# A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate
## 4 Integer-Order Nonlinear Creep Model  
### Integer-Order Nonlinear Viscoelastic-Plastic Creep Model  
Based on the creep curves of coal samples presented in Section 3, four characteristic deformation stages are identified:  
(1) Instantaneous deformation. The instantaneous behavior of coal-rock deformation can be described by the Hookean elastic body. (2) Decelerated creep. Among existing rheological models, both the Kelvin body and Muruyama body are capable of characterizing the decelerated creep behavior of coal. A critical distinction lies in their activation mechanisms: the Kelvin body exhibits creep deformation under all stress conditions, whereas the Muruyama body incorporates a Saint-Venant element to govern a stress threshold, permitting creep initiation only when the applied stress exceeds this critical value. This study adopts the fractional-order Muruyama body to model the decelerated creep phase. (3) Steady creep. When the stress level exceeds a critical threshold, a steady creep process emerges in the creep curves of coal pillars. This phase can be characterized by the Bingham body. (4) Accelerated creep. At stress levels exceeding critical stability limits, the creep curves enter a phase of rapidly increasing deformation rates. Conventional viscous damping coefficients fail to capture this nonlinear acceleration, necessitating the introduction of a damage-coupled viscous coefficient to quantify the instability mechanisms. This study establishes a nonlinear integer-order viscoelastic-plastic creep model, which is composed of four elements connected in series: a Hook body, a Muruyama body, a Bingham body, and a modified Bingham body (Figure 6). During the triaxial creep process of coal samples, the decelerated creep and steady creep stages can be described using the Hook body, Muruyama body, and Bingham body. The physical significance of these models is well-established and thus not elaborated upon here. In the accelerated creep phase characterized by higher creep rates, the conventional Bingham body fails to adequately characterize accelerated creep behavior. This study implements a nonlinear modification to the viscous coefficient of the Bingham body, specifically  
\[\eta_{3}(\sigma,t)=\eta_{c}e^{-at} \tag{2}\]  
where \(\sigma\) denotes stress, \(\eta\) is the viscosity coefficient, and a is \(a\) material mechanical parameter with \(a>0\). The constitutive equation of the modified Bingham body is  
\[\sigma-\sigma_{\rm s}=\eta_{c}e^{-at}\dot{\varepsilon} \tag{3}\]  
Figure 6: Integer-order nonlinear viscoelastic-plastic creep model [49; 50]. where \(\varepsilon\) denotes strain, when \(\sigma=\sigma_{c}\), the creep equation of the modified Bingham body is  
\[\varepsilon=\frac{\sigma_{c}-\sigma_{s}}{a\eta_{c}}\varepsilon^{at} \tag{4}\]  
By substituting the relevant parameters into Equation (4), the influence of parameter a on creep acceleration is obtained as shown in the figure. It can be observed that an increase in parameter a leads to a higher creep rate in the curves. This demonstrates that the accelerated creep stage can be quantitatively characterized by controlling the magnitude of parameter a, as can be observed in Figure 7.  
It is evident that the four components of the nonlinear integer-order viscoelastic-plastic creep model exhibit well-defined physical significance and parametric characteristics. The complete rheological constitutive equations and creep equations for this model are formulated as follows:  
(1) Rheological constitutive equations. When \(\sigma<\sigma_{s1}\), the rheological model contains only the Hook body, and its constitutive equation is  
\[\sigma=E_{1}\varepsilon \tag{5}\]  
where \(E\) is the elastic modulus, when \(\sigma_{s1}\leq\sigma<\sigma_{s2}\), the rheological model is composed of a Hookean body and a Muruyama body, and each element in the rheological model satisfies the following equations:  
\[\left\{