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<p>Fractional calculus has proven to be highly effective in simulating the viscoelastic and memory-dependent behavior of materials. This paper presents a three-dimensional fractional derivative standard linear solid (FSLS) model and its numerical implementation. By introducing the Caputo fractional operator to define a new Koeller spring-pot element, a one-dimensional FSLS model is developed, which can degenerate into other fractional derivative models, including the fractional derivative Maxwell (FM) and fractional derivative Kelvin-Voigt (FKV) models. The three-dimensional constitutive relationships are derived using a tensor generation method, and high-precision difference equations are formulated using the <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$L_{1}$</EquationSource> </InlineEquation> time-discretization scheme. The model is implemented in ABAQUS using the UMAT interface and validated through creep and relaxation tests. The results indicate that: (1) The FSLS model based on the Caputo fractional operator is better suited for finite element computations because it avoids singularity issues, preventing computational difficulties; (2) The <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$L_{1}$</EquationSource> </InlineEquation> discretization formula, with improved accuracy of order 2-<InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> <EquationSource Format="TEX">$\alpha $</EquationSource> </InlineEquation>, can be employed to develop three-dimensional fractional derivative models and facilitate their numerical implementation in engineering applications; (3) A comparison between model predictions with experimental data demonstrates that the proposed model effectively captures the time-dependent behavior of geotechnical materials. The proposed model and discretization method provide valuable support for understanding and simulating the time-dependent deformation of geomaterials.</p>

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Three-dimensional fractional viscoelastic constitutive modeling and numerical implementation using the \(L_{1}\) time-discretization scheme

  • 期刊论文

  • 发表日期：

    2026年1月9日

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  • Pan Ding,
  • Riqing Xu,
  • Minjie Wen,
  • Menghuan Chen,
  • Ji Peng,
  • Yuan Tu

摘要

Fractional calculus has proven to be highly effective in simulating the viscoelastic and memory-dependent behavior of materials. This paper presents a three-dimensional fractional derivative standard linear solid (FSLS) model and its numerical implementation. By introducing the Caputo fractional operator to define a new Koeller spring-pot element, a one-dimensional FSLS model is developed, which can degenerate into other fractional derivative models, including the fractional derivative Maxwell (FM) and fractional derivative Kelvin-Voigt (FKV) models. The three-dimensional constitutive relationships are derived using a tensor generation method, and high-precision difference equations are formulated using the L 1 $L_{1}$ time-discretization scheme. The model is implemented in ABAQUS using the UMAT interface and validated through creep and relaxation tests. The results indicate that: (1) The FSLS model based on the Caputo fractional operator is better suited for finite element computations because it avoids singularity issues, preventing computational difficulties; (2) The L 1 $L_{1}$ discretization formula, with improved accuracy of order 2- α $\alpha $ , can be employed to develop three-dimensional fractional derivative models and facilitate their numerical implementation in engineering applications; (3) A comparison between model predictions with experimental data demonstrates that the proposed model effectively captures the time-dependent behavior of geotechnical materials. The proposed model and discretization method provide valuable support for understanding and simulating the time-dependent deformation of geomaterials.