Strong-Weak Non-Local Medium Constitutive Modeling Based on the Spatial Fractional Derivative

FANG Jun; WU Yishi

doi:10.21656/1000-0887.450073

Volume 46 Issue 6

Jun. 2025

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FANG Jun, WU Yishi. Strong-Weak Non-Local Medium Constitutive Modeling Based on the Spatial Fractional Derivative[J]. Applied Mathematics and Mechanics, 2025, 46(6): 764-780. doi: 10.21656/1000-0887.450073

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Strong-Weak Non-Local Medium Constitutive Modeling Based on the Spatial Fractional Derivative

doi: 10.21656/1000-0887.450073

FANG Jun^{1
,
,},
WU Yishi²

1. Department of General Foundation Requisite, Nanjing Vocational University of Industry Technology,Nanjing 210023, P.R.China;
2. China Huadian Corporation Shanghai Branch, Shanghai 200100, P.R.China

Received Date: 2024-03-22
Rev Recd Date: 2024-05-06

Available Online: 2025-06-30

Abstract

Abstract

The non-local medium constitutive modeling method based on the spatial fractional derivative was studied, which provides a theoretical guidance for studying the mechanical properties of complex non-local materials. Firstly, the definition of the Chen-Holm fractional-order Laplace operator was extended, to obtain a new-type 0th- to 4th-order spatial fractional derivative operator. Then the constitutive relation of the non-local medium containing the new operator was established based on the strong-weak non-local continuum theory, with some new mechanical elements constructed. Through different combinations of mechanical elements, several types of non-local fractional derivative constitutive models were obtained: the Kelvin model, the Maxwell model and the Zener model. Based on the correlation between the scattered wave equations and the medium constitutive equations, the expressions and physical meanings of the model parameters were determined, and the creep and stress relaxation of some models were studied. Finally, the effectiveness of the non-local Kelvin model was verified by the case study of creep in sand-bearing soft soil.
- fractional derivative,
- non-local medium,
- constitutive modeling,
- scattering wave equation,
- creep flexibility

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References(34)

References

陈荟键, 朱清锋, 苗鸿臣, 等. 受载结构中SH₀波与裂纹作用的非线性散射场的数值研究[J]. 应用数学和力学, 2023,44(4): 367-380.

(CHEN Huijian, ZHU Qingfeng, MIAO Hongchen, et al. Numerical study of nonlinear scattering characteristics of SH₀ waves encountering cracks in prestressed plates[J].Applied Mathematics and Mechanics,2023,44(4): 367-380. (in Chinese))

[2]邓健, 肖鹏程, 王增贤, 等. 基于黏聚区模型的ENF试件层间裂纹扩展分析[J]. 应用数学和力学, 2022,43(5): 515-523. (DENG Jian, XIAO Pengcheng, WANG Zengxian, et al. Interlaminar crack propagation analysis of ENF specimens based on the cohesive zone model[J].Applied Mathematics and Mechanics,2022,43(5): 515-523. (in Chinese))

[3]MENG C, WEI H, CHEN H, et al. Modeling plasticity of cubic crystals using a nonlocal lattice particle method[J].Computer Methods in Applied Mechanics and Engineering,2021,385: 114069.

[4]FU L, ZHOU X P, BERTO F. A three-dimensional non-local lattice bond model for fracturing behavior prediction in brittle solids[J].International Journal of Fracture,2022,234(1/2): 297-311.

[5]SHEYKHI M, ESKANDARI A, GHAFARI D, et al. Investigation of fluid viscosity and density on vibration of nano beam submerged in fluid considering nonlocal elasticity theory[J].Alexandria Engineering Journal,2023,65: 607-614.

[6]TARASOV V E. General non-local electrodynamics: equations and non-local effects[J].Annals of Physics,2022,445: 169082.

[7]ESEN I, DAIKH A A, ELTAHER M A. Dynamic response of nonlocal strain gradient FG nanobeam reinforced by carbon nanotubes under moving point load [J].The European Physical Journal Plus,2021,136(4): 1-22.

[8]田娅, 秦瑶, 向晶. 一类带有变指数非局部项的反应扩散方程解的爆破行为[J]. 应用数学和力学, 2022,43(10): 1177-1184. (TIAN Ya, QIN Yao, XIANG Jing. Blow-up behaviors of solutions to reaction-diffusion equations with nonlocal sources and variable exponents[J].Applied Mathematics and Mechanics,2022,43(10): 1177-1184. (in Chinese))

[9]KRUMHANSL J A. Generalized continuum field representations for lattice vibrations[J].Solid State Communications,1963,1(6): 198.

[10]ERINGEN A C, WEGNER J L. Nonlocal continuum field theories[J].Applied Mechanics Reviews,2003,56(2): B20-B22.

[11]AIFANTIS E C. On the role of gradients in the localization of deformation and fracture[J].International Journal of Engineering Science,1992,30(10): 1279-1299.

[12]LIM C W, ZHANG G, REDDY J N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation[J].Journal of the Mechanics and Physics of Solids,2015,78: 298-313.

[13]LAZOPOULOS K A. Non-local continuum mechanics and fractional calculus[J].Mechanics Research Communications,2006,33(6): 753-757.

[14]CARPINTERI A, CORNETTI P, SAPORA A. Nonlocal elasticity: an approach based on fractional calculus[J].Meccanica,2014,49(11): 2551-2569.

[15]VAIYAPURI S. Fractional derivative analysis of wave propagation studies using eringen’s nonlocal model with elastic medium support[J].Journal of Vibration Engineering & Technologies,2022,11(8): 3677-3685.

[16]庞国飞, 陈文. 基于Riesz势空间分数阶算子的非局部粘弹性力学元件[J]. 固体力学学报, 2017,38(1): 47-54. (PANG Guofei, CHEN Wen. Nonlocal viscoelastic elements based on Riesz potential space-fractional operator[J].Chinese Journal of Solid Mechanics,2017,38(1): 47-54. (in Chinese))

[17]TARASOV V E, AIFANTIS E C. Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality[J].Communications in Nonlinear Science and Numerical Simulation,2015,22(1/2/3): 197-227.

[18]TARASOV V E. General non-local continuum mechanics: derivation of balance equations[J].Mathematics,2022,10(9): 1427.

[19]WU X, WEN Z, JIN Y, et al. Broadband Rayleigh wave attenuation by gradient metamaterials[J].International Journal of Mechanical Sciences,2021,205: 106592.

[20]HOLM S, NASHOLM S P. Comparison of fractional wave equations for power law attenuation in ultrasound and elastography[J].Ultrasound in Medicine & Biology,2014,40(4): 695-703.

[21]HOLM S, NASHOLM S P. A causal and fractional all-frequency wave equation for lossy media[J].The Journal of the Acoustical Society of America,2011,130(4): 2195-2202.

[22]SAMKO S G, KILBAS A A, MARICEV O I. Fractional integrals and derivations and some applications[J].Minsk: Nauka I Tekhnika,1993,3: 397-414.

[23]CHEN W, HOLM S. Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency[J].The Journal of the Acoustical Society of America,2004,115(4): 1424-1430.

[24]BUCKINGHAM M J. Wave-speed dispersion associated with an attenuation obeying a frequency power law[J].The Journal of the Acoustical Society of America,2015,138(5): 2871-2884.

[25]AIFANTIS E C. On the gradient approach-relation to Eringen’s nonlocal theory[J].International Journal of Engineering Science,2011,49(12): 1367-1377.

[26]ASRARI R, EBRAHIMI F, KHEIRIKHAH M M, SAFARI K H. Buckling analysis of heterogeneous magneto-electro-thermo-elastic cylindrical nanoshells based on nonlocal strain gradient elasticity theory[J].Mechanics Based Design of Structures and Machines,2022,50(3): 817-840.

[27]AIFANTIS E C. Fractional generalizations of gradient mechanics[M]//TARASOV V E.Handbook of Fractional Calculus With Applications,2019: 241-262.

[28]CHEN W, FANG J, PANG G, et al. Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation[J].The Journal of the Acoustical Society of America,2017,141(1): 244-253.

[29]TREEBY B E, COX B T. Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian[J].The Journal of the Acoustical Society of America,2010,127(5): 2741-2748.

[30]HOLM S, NSHOLM S P, PRIEUR F, et al. Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations[J].Computers & Mathematics With Applications,2013,66(5): 621-629.

[31]KHOKHLOV N, FAVORSKAYA A, STETSYUK V, et al. Grid-characteristic method using Chimera meshes for simulation of elastic waves scattering on geological fractured zones[J].Journal of Computational Physics,2021,446: 110637.

[32]OHARA Y, REMILLIEUX M C, ULRICH T J, et al. Exploring 3D elastic-wave scattering at interfaces using high-resolution phased-array system[J].Scientific Reports,2022,12: 8291.

[33]方俊. 超声无损检测中声波散射衰减的分数阶导数建模研究[J]. 冶金与材料, 2019,39(1): 7-11. (FANG Jun. Fractional derivative modeling of frequency dependent scattering attenuation in ultrasonic non-destructive testing[J].Metallurgy and Materials,2019,39(1): 7-11. (in Chinese))

[34]孙海忠, 张卫. 一种分析软土黏弹性的分数导数开尔文模型[J]. 岩土力学, 2007,28(9): 1983-1986. (SUN Haizhong, ZHANG Wei. Analysis of soft soil with viscoelastic fractional derivative Kelvin model[J].Rock and Soil Mechanics,2007,28(9): 1983-1986. (in Chinese))

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