A Fractional Derivative Insight into Full-Stage Creep Behavior in Deep Coal

Abstract

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage creep model is constructed through serial connection of elastic, viscous, viscoelastic, and viscoelastic–plastic components. Based on this model, both one-dimensional and three-dimensional (3D) fractional creep damage constitutive equations are acquired. Model parameters are identified using experimental data from deep coal samples in the mining area. The result curves of the improved model coincide with experimental data points, accurately describing the deceleration creep stage (DCS), steady-state creep stage (SCS), and accelerated creep stage (ACS). Furthermore, a sensitivity analysis elucidates the impact of model parameters on coal creep behavior, thereby confirming the model’s robustness and applicability. Consequently, the proposed model offers a solid theoretical basis for evaluating the sustained stability of deep coal mining and has great application potential in deep underground engineering.

1. Introduction

As coal mining shifts more towards deeper regions [1], the geological environments and coal seam occurrence conditions become increasingly complex, and multi-physics coupling such as high ground stress, high geothermal temperature [2], and high osmotic pressure significantly exacerbate various disaster risks [3]. Under such complex mining conditions, the stability of surrounding rocks in deep coal mine tunnels faces severe challenges. Research has shown that with increasing mining depth, shallow hard rock gradually transforms into deep soft rock, and the elastic behavior gradually evolves into plastic behavior; this transformation increases the uncertainty of the geological environment [4]. Especially in soft coal tunnels, owing to the low strength, poor cementation, and pronounced rheological characteristics of the coal mass, the synergistic effect of mining stress and aging deformation often lead to accelerated creep instability and roof subsidence [5]. Therefore, the development of creep constitutive models capable of accurately capturing the mechanical response of coal is essential. This advancement provides critical technical support for ensuring the safety and efficiency of deep coal resource extraction.

Research on various creep constitutive models typically includes empirical modeling, component modeling, and damage-based creep modeling approaches [6,7]. These models each have their own advantages, among which the component models not only effectively describe the mechanical characteristics of coal and rock creep but also provide clear physical interpretations of model parameters. The constitutive modeling of coal creep behavior is typically based on elastic, viscous and plastic components. By combining these basic components, a classical component model can be constructed. Such component models are capable of describing both the deceleration creep stage (DCS) and steady-state creep stage (SCS) of material creep; however, it is difficult to precisely capture the accelerated creep stage (ACS) using classical component models. When the applied stress surpasses the coal’s strength threshold, damage will occur inside the material, making it more prone to ACS. Relevant researchers have made many improvements to classical models, further enhancing the creep model. Qi et al. [8] introduced a nonlinear strain triggering module, deriving a three-dimensional (3D) creep constitutive equation that can capture the ACS of sandstone. The creep model developed by Yang [9], incorporating a nonlinear viscoplastic component, effectively characterizes typical full-stage creep behavior. Fahimifar et al. [10] modified the viscoelastic-plastic model and built a viscoelastic hardening model that can predict delayed deformation of samples at different stress levels. Damage variables are usually introduced into the constitutive relationship to describe the nonlinear acceleration stage before material failure [11,12]. The influence of damage on moisture transport in brittle materials was studied by Kruis et al. [13], with a particular focus on its consequences for permeability. Hou et al. [14] proposed a model which can describe the three-stage creep and also captures the influence of pre-existing damage on the stress at creep failure. By investigating nonlinear creep models, a fractional derivative has been widely introduced as an effective tool for studying nonlinear problems in creep constitutive models [11,15,16,17,18]. Based on the theoretical framework of Nishihara’s model, Zhou et al. [7] introduced an Abel dashpot and constructed a fractional-order (FO) constitutive model. To enhance the accuracy of modeling tensile creep in concrete, Huang et al. [19] proposed an FO model incorporating hydration effects. Peng et al. [20] presented an FO model that accounts for the initial damage. Xiang et al. [21] employed the fractional flow approach to model and analyze the complete creep process of soft soil. Kamdem et al. [22] constructed an FO rheological model achieving adaptive characterization of three-stage creep. By embedding the FO viscoelastic model into peridynamic theory, Wang et al. [23] achieved an effective description of the deformation evolution from continuous to discontinuous states under creep and stress relaxation.

However, current models consider damage in a relatively limited way, mainly focusing on the degradation of viscous components and ignoring the simultaneous degradation of elastic stiffness during ACS, thereby oversimplifying the characterization of damage. Research on the coupled viscoelastic damage mechanism of third stage creep behavior is also limited. In summary, fractional calculus incorporates historical dependence or time memory effects, so applying it to time-dependent constitutive models has good applicability. The creep process of coal is time-dependent; therefore, using the fractional derivative has great advantages.

This work focuses on the nonlinear evolution properties of coal creep, particularly during ACS. To characterize these behaviors, this article systematically enhances the classical Burgers model by integrating fractional calculus and damage mechanics. A viscoelastic-plastic component is introduced in series with the Burgers model, wherein traditional Newtonian dashpots are replaced by fractional-order viscous elements. By coupling these with Weibull distribution damage variables, an FO damage creep model which characterizes the full-stage creep properties of deep coal is derived. The accuracy and reliability of the FO model are proved through experimental data fitting and a sensitivity analysis of the model parameters.

2. Modeling

2.1. Fundamentals of Fractional Calculus

As an extension of classical calculus, the order of fractional calculus operators can take real or complex values, rather than being limited to integers. The definition of the fractional derivative includes various forms, and its mathematical properties differ significantly from practical application scenarios. Common fractional derivatives include the Grünwald–Letnikov derivatives [24], Riemann–Liouville derivative [25] and Caputo derivative [26,27]. Grünwald–Letnikov’s derivative is based on the definition form of the discrete difference limit, which requires historical data, and the accumulation of high-order derivative errors is significant. The initial condition for the Riemann–Liouville derivative needs to be expressed through fractional integral values, which have unclear physical meanings and non-zero derivatives of constant functions. The Caputo derivative of a constant value is zero, allowing it to adopt the same physical and engineering initial conditions as classical integer-order derivative models, making it more flexible and convenient in dealing with practical problems [28,29]. For example, Yang et al. [30] analyzed the stability of FO systems incorporating mixed time-varying delays through the application of the Caputo-type derivative. Liu et al. [31] established a variable order creep model within the Caputo fractional derivative framework and analyzed the role of relaxation time in determining creep evolution. The creep experiment started at t = 0, based on the above analysis; in this work, the Caputo derivative is invoked, which is denoted by

$$\frac{d^{\gamma }f(t)}{d{t}^{\gamma }}=\frac{1}{\mathsf{\Gamma }(1-\gamma )}{\int }_0^t\frac{f^{\prime }(\tau )}{{(t-\tau )}^{\gamma }}d\tau ,$$

(1)

where 0 < γ ≤ 1 is a fractional order, and Γ(·) represents the Gamma function which is defined as $\mathsf{\Gamma }(w)={{\displaystyle \int }}_0^{\infty }{e}^{-t}{t}^{w-1}dt$. f′(τ) represents the first-order derivative.

The FO theory exhibits unique advantages, its memory inheritance property achieves weighted representation of historical states through integral kernel functions, significantly reducing the number of parameters required for traditional integer-order models [32,33]. The fractional derivative enables non-local physical processes to be accurately described through compact differential equations [34]. These theoretical advantages directly improve the accuracy of fractional calculus in describing coal creep processes.

2.2. Weibull Distribution Damage

The statistical theory of strength [35] suggests that the strength of a representative volume unit of a material follows a specific random distribution pattern. Coal will generate cracks under external loads, which give it statistical strength characteristics. The process of crack generation and propagation also exhibits randomness [36]. Therefore, statistical methods should be employed to analyze the damage evolution and deformation characteristics [37]. During the exploration of damage in coal rock materials, the Weibull distribution based on the weakest chain theory is commonly used to describe the strength distribution of the material [38]. The probability density can be denoted by

$$p(\epsilon )={\displaystyle \frac{m}{F}}{\left({\displaystyle \frac{\epsilon }{F}}\right)}^{m-1}\exp \left[-{\left({\displaystyle \frac{\epsilon }{F}}\right)}^m\right],$$

(2)

where m is the Weibull modulus which represents the response characteristics of coal mass to external loads, F is a parameter associated with the physical and mechanical features of coal, ε represents the random distribution in microelement strength—due to the use of strain strength theory here— and ε represents the strain amount of the coal mass.

When the coal mass is subjected to a load, some microelements will gradually fail. Based on the statistical distribution properties of microelements, the gradual change can be described. The damage variable of coal under load is [39]

$$\mathrm{D}={\displaystyle \frac{n}{N}},$$

(3)

where the value range of D is from 0 to 1, and n and N represent the number of broken microelements and initial microelements when no damage has occurred in the coal mass, respectively.

Given an interval (x, x + dx), when the coal follows a Weibull distribution, the number of failed microelements within this interval is Np(x)dx. At a certain strain level ε, the failure count of microelements is

$$n={{\displaystyle \int }}_0^{z}Np(x)dx=N\left(1-\exp \left[-{\left({\displaystyle \frac{\epsilon }{F}}\right)}^m\right]\right).$$

(4)

The expression for D can be obtained by solving Equation (4),

$$D=1-\exp \left[-{\left({\displaystyle \frac{\epsilon }{F}}\right)}^m\right].$$

(5)

The primary objective of incorporating damage into the FO model is to capture the nonlinear mechanical behavior of deep coal creep. For improved computational efficiency, set the relationship between ε and t as ε = Ct^(1/m) (C is a constant), and assign the coefficient of t to α again. Equation (5) is rewritten as

$$\mathrm{D}=1-e^{-\alpha t},$$

(6)

where α is the damage factor.

2.3. Establishment of FO Damage Creep Model

Spring, dashpot, and slider characterize elastic response, viscous flow, and plastic deformation, respectively. These components can represent the coupling effect of multi-scale mechanical mechanisms, and damage can represent the cumulative damage during the ACS. By connecting these basic elements in series and parallel, and adding damage factors based on the creep mechanism, components that can reflect the ACS can be constructed. Therefore, a novel creep constitutive modeling method is proposed in this work by extending the Burgers model. By embedding damage variables into the viscoelastic-plastic elements and reconstructing the creep response mechanism of traditional Newtonian dashpots using the Caputo FO derivative, an FO damage creep model is developed to capture the full-stage mechanical behavior of deep coal creep.

2.3.1. Basic Mechanical Components and Composition

As shown in Figure 1, k₃ corresponds to the elastic modulus of the Damaged Hooke element. η₁, η₃ are the viscosity coefficients of the Abel and damaged Abel dashpots, respectively, and γ defines their fractional orders. α is related to damage.

(1) Figure 2 shows the Burgers model. k₁ and k₂ denote the elastic moduli, and η₁ and η₂ denote the viscosity coefficients.

The creep constitutive relationship of this model is

$$\epsilon \left(t\right)={\displaystyle \frac{\sigma }{k_1}}+{\displaystyle \frac{\sigma }{{\eta }_1}}t+{\displaystyle \frac{\sigma }{k_2}}\left(1-e^{-{\displaystyle \frac{k_2}{{\eta }_2}}t}\right),$$

(7)

where ε, σ represent strain and stress, respectively.

(2) Extend the first-order derivative of the Newtonian dashpot’s constitutive equation to the fractional derivative, and the FO equation is derived.

$$\sigma ={\eta }_1{\displaystyle \frac{d^{\gamma }\epsilon \left(t\right)}{d{t}^{\gamma }}}\Rightarrow \epsilon \left(t\right)={\displaystyle \frac{\sigma }{{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}.$$

(8)

In the figure, k₂ and k₃ denote the initial elastic modulus, η₂ and η₃ denote the initial viscosity coefficients, σ_s is the yield limit, and D is defined in Section 2.2. k₃ (D) = k₃ (1−D) and η₃ (D) = η₃ (1−D).

By connecting the Abel dashpot and Hooke body in parallel, an FO Kelvin body can be obtained. By adding damage to the Abel dashpot and connecting it in parallel with the damaged Hooke spring and slider, an FO damaged viscoelastic-plastic component can be obtained, as shown in Figure 3.

2.3.2. FO Damaged Creep Model

The traditional Burgers model is capable of capturing both DCS and SCS, which makes it highly accurate in describing complex mechanical behavior. It combines the advantages of the Maxwell and Kelvin models. However, it may not be accurate enough in describing nonlinear viscoelastic behavior. If the stress level rises above the yield limit, the coal is subjected to damage and accelerates the creep stage rapidly over time, with significant nonlinear deformation. To describe the third stage, a damage variable is introduced and expressed in the fractional derivative, which has a good memory effect and significant advantages for nonlinear creep.

In this section, the Abel dashpot is used instead of the Newtonian dashpot, and the viscoelastic-plastic element from Figure 3 is connected in series with the FO Burgers model to capture the ACS. Figure 4 illustrates the newly developed creep model, which integrates elastic, viscous, viscoelastic, and viscoelastic-plastic elements.

The total strain is

$$\epsilon ={\epsilon }_1+{\epsilon }_2+{\epsilon }_3+{\epsilon }_4,$$

(9)

where ε₁, ε₂, ε₃ and ε₄ correspond to the strain in Figure 4.

(1) The spring element characterizes the initial elastic response of the material, and for the Hooke element, its constitutive relationship is

$${\epsilon }_1={\displaystyle \frac{\sigma }{k_1}}.$$

(10)

(2) The constitutive relationship for the Abel dashpot has been explained in the previous section.

$${\epsilon }_2={\displaystyle \frac{\sigma }{{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}.$$

(11)

(3) The constitutive relationship of the viscoelastic body is

$$\sigma =k_2{\epsilon }_3+{\eta }_2{\displaystyle \frac{d^{\gamma }{\epsilon }_3}{d{t}^{\gamma }}},$$

(12)

Equation (12) can be written as

$${\displaystyle \frac{d^{\gamma }{\epsilon }_3}{d{t}^{\gamma }}}={\displaystyle \frac{\sigma }{{\eta }_2}}-{\displaystyle \frac{k_2}{{\eta }_2}}{\epsilon }_3.$$

(13)

Applying Laplace transform to Equation (13), the result is

$$E\left(s\right)={\displaystyle \frac{\sigma }{{\eta }_2}}{\displaystyle \frac{1}{s}}{\displaystyle \frac{1}{s^{\gamma }+\frac{k_2}{{\eta }_2}}}.$$

(14)

Applying the inverse Laplace transform to Equation (14) yields

$${\epsilon }_3={\displaystyle \frac{\sigma }{k_2}}\left[1-E_{\gamma ,1}\left(-{\displaystyle \frac{k_2}{{\eta }_2}}{t}^{\gamma }\right)\right],$$

(15)

where E_γ,1(·) is the two-parameter Mittag–Leffler function, expressed as follows:

$$E_{\alpha ,\beta }\left(z\right)={\displaystyle \sum _{k=0}^{\infty }\frac{z^k}{\Gamma \left(\alpha k+\beta \right)}}.$$

(16)

(4) When the axial load surpasses the long-term strength, the coal mass undergoes damage, leading to time-dependent variation.

If σ < σ_s, the strain of the viscoelastic-plastic element is 0.

If σ ≥ σ_s, the constitutive relationship of the viscoelastic-plastic element is

$$\sigma =k_3\left(1-\mathrm{D}\right){\epsilon }_4+{\eta }_3\left(1-\mathrm{D}\right){\displaystyle \frac{d^{\gamma }{\epsilon }_4}{d{t}^{\gamma }}}+{\sigma }_s.$$

(17)

By substituting the damage variable D = 1 − e^−αt, the following can be obtained:

$${\displaystyle \frac{d^{\gamma }{\epsilon }_4}{d{t}^{\gamma }}}={\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_3}}{e}^{\alpha t}-{\displaystyle \frac{k_3}{{\eta }_3}}{\epsilon }_4.$$

(18)

Appling Laplace transform to Equation (18), the following can be obtained:

$$E\left(s\right)={\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_3}}{\displaystyle \frac{1}{s-\alpha }}{\displaystyle \frac{1}{s^{\gamma }+\frac{k_3}{{\eta }_3}}}.$$

(19)

According to the properties of the Laplace transform, the transform of the convolution of two time-domain functions is equivalent to the product of their individual Laplace transforms, as shown in Equation (20). Therefore, the expression can be decomposed into the product of two components. By determining their respective inverse Laplace transforms and performing convolution in the time domain, the original function can be obtained [40].

$$\mathrm{L}\left\{f\left(t\right)\ast g\left(t\right)\right\}=\mathrm{L}\left\{f\left(t\right)\right\}\cdot \mathrm{L}\left\{g\left(t\right)\right\}=F\left(s\right)\cdot G\left(s\right).$$

(20)

where $f\left(t\right)\ast g\left(t\right)={\displaystyle {\int }_0^{t}f\left(t-\tau \right)g\left(\tau \right)}d\tau$.

Given the Laplace domain solution,

$$E\left(s\right)={\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_3}}\underset{F\left(s\right)}{\underbrace{{\displaystyle \frac{1}{s-\alpha }}}}\underset{G\left(s\right)}{\underbrace{{\displaystyle \frac{1}{s^{\gamma }+\frac{k_3}{{\eta }_3}}}}}.$$

(21)

Applying Laplace inverse transform on F(s) and G(s), respectively, two equations can be obtained:

$${\mathrm{L}}^{-1}\left\{{\displaystyle \frac{1}{s-\alpha }}\right\}=e^{\alpha t},$$

(22)

$${\mathrm{L}}^{-1}\left\{{\displaystyle \frac{1}{s^{\gamma }+\frac{k_3}{{\eta }_3}}}\right\}=t^{\gamma -1}{E}_{\gamma ,\gamma }\left(-{\displaystyle \frac{k_3}{{\eta }_3}}{t}^{\gamma }\right).$$

(23)

According to the convolution theorem, the original function is a convolution of two time-domain functions:

$${\epsilon }_4\left(t\right)={\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_3}}\left[e^{\alpha t}\ast \left(t^{\gamma -1}{E}_{\gamma ,\gamma }\left(-{\displaystyle \frac{k_3}{{\eta }_3}}{t}^{\gamma }\right)\right)\right].$$

(24)

Expanding the convolution integral, it can be written as

$${\epsilon }_4\left(t\right)={\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_3}}{\displaystyle {\int }_0^t{e}^{\alpha \left(t-\tau \right)}{\tau }^{\gamma -1}{E}_{\gamma ,\gamma }\left(-\frac{k_3}{{\eta }_3}{\tau }^{\gamma }\right)d\tau }.$$

(25)

Therefore, the constitutive equation of the FO damaged creep model is

$$\epsilon \left(t\right)=\left\{\begin{array}{l}{\displaystyle \frac{\sigma }{k_1}}+{\displaystyle \frac{\sigma }{{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(\gamma +1\right)}}+{\displaystyle \frac{\sigma }{k_2}}\left(1-E_{\gamma }\left(-{\displaystyle \frac{k_2}{{\eta }_2}}{t}^{\gamma }\right)\right),\sigma <{\sigma }_s\\ {\displaystyle \frac{\sigma }{k_1}}+{\displaystyle \frac{\sigma }{{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(\gamma +1\right)}}+{\displaystyle \frac{\sigma }{k_2}}\left(1-E_{\gamma }\left(-{\displaystyle \frac{k_2}{{\eta }_2}}{t}^{\gamma }\right)\right)+{\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_3}}\left[e^{\alpha t}\ast \left(t^{\gamma -1}{E}_{\gamma ,\gamma }\left(-{\displaystyle \frac{k_3}{{\eta }_3}}{t}^{\gamma }\right)\right)\right],\sigma \ge {\sigma }_s,\end{array}\right.$$

(26)

There is a close relationship between mechanical behavior and energy dissipation [41]. The convolutional kernel function describes the energy dissipation during the creep process and reflects the sustained influence of the initial damage state on the rate of crack propagation inside the coal, exhibiting both memory and nonlinear damping effects. The synergistic influence of the rate dependence and historical correlation in crack propagation mimics the accelerated creep behavior attributable to damage.

2.3.3. Triaxial Creep Constitutive Model

Considering that coal withstands triaxial stresses in deep mining operations, the development of a 3D creep constitutive equation is required. The total strain under this condition can be formulated as follows:

$${\epsilon }_{ij}={\epsilon }_{ij}^1+{\epsilon }_{ij}^2+{\epsilon }_{ij}^3+{\epsilon }_{ij}^4,$$

(27)

where ${\epsilon }_{ij}$ represents the overall strain tensor, and ${\epsilon }_{ij}^1$, ${\epsilon }_{ij}^2$, ${\epsilon }_{ij}^3$ and ${\epsilon }_{ij}^4$ are the strain tensors associated with the elastic, viscous, viscoelastic, and viscoelastic-plastic bodies, respectively.

In accordance with the principles of Hooke’s law, the 3D equation of the elastic element is expressed as

$${\epsilon }_{ij}^1={\displaystyle \frac{1}{2G_1}}{S}_{ij},$$

(28)

where G₁ denotes the shear modulus, and S_ij denotes the deviatoric stress tensor.

In a 3D condition, the constitutive equation of viscous element is

$${\epsilon }_{ij}^2={\displaystyle \frac{S_{ij}}{2{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}.$$

(29)

The 3D constitutive equation of the viscoelastic element is written as

$${\epsilon }_{ij}^3={\displaystyle \frac{S_{ij}}{2G_2}}\left[1-E_{\gamma }\left(-{\displaystyle \frac{G_2}{{\eta }_2}}{t}^{\gamma }\right)\right],$$

(30)

where G₂ represents the shear modulus.

The constitutive equation of the viscoelastic-plastic element under 3D stress conditions incorporates yield criteria and plastic potential theory. It can be described as follows:

$${\epsilon }_{ij}^4={\displaystyle \frac{1}{2{\eta }_3}}〈\phi {\left({\displaystyle \frac{F}{F_0}}\right)}^n〉{\displaystyle \frac{\partial F}{\partial {\sigma }_{ij}}}\left[e^{\alpha t}\ast \left(t^{\gamma -1}{E}_{\gamma ,\gamma }\left(-{\displaystyle \frac{G_3}{{\eta }_3}}{t}^{\gamma }\right)\right)\right],$$

(31)

where

$$〈\phi \left({\displaystyle \frac{F}{F_0}}\right)〉=\left\{\begin{array}{l}0,\left(F<0\right)\\ \phi \left({\displaystyle \frac{F}{F_0}}\right),\left(F\ge 0\right),\end{array}\right.$$

(32)

where F corresponds to the yield function, F₀ represents its initial value, and G₃ is the shear modulus. ϕ(·) is characterized by a power function expression. Usually, n = 1 is taken.

Assuming isotropy of the coal mass, Poisson’s ratio in the creep process is unchanging, and neglecting volumetric creep effects, the 3D creep equation is

$${\epsilon }_{ij}\left(t\right)=\left\{\begin{array}{l}{\displaystyle \frac{S_{ij}}{2G_1}}+{\displaystyle \frac{S_{ij}}{2{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}+{\displaystyle \frac{S_{ij}}{2G_2}}\left[1-E_{\gamma }\left(-{\displaystyle \frac{G_2}{{\eta }_2}}{t}^{\gamma }\right)\right],\left(F<0\right)\\ {\displaystyle \frac{S_{ij}}{2G_1}}+{\displaystyle \frac{S_{ij}}{2{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}+{\displaystyle \frac{S_{ij}}{2G_2}}\left[1-E_{\gamma }\left(-{\displaystyle \frac{G_2}{{\eta }_2}}{t}^{\gamma }\right)\right]\\ +{\displaystyle \frac{1}{2{\eta }_3}}\left({\displaystyle \frac{F}{F_0}}\right){\displaystyle \frac{\partial F}{\partial {\sigma }_{ij}}}\left[e^{\alpha t}\ast \left(t^{\gamma -1}{E}_{\gamma ,\gamma }\left(-{\displaystyle \frac{G_3}{{\eta }_3}}{t}^{\gamma }\right)\right)\right],\left(F\ge 0\right).\end{array}\right.$$

(33)

The functional form of the yield condition is

$$F=\sqrt{J_2}-{\sigma }_s/\sqrt{3},$$

(34)

where $J_2={\displaystyle \frac{1}{6}}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]$; it represents the second fixed deviatoric stress tensor.

In fact, the confining pressure is equal in the triaxial creep test, so σ₂ = σ₃, σ_m is the spherical strain tensor, and then,

$$\left\{\begin{array}{l}{\sigma }_m={\displaystyle \frac{1}{3}}\left({\sigma }_1+2{\sigma }_3\right),\sqrt{J_2}={\displaystyle \frac{{\sigma }_1-{\sigma }_3}{\sqrt{3}}}\\ S_{11}={\sigma }_1-{\sigma }_m={\displaystyle \frac{2}{3}}\left({\sigma }_1-{\sigma }_3\right).\end{array}\right.$$

(35)

By substituting Equations (34) and (35) into Equation (33) and making F₀ = 1, the 3D FO derivative damage creep equation can be obtained.

$$\epsilon \left(t\right)=\left\{\begin{array}{l}{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_1}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_2}}\left[1-E_{\gamma }\left(-{\displaystyle \frac{G_2}{{\eta }_2}}{t}^{\gamma }\right)\right],\left(F<0\right)\\ {\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_1}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3{\eta }_1}}{\displaystyle \frac{t^{\gamma }}{\Gamma \left(1+\gamma \right)}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_2}}\left[1-E_{\gamma }\left(-{\displaystyle \frac{G_2}{{\eta }_2}}{t}^{\gamma }\right)\right]\\ +{\displaystyle \frac{{\sigma }_1-{\sigma }_3-{\sigma }_s}{6{\eta }_3}}\left[e^{\alpha t}\ast \left(t^{\gamma -1}{E}_{\gamma ,\gamma }\left(-{\displaystyle \frac{G_3}{{\eta }_3}}{t}^{\gamma }\right)\right)\right],\left(F\ge 0\right)\end{array}\right.$$

(36)

The 3D constitutive equation of the Burgers model is

$$\epsilon \left(t\right)={\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_1}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3{\eta }_1}}t+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_2}}\left(1-e^{-{\displaystyle \frac{G_2}{{\eta }_2}}t}\right)$$

(37)

3. Results and Discussions

3.1. Parameter Identification

The model’s ability to achieve high-precision fitting with creep data serves as a critical criterion for evaluating the reliability of the FO creep damage model. The coal blocks used in this study were obtained by Zhang et al. [42] from the No. 12 coal mine in Pingdingshan area, Henan Province. The axial pressure of the triaxial experiment was 30 MPa, the confining pressure was 5 MPa, the experimental temperature of UCT-3 was 70 °C, and the experimental temperature of UCT-1 was 30 °C. Temperature is also an important factor affecting creep, and the model in this article does not consider temperature. The effect of temperature is considered as the influence on damage accumulation.

Table 1. Coal sample properties [42].

Specimen ID	Diameter (mm)	Height (mm)	Mass (g)	Volume (cm3)	Density (g/cm3)	Uniaxial Compressive Strength (MPa)
UCT-1	49.64	100.48	258.49	194.34	1.33	15.60
UCT-3	49.75	100.30	262.52	194.87	1.35	16.38

lists the dimensional and physical features of the coal block.

The experimental data show that UCT-1 does not exhibit an ACS, whereas UCT-3 does. The parameters were obtained via the nonlinear least squares method, with the resulting fitting values listed in

Table 2. Fractional-order model fitting parameters.

Specimen ID	G1 (MPa)	G2 (MPa)	G3 (MPa)	η1 (MPa·hγ)	η2 (MPa·hγ)	η3 (MPa·hγ)	α	γ
UCT-1	4.19 × 103	8.09 × 103	-	6.89 × 105	551.91	-	-	0.9386
UCT-3	3.33 × 103	6.14 × 103	2.93 × 106	1.30 × 106	4.91 × 103	3.38 × 103	0.4583	0.6820
UCT-4	1.16 × 104	3.06 × 104	1.47 × 107	2.10 × 106	1.50 × 104	8.96 × 106	0.5645	0.9014

. From Figure 5, there is strong conformity between the FO model output and the experimental creep data, with the correlation coefficient nearly equal to 1 and a small root mean square error. The obtained curves indicate that the established FO damaged creep model can perfectly describe the full-stage creep characteristics in coal samples. Figure 5 also shows the fitting prediction results of the traditional Burgers model, from which it can be seen that the Burgers model can capture the creep data of the first two stages of UCT-3 well. The correlation coefficients of the FO Burgers model and traditional Burgers model for the UCT-1 coal sample are 0.9927 and 0.9913, respectively. Both match the creep data which do not exhibit an ACS well.

To further verify the effectiveness of the model, triaxial creep experimental data for rock after 10 freeze-thaw cycles were selected for analysis [43]. This sample was labelled as UCT-4. The axial stress of the experiment was 40 MPa, the confining pressure was 5 MPa, and the long-term strength of the rock was 32.17 MPa. The micro crack propagation behavior induced by freeze-thaw cycles inside rocks conforms to the Weibull distribution characteristics, and its damage evolution is consistent with the theory of this model. The model validation results of the fractional derivative model and Burgers model are shown in Figure 6, which proves that the FO model fits the experimental data from rock freeze-thaw cycles well, further verifying the effectiveness of the model. The correlation coefficient fitted by this model is higher than that of the original article. The fitting parameters are shown in Table 2.

3.2. Parameter Sensitivity Analysis

The model developed in this work effectively reflects the full-stage creep behavior of experimental data in deep coal. Based on this, in this section, sensitivity analysis was conducted on some parameters to explore their influence on creep behavior.

In Figure 7, variations in the strain curve with the fractional order γ indicates that the change in γ significantly affects the SCS and the ACS. In SCS, higher values of γ result in higher creep strain. During the ACS, creep strains with lower γ values are higher and exhibit significant nonlinear characteristics. Therefore, fractional derivatives have unique order continuous adjustment characteristics due to their memory effect, demonstrating significant advantages in characterizing nonlinear creep behavior.

When the applied stress surpasses the long-term strength, damage will accumulate progressively, leading to ruin. In Figure 8, according to sensitivity analysis, the damage factor has a great impact on the strain rate during the ACS, and an increase in the damage factor significantly intensifies nonlinear deformation.

Figure 9 presents the effect of the shear modulus on the creep process. The shear modulus G₁ dominates the instantaneous elastic response stage: as it increases, the corresponding initial elastic strain decreases. G₂ mainly affects the transition of strain from DCS to SCS, with higher G₂ resulting in lower steady-state creep strain. G₃ affects the ACS, and the accelerated creep rate is related to the size of G₃. The smaller the G₃, the greater the accelerated creep rate.

Figure 10 shows the sensitivity analysis of the viscosity coefficient to creep strain. Increasing η₁ strengthens the viscous damping effect. This action inhibits the viscous flow rate and diminishes the SCS strain. The variation in the creep strain curve with the viscosity coefficient η₂ indicates that the DCS and SCS are affected by η₂, while the final creep strain is basically unaffected by it. High η₂ results in low transient strain rates and lower steady-state creep strains. η₃ mainly affects the third stage, where an increase in η₃ value directly suppresses the ACS rate.

By performing sensitivity analysis on the model parameters, a deeper insight into their influence on creep behavior can be achieved. The effects of different parameters are both individual and interrelated in the creep process. The strain rates of ACS in Figure 5 and Figure 6 are different. According to Table 2, the fractional order is similar, and the fitted G₃, η₃, and α values in Figure 6 are significantly higher than those in Figure 5. The sensitivity analysis results not only reveal the dominant role of key parameters in creep behavior, but also further verify the stability and accuracy of the model under different parameter combinations. However, the current model does not take into account the effects of other geological conditions such as temperature and initial damage. Temperature is also an important factor affecting creep rate. Subsequent work will focus on deepening the model structure.

4. Conclusions

Based on the theories of fractional calculus and damage mechanics, an FO damage creep model was established and validated. The principal research outcomes include the following:

(1)

A viscoelastic-plastic component considering damage has been constructed, which can simultaneously characterize the decline in viscosity coefficient and the degradation of the elastic modulus. This component was connected in series with elastic, viscous, and viscoelastic components. Then, Abel dashpots were introduced to replace Newton dashpots in the model, thus establishing a fractional damage creep model that can describe the ACS.

(2)

Nonlinear least squares fitting was performed on experimental data to determine model parameters and compare this model with the traditional Burgers model. The findings demonstrate that the established creep model can perfectly describe the full-stage mechanical behavior of deep coal during the creep process, which also shows that the proposed viscoelastic-plastic component can effectively capture the nonlinear growth phenomenon when stress exceeds long-term strength. In addition, rock freeze-thaw cycle triaxial compression experimental data were used for model validation, and the results showed a strong agreement between the model predictions and the experimental data, further confirming the reliability of the model.

(3)

Considering the availability of the model for other geological conditions, a detailed sensitivity analysis was performed to explore how each parameter affects the creep behavior, and the role of parameters in different creep stages was clarified, further supporting the credibility and broad applicability of the model. This study provides theoretical guidance for practical engineering applications.