefficient, respectively.
Substituting Equations (4) and (7) into Equation (8), the constitutive equation describing the damaged viscoelastic body can be obtained
d γ ε v e d t γ + E v e η v e ε v e = σ η v e e α t + σ η v e D T 1 D T .
Applying the Laplace transform to both sides of Equation (9), the following results can be obtained
s γ E s + E v e η v e E s = σ η v e 1 s α + σ η v e D T 1 D T 1 s ,
where E(s) = L[εve] is Laplace transform. By solving Equation (10), the following relation can be obtained
E s = σ η v e 1 s α 1 s γ + E v e η v e + σ η v e D T 1 D T 1 s s γ + E v e η v e .
Applying the convolution theorem L−1[F(s)G(s)] = (fg)(t) and the inverse Laplace transform, ‘⁎’ denotes the convolution operation, based on the initial conditions εve = 0 at time t = 0, the following expression can be obtained
ε v e t = σ η v e 0 t e α t τ τ γ 1 E γ , γ E v e η v e τ γ d τ + σ η v e D T 1 D T 0 t τ γ 1 E γ , γ E v e η v e τ γ d τ .
After simplification, Equation (12) becomes
ε v e t = σ η v e e α t t γ 1 E γ , γ E v e η v e t γ + σ η v e D T 1 D T t γ E γ , γ + 1 E v e η v e t γ .

2.3.2. Multiscale Damage Viscoplastic Body

In Figure 3, the constitutive equation of the damaged viscoplastic body is as follows
ε v p = ε d = ε p σ = σ d + σ p ,
where εd and εp represent the strains of the damaged Abel element and plastic element, the stresses on the damaged Abel dashpot and the plastic element are represented by σd and σp respectively,
σ p = σ , σ < σ s σ s , σ σ s ,
where σs is the long-term strength of sandstone.
Based on the expression of the multiscale damage in Equation (4), and the expression of the damaged viscosity coefficient ηvp(D) in Equation (7), the strain formula of the damaged viscoplastic body can be derived
When σ < σs,
ε v p t = 0 ,
When σ ≥ σs, according to Equation (14) can be derived
σ = η v p D d γ ε v p d t γ + σ s .
Substituting Equations (4) and (7) into Equation (17) and simplify to obtain
d γ ε v p d t γ = σ σ s η v p e β t + D T 1 D T .
Performing Laplace transform on both sides simultaneously can be calculated
s γ E s = σ σ s η v p 1 s β + σ σ s η v p D T 1 D T 1 s ,
E s = σ σ s η v p 1 s β 1 s γ + σ σ s η v p D T 1 D T 1 s γ + 1 .
Applying the inverse Laplace transform can be derived
ε v p = σ σ s η v p t γ E 1 , γ + 1 β t + σ σ s η v p D T 1 D T t γ Γ γ + 1 .
Based on the above derivations, the total creep strain of the fractional derivative creep model considering damage is
ε t = σ E e + σ η v e e α t t γ 1 E γ , γ E v e η v e t γ + σ η v e D T 1 D T t γ E γ , γ + 1 E v e η v e t γ σ < σ s σ E e + σ η v e e α t t γ 1 E γ , γ E v e η v e t γ + σ η v e D T 1 D T t γ E γ , γ + 1 E v e η v e t γ + σ σ s η v p t γ E 1 , γ + 1 β t + σ σ s η v p D T 1 D T t γ Γ γ + 1 σ σ s .
In the case of γ = 1, Equation (22) reduced to an integer-order damage creep model
ε t = σ E e + σ E e + α η v e e α t σ E e + α η v e + σ E v e D T 1 D T e E v e η v e t + σ E v e D T 1 D T σ < σ s σ E e + σ E e + α η v e e α t σ E e + α η v e + σ E v e D T 1 D T e E v e η v e t + σ E v e D T 1 D T + σ σ s η v p e β t 1 + σ σ s η v p D T 1 D T t σ σ s .

3. Results and Discussion

3.1. Model Validation

To verify the effectiveness of the proposed fractional-order multiscale damage creep model, this work conducts a validation analysis. Hou et al. [56] previously employed the Burgers model to fit the creep data of sandstone containing macro damage, aiming to predict its long-term strength and failure time. To further assess the applicability and accuracy of the proposed model in describing creep behavior, the same experimental dataset was analyzed in this study using the least squares method. Three sets of creep data with different macro damage values (DT = 0.227, 0.383, and 0.505) were selected as the research objects. Considering the need for comparative analysis with the conventional integer-order model, a special case where the fractional order γ = 1 was also investigated. In this case, the proposed fractional order model degenerates into the corresponding integer-order form, which can be expressed as Equation (23). Table 1 presents the key parameters obtained from the fitting results of the fractional order model and the integer-order model. Particular attention is given to the differences in the fractional order, elastic modulus, viscosity coefficients, and the decay parameters between the two models to facilitate a comparative analysis. To quantitatively evaluate the fitting performance of the models, the correlation coefficient was employed as the primary evaluation metric.
In Figure 4a, when the macro damage reaches 0.227, the internal structure of the sandstone has already deteriorated. During the decelerated creep stage, the fractional-order model demonstrates superior fitting performance due to the flexibility of fractional calculus. As time progresses, the continuous accumulation of micro damage within the rock leads to a significant nonlinear increase in strain during the steady-state and accelerated creep stages. Although the integer-order model can reflect the overall trend of strain development in these stages, its fitting performance is clearly insufficient and fails to accurately capture the nonlinear response induced by damage evolution. In contrast, the fractional-order model more effectively represents the damage accumulation effect in these stages, exhibiting higher fitting performance and indicating its stronger applicability in characterizing complex creep processes. As shown in Figure 4b, when the macro damage reaches 0.383, the physical properties of the rock undergo significant changes. During the decelerated and steady-state creep stages, the fitting performance of the integer-order model begins to decline noticeably, while the fractional-order model, benefiting from the memory effect of the Abel dashpot, still maintains good fitting performance. In the accelerated creep stage, the higher level of macro damage leads to a more pronounced nonlinear increase in strain, and the fractional-order model exhibits a more evident strain acceleration trend compared to the integer-order model. As illustrated in Figure 4c, when the macro damage is large, the physical properties of the rock undergo significant changes. The integer-order model still maintains a certain level of fitting performance in the decelerated creep stage, but in the steady-state and accelerated creep stages, it fails to accurately describe the nonlinear growth behavior, and the fitting performance decreases significantly. In contrast, the fractional-order model continues to exhibit good trend representation and fitting performance in all stages, indicating that it is more applicable under complex stress conditions.
Overall, the fractional order multiscale damage creep model achieves correlation coefficients exceeding 0.99 with the experimental data, demonstrating excellent fitting performance. Compared with the integer-order multiscale damage model, the fractional model more accurately captures the nonlinear growth characteristics during the steady-state and accelerated creep stages. Although the model includes the fractional order parameter γ, this study did not perform a parametric sweep or range analysis of γ. Instead, γ was treated as the optimal fitting parameter for each dataset and independently determined using the least squares method. For different damage levels, the optimal γ values were identified as 0.79, 0.82, and 0.57, corresponding to the highest correlation coefficients for each respective dataset. These findings further demonstrate the flexibility and adaptability of the fractional model in characterizing complex time-dependent deformation behaviors of rocks under varying damage conditions.

3.2. Parameter Sensitivity Analysis

The fractional multiscale damage creep model proposed in this paper effectively reflects the overall stage of sandstone creep experimental data. Based on this, this section conducts a sensitivity analysis of the important parameters in the model and discusses the influence of the parameters on the creep process.
As shown in Figure 5, the creep strain curve varies with the fractional-order γ, indicating that γ significantly influences both the steady-state and accelerated creep stages. In the steady-state creep stage, a higher γ results in greater creep strain, while in the accelerated creep stage, the strain exhibits pronounced nonlinear characteristics. Therefore, due to the flexibility provided by the fractional-order parameter, the fractional model demonstrates clear advantages in capturing complex creep behavior.
Figure 6 illustrates the effect of the decay parameter α in the damaged viscoelastic element on the strain curve. The creep strain is highly sensitive to changes in α, particularly during the steady-state and accelerated creep stages. The decay parameter modulates the transition from steady-state to accelerated creep. As α increases, the nonlinear growth of creep strain during this transition becomes less pronounced.
Changing the elastic modulus Eve and viscosity coefficient ηve of the viscoelastic body affects the evolution of the creep strain curve, as shown in Figure 7a,b. Larger values of Eve and ηve slow down the growth rate of steady-state creep strain because higher elastic modulus and viscosity coefficient make creep more difficult to occur.
When the stress exceeds the long-term strength of the rock, the viscoplastic body begins to produce strain. In this stage, altering the viscosity coefficient ηvp of the viscoplastic body significantly influences the creep strain, especially during the accelerated creep phase. A higher ηvp reduces the rate of damage accumulation, thereby slowing down the nonlinear increase in creep strain, as shown in Figure 8.
Damage in the viscoplastic body is primarily evident during the accelerated creep stage. As illustrated in Figure 9, changing the decay parameter β of the viscoplastic body affects the evolution characteristics of this stage. The larger β leads to an earlier onset of accelerated creep, a faster rate of damage accumulation, and more pronounced strain growth during this phase.

4. Conclusions

To investigate the long-term mechanical behavior of coal seam roof surrounding rock under complex geological conditions, this work proposes and verifies a new fractional-order multiscale damage creep model based on fractional-order theory and damage mechanics and analyzes the sensitivity of key parameters. The main research conclusions are as follows:
(1)
By introducing fractional calculus and multiscale damage theory, the Nishihara model is improved by replacing the Newtonian dashpot with an Abel dashpot, thereby establishing a fractional-order multiscale damage creep model. Compared with the traditional integer-order multiscale damage model, the proposed model demonstrates better fitting performance, especially in capturing the accelerated creep stage. This advantage is mainly attributed to the higher fitting degrees of freedom of the fractional-order model and its realistic characterization of time dependence. The results confirmed the effectiveness and broad applicability of the proposed model in describing the multiscale damage creep behavior of rock masses, providing theoretical support for its practical engineering applications.
(2)
The creep data were fitted using the least squares method to determine the model parameters, which were then compared with the integer-order damage creep model. The results showed that the established fractional-order damage creep model can more accurately reflect the complete creep process. The fractional-order has more advantages than the integer-order in terms of fitting degrees of freedom, and the fractional-order damage viscoplastic elements can better describe the accelerated creep stage.
(3)
Through a sensitivity analysis of the model parameters, the specific effects of each parameter on the creep behavior of the rock mass during different stages of the creep process, including the decelerated, steady-state, and accelerated stages, were thoroughly examined. This analysis not only revealed the sensitivity of the model outputs to variations in each parameter but also helped to clarify the mechanisms by which these parameters regulate the creep response.
In summary, the proposed fractional order multiscale damage creep model effectively captures the nonlinear creep behavior and damage evolution of sandstone in deep coal seam roofs under sustained loading. The model shows strong potential for practical engineering applications, including predicting roof failure time, evaluating residual strength, and optimizing support system design. It also offers theoretical guidance for intelligent monitoring and disaster prevention in deep underground coal mining. Additionally, the model can be extended to high-temperature material creep analysis and safety assessments of geological repositories for nuclear waste. These applications highlight the model’s versatility and value in supporting long-term performance prediction and safety evaluation in complex engineering environments.

Author Contributions

Conceptualization, S.Y. and S.X.; Software, W.Z. and H.S.; Validation, S.Y., W.Z., S.X., B.L. and H.S.; Investigation, W.Z.; Writing—original draft, W.Z.; Writing—review & editing, S.Y., S.X. and B.L.; Supervision, S.Y. and S.X.; Funding acquisition, S.Y. All authors have read and agreed to the published version of the manuscript.

Funding

The authors would like to acknowledge the financial supports from the National Natural Science Foundation of China (52204110, 52121003), Deep Earth Probe and Mineral Resources Exploration-National Science and Technology Major Project (2024ZD1003902), the European Commission Horizon Europe Marie Skłodowska-Curie Actions Staff Exchanges Project—LOC3G (101129729), and the Natural Science Foundation of Shaanxi Province (2025JC-YBQN-571).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Fractional multiscale damage creep model.
Figure 1. Fractional multiscale damage creep model.
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Figure 2. Fractional damage viscoelastic body.
Figure 2. Fractional damage viscoelastic body.
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Figure 3. Fractional damage viscoplastic body.
Figure 3. Fractional damage viscoplastic body.
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Figure 4. Fitting of creep curves for sandstone with varying macro damage using Equations (22) and (23) (a) DT 0.227 (b) DT 0.383 (c) DT 0.505.
Figure 4. Fitting of creep curves for sandstone with varying macro damage using Equations (22) and (23) (a) DT 0.227 (b) DT 0.383 (c) DT 0.505.
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Figure 5. The influence of the fractional-order on creep strain.
Figure 5. The influence of the fractional-order on creep strain.
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Figure 6. The influence of the decay parameter in viscoelastic materials.
Figure 6. The influence of the decay parameter in viscoelastic materials.
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Figure 7. The different parameters in a viscoelastic body. (a) Elastic modulus of viscoelastic materials (Eve). (b) Viscosity coefficient of viscoelastic materials ηve.
Figure 7. The different parameters in a viscoelastic body. (a) Elastic modulus of viscoelastic materials (Eve). (b) Viscosity coefficient of viscoelastic materials ηve.
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Figure 8. The viscosity coefficient of viscoplastic materials.
Figure 8. The viscosity coefficient of viscoplastic materials.
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Figure 9. The influence of the decay parameter in viscoplastic materials.
Figure 9. The influence of the decay parameter in viscoplastic materials.
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Table 1. Parameters comparison.
Table 1. Parameters comparison.
DTγαEe/GPaEve/GPaηve/GPa.hηvp/GPa.hβR2
0.2270.790.10027.3346.7362.41875.870.61650.9976
10.05047.3040.7677.56977.970.59220.9897
0.3830.820.09237.8275.0662.02440.650.62220.9971
10.10027.7573.1981.34416.050.42010.9863
0.5050.570.01018.23177.32541.758888.910.36860.9910
10.00828.25244.13570.726474.220.30020.9580
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Yang, S.; Zhou, W.; Xie, S.; Lei, B.; Song, H. Fractional Order Analysis of Creep Characteristics of Sandstone with Multiscale Damage. Mathematics 2025, 13, 2551. https://doi.org/10.3390/math13162551

AMA Style

Yang S, Zhou W, Xie S, Lei B, Song H. Fractional Order Analysis of Creep Characteristics of Sandstone with Multiscale Damage. Mathematics. 2025; 13(16):2551. https://doi.org/10.3390/math13162551

Chicago/Turabian Style

Yang, Shuai, Wentao Zhou, Senlin Xie, Bo Lei, and Hongchen Song. 2025. "Fractional Order Analysis of Creep Characteristics of Sandstone with Multiscale Damage" Mathematics 13, no. 16: 2551. https://doi.org/10.3390/math13162551

APA Style

Yang, S., Zhou, W., Xie, S., Lei, B., & Song, H. (2025). Fractional Order Analysis of Creep Characteristics of Sandstone with Multiscale Damage. Mathematics, 13(16), 2551. https://doi.org/10.3390/math13162551

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