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fractional derivative statistical damage model of unsaturated expansive soil based on unified hydraulic effect

山地科学学报(英文) 2023年第9期20卷 2769-2782页

作者： ZHANG Hua WANG Peng School of Architecture and Civil Engineering Chengdu UniversityChengdu 610106China Sichuan Provincial Engineering Research Center of City Solid Waste Energy and Building Materials Conversion and Utilization Technology Chengdu UniversityChengdu 610106China Geotechnical Engineering Institute Sichuan Institute of Building ResearchChengdu 610081China Sichuan Engineering Research Center for Mechanical Properties and Engineering Technology of Unsaturated Soils Chengdu UniversityChengdu 610106China

Unsaturated expansive soil is widely distributed in China and has complex engineering properties.This paper proposes the unified hydraulic effect shear strength theory of unsaturated expansive soil based on the effective stress principle,swelling force principle,and soil–water characteristics.Considering the viscoelasticity and structural damage of unsaturated expansive soil during loading,a fractional hardening–damage model of unsaturated expansive soil was established.The model parameters were established on the basis of the proposed calculation method of shear strength and the triaxial shear experiment on unsaturated expansive soil.The proposed model was verified by the experimental data and a traditional damage model.The proposed model can satisfactorily describe the entire process of the strain-hardening law of unsaturated expansive soil.Finally,by investigating the damage variables of the proposed model,it was found that:(a)when the values of confining pressure and matric suction are close,the coupling of confining pressure and matric suction contributes more to the shear strength;(b)there is a damage threshold for unsaturated expansive soil,and is mainly reflected by strength criterion of infinitesimal body;(c)the strain hardening law of unsaturated expansive soil is mainly reflected by fractional derivative operator.

关键词： Unsaturated expansive soil Unified hydraulic effect Shear strength theory Hardening-damage model fractional derivative

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A creep model for ultra-deep salt rock considering thermal-mechanical damage under triaxial stress conditions

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Journal of Rock Mechanics and Geotechnical Engineering 2024年第2期16卷 588-596页

作者： Chao Liang Jianfeng Liu Jianxiong Yang Huining Xu Zhaowei Chen Lina Ran College of Water Resource and Hydropower Sichuan UniversityChengdu610065China CNPC Key Laboratory of Oil and Gas Underground Storage Engineering Langfang065007China CNPC Engineering Technology R&D Company Limited Beijing102206China

To investigate the specific creep behavior of ultra-deep buried salt during oil and gas exploitation,a set of triaxial creep experiments was conducted at elevated temperatures with constant axial pressure and unloading confining pressure conditions.Experimental results show that the salt sample deforms more significantly with the increase of applied temperature and deviatoric loading.The accelerated creep phase is not occurring until the applied temperature reaches 130℃,and higher temperature is beneficial to the occurrence of accelerated creep.To describe the specific creep behavior,a novel three-dimensional(3D)creep constitutive model is developed that incorporates the thermal and mechanical variables into mechanical elements.Subsequently,the standard particle swarm optimization(SPSO)method is adopted to fit the experimental data,and the sensibility of key model parameters is analyzed to further illustrate the model function.As a result,the model can accurately predict the creep behavior of salt under the coupled thermo-mechanical effect in deep-buried condition.Based on the research results,the creep mechanical behavior of wellbore shrinkage is predicted in deep drilling projects crossing salt layer,which has practical implications for deep rock mechanics problems.

关键词： Creep experiments Creep model Thermal and mechanical damage fractional derivative

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Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

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Acta Mechanica Sinica 2021年第3期37卷 507-515,I0004页

作者： Ya-Hui Sun Yong-Ge Yang Wei Xu School of Applied Mathematics Guangdong University of TechnologyGuangzhou 510520China Department of Applied Mathematics Northwestern Polytechnical UniversityXi'an 710072China

This paper investigates the stochastic P-bifurcation(SPB)of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise.Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function(PDF)of the system response.Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function.Numerical results are given to show the effectiveness of the proposed approach.Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions.

关键词： Stochastic P-bifurcation fractional derivative Stochastic averaging method Gaussian white noise

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UNSTEADY FLOWS OF A GENERALIZED SECOND GRADE FLUID WITH THE fractional derivative MODEL BETWEEN TWO PARALLEL PLATES

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Acta Mechanica Sinica 2004年第5期20卷 471-476页

作者：谭文长徐明喻 1 Department of Mechanics and Engineering Science & State Key Laboratory for Turbulence and Complex System Research Peking UniversityBeijing 100871China Institute of Mathematics and Systematical Science Shandong UniversityJinan 250100China

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus.The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates.In addition,the solutions of the shear stresses at the plates are also determined.

关键词： fractional derivative unsteady flows generalized second grade fluid parallel plates

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Conservatory of Kaup-Kupershmidt Equation to the Concept of fractional derivative with and without Singular Kernel

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Acta Mathematicae Applicatae Sinica 2018年第2期34卷 351-361页

作者： Abdon ATANGANA Emile Franc Doungmo GOUFO Institute for Groundwater Studies University of the Free State 9300 Bloemfontein South Africa Department of Mathematical Sciences University of South Africa Florida 0003 South Africa

Making use of the traditional Caputo derivative and the newly introduced Caputo-Fabrizio deriva- tive with fractional order and no singular kernel, we extent the nonlinear Kaup-Kupershmidt to the span of fractional calculus. In the analysis, different methods of fixed-point theorem together with the concept of piccard L-stability are used, allowing us to present the existence and uniqueness of the exact solution to models with both versions of derivatives. Finally, we present techniques to perform some numerical simulations for both non-linear models and graphical simulations are provided for values of the order α = 1.00; 0.90. Solutions are shown to behave similarly to the standard well-known traveling wave solution of Kaup-Kupershmidt equation.

关键词： Kaup-Kupershmidt equation fractional derivative fixed-point theorem numerical simulations

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Well-Posedness of Equations with fractional derivative

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Acta Mathematica Sinica,English Series 2010年第7期26卷 1223-1232页

作者： Shang Quan BU Department of Mathematical Science Tsinghua University Beijing 100084 P. R. China

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t）＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.

关键词： well-posedness fractional derivative fractional Sobolev spaces Fourier multipliers

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Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative

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Earthquake Engineering and Engineering Vibration 2003年第1期2卷 133-140页

作者：叶昆李黎唐家祥 School of Civil Engineering Huazhong University of Science and TechnologyWuhan 430074China Doctoral Candidate School of Civil Engineering Huazhong University of Science and TechnologyWuhan 430074China Professor

Viscoelastic dampers,as spplementary energy dissipation devices,have been used in building structures un- der seismic excitation or wind loads.Different analytical models have been proposed to describe their dynamic force deform- ation characteristics.Among these analytieal models,the fractional derivative models have attracted more attention as they can capture the frequency dependence of the material stiffness and damping properties observed from tests very well.In this paper,a Fourier-transform-based technique is presented to obtain the fractional unit impulse function and the response of structures with added viscoelastic dampers whose foree-detormation relationship is described by a fractional derivative mod- el.Then,a Duhamel integral-type expression is suggested for the response analysis of a fractional damped dynamie system subjected to deterministic or random excitation.Through numerical verification,it is shown that viscoelastic dampers are ef- fective in reducing structural responses over a wide frequency range,and the proposed schmnes can be used to accurately predict the stochastic seismic response of structures with added viscoelastic dampers described by a Kelvin model wills frac- tional derivative.

关键词： fractional derivative viscoelastic damper stochastic seismic response

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Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior

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International Journal of Modeling, Simulation, and Scientific Computing 2021年第4期12卷 92-119页

作者： Soufiane Bentout Behzad Ghanbari Salih Djilali Lakshmi Narayan Guin Department of Mathematics and Informatics Belhadj Bouchaib University Center of Ain Temouchent BP 284 RP 46000Algeria Laboratoire d’Analyse Non Linaire et Mathmatiques Appliques Universit de Tlemcen TlemcenAlgeria Department of Engineering Science Kermanshah University of Technology KermanshahIran Department of Biomedical Engineering Faculty of Engineering and Naturel Sciences Bah¸ce¸sehir University 34349 IstanbulTurkey Faculty of Exact and Computer Sciences Mathematic Department Hassiba Benbouali University ChlefAlgeria Department of Mathematics Visva-Bharati Santiniketan-731235 West BengalIndia

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.

关键词： Predator–prey model herd behavior fractional derivative numerical schema backward bifurcation eco-epidemiological model

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Well-Posedness of Equations with fractional derivative via the Method of Sum

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Acta Mathematica Sinica,English Series 2012年第1期28卷 37-44页

作者： Shang Quan BU Department of Mathematical Science University of Tsinghua

We study the well-posedness of the equations with fractional derivative D^αu（t）= Au（t）＋ f（t）（0 ≤ t ≤ 2π）, where A is a closed operator in a Banach space X, 0 〈 α 〈 1 and D^αis the fractional derivative in the sense of Weyl. Although this problem is not always well-posed in L^P（0, 2π; X） or periodic continuous function spaces Cper（[0, 2π]; X）, we show by using the method of sum that it is well-posed in some subspaces of L^P（0, 2π; X） or Cper （[0, 2π]; X）.

关键词： Well-posedness fractional derivative the method of sum real interpolation spaces fractional Sobolev spaces

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fractional order distance regularized level set method with bias correction

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The Journal of China Universities of Posts and Telecommunications 2024年第1期31卷 64-82页

作者： Cai Xiumei He Ningning Wu Chengmao Liu Xiao Liu Hang School of Automation Xi'an University of Posts and TelecommunicationsXi'an 710121China School of Electronic Engineering Xi'an University of Posts and TelecommunicationsXi'an 710121China

The existing level set segmentation methods have drawbacks such as poor convergence,poor noise resistance,and long iteration times.In this paper,a fractional order distance regularized level set segmentation method with bias correction is proposed.This method firstly introduces fractional order distance regularized term to punish the deviation between the level set function(LSF)and the signed distance function.Secondly a series of covering template is constructed to calculate fractional derivative and its conjugate of image pixel.Thirdly introducing the offset correction term and fully using the local clustering property of image intensity,the local clustering criterion of image intensity is defined and integrated with the neighborhood center to obtain the global criterion of image segmentation.Finally,the fractional distance regularization,offset correction,and external energy constraints are combined,and the energy optimization segmentation method for noisy image is established by level set.Experimental results show that the proposed method can accurately segment the image,and effectively improve the efficiency and robustness of exiting state of the art level set related algorithms.

关键词： image segmentation fractional order distance regularization level set function fractional derivative bias correction

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