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<p>Deep coal rocks exhibit complex nonlinear rheological properties under high-stress environments, so conventional integer-order creep models cannot describe the full creep process, especially the accelerated phase. This paper introduces a novel variable-order fractal dashpot into the constitutive modeling of coal rocks. The fractal derivative is a strictly local operator that avoids the convolutional integration required by fractional derivatives, offering closed-form analytical solutions, the ability to accommodate variable-order and damage-coupled parameters without inflating computation, and concise expressions free of special functions. Defining the fractal order as a function of time and cleat-damage evolution lets the dashpot capture the progressive deterioration of the rock’s mechanical properties. Replacing the Newtonian dashpot in the classical Maxwell model with this element yields a new nonlinear creep damage model. An intelligent parameter-identification method based on Adaptive Particle Swarm Optimization (APSO) is proposed for the hard-to-invert nonlinear equation. Analytical solutions are derived and validated against cited triaxial creep data of coal rocks. The APSO-based fitting shows that the model reproduces the primary, steady-state, and highly nonlinear tertiary creep stages with physically reasonable parameters (<InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.9984</mn> </math></EquationSource> <EquationSource Format="TEX">$R^{2}=0.9984$</EquationSource> </InlineEquation>), outperforming the integer-order Nishihara model and a constant-order fractional model. A quantitative comparison further shows that the local fractal operator is roughly two orders of magnitude faster than the global fractional operator, with the gap widening as the number of evaluation points increases. The scope and limitations are discussed explicitly.</p>

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Variable-order fractal derivative-based nonlinear creep damage model for coal rock with intelligent parameter identification

  • 期刊论文

  • 发表日期：

    2026年7月3日

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  • Cong Lu,
  • Zhile Han,
  • Jianchun Guo,
  • Shouxin Wang,
  • Qijun Zeng,
  • Qiuyue Li,
  • Chi Chen,
  • Shiqian Xu

摘要

Deep coal rocks exhibit complex nonlinear rheological properties under high-stress environments, so conventional integer-order creep models cannot describe the full creep process, especially the accelerated phase. This paper introduces a novel variable-order fractal dashpot into the constitutive modeling of coal rocks. The fractal derivative is a strictly local operator that avoids the convolutional integration required by fractional derivatives, offering closed-form analytical solutions, the ability to accommodate variable-order and damage-coupled parameters without inflating computation, and concise expressions free of special functions. Defining the fractal order as a function of time and cleat-damage evolution lets the dashpot capture the progressive deterioration of the rock’s mechanical properties. Replacing the Newtonian dashpot in the classical Maxwell model with this element yields a new nonlinear creep damage model. An intelligent parameter-identification method based on Adaptive Particle Swarm Optimization (APSO) is proposed for the hard-to-invert nonlinear equation. Analytical solutions are derived and validated against cited triaxial creep data of coal rocks. The APSO-based fitting shows that the model reproduces the primary, steady-state, and highly nonlinear tertiary creep stages with physically reasonable parameters ( R 2 = 0.9984 $R^{2}=0.9984$ ), outperforming the integer-order Nishihara model and a constant-order fractional model. A quantitative comparison further shows that the local fractal operator is roughly two orders of magnitude faster than the global fractional operator, with the gap widening as the number of evaluation points increases. The scope and limitations are discussed explicitly.