K-Hilfer Prabhakar Variable Order Fractional Creep Model And Its Numerical Method For Cemented Backfill Materials

Backfill mining is a large-scale coal mining method,which meets the objective requirements of energy conservation and environmental protection in China.Cemented backfill materials are widely used in backfill mining due to their good bearing capacity and creep properties.The study on the creep characteristics of cemented backfill materials is an important prerequisite for the accurate prediction of the displacement of overlying strata and surface subsidence during mining.The growing types of creep models has laid a theoretical foundation for the study of the creep properties of backfill.Some effective measures like increasing the number of elements,introducing damage factors and piecewise formulation of constitutive equations have taken in order to describe the variation characteristics of the strain deviation and volumetric strain of cemented backfill materials in the deceleration creep stage,steady-state creep stage and accelerated creep stage.How to use the least elements to construct the constitutive equation with the same form in different creep stages is the focus of the research on the creep model of cemented backfill materials.In this thesis,the Maxwell model is modified according to the variation characteristics of the deviational strain rate of cemented backfill materials coupled with the memory and heredity of k-HilferPrabhakar fractional derivative,and the creep model of variable order fractional derivative is systematically studied by using laboratory tests,theoretical analysis and numerical simulation.The main research results are as follows:(1)Coal gangue from a mine in Anhui Province has been used as aggregate and composite silicate cement(PC32.5R)and fly ash as cementitious materials to prepare the cemented backfill specimens.Uniaxial compression(creep)tests were completed using the rock mechanics test system MTS815.02.The effect of the aggregate particle size distribution’s fractal dimension on the strength properties,dilatancy deformation,dilatancy initiation stress,and damage zone of the cemented backfill materials was analyzed using uniaxial and triaxial compression tests.Conventional triaxial compression graded loading creep tests were used to investigate the effect of the fractal dimension of the aggregate particle size distribution and the confining pressure on the creep rate of the cemented backfill materials.The microstructure of the cemented backfill specimens with varying fractal dimensions of the aggregate particle size distribution was observed using scanning electron microscopy to analyze the structural characteristics of the cemented backfill specimens’ voids(pores and fractures).(2)Based on the creep test results of conventional triaxial creep loading test,the variation characteristics of the deviatoric strain rate of cemented backfill materials are analyzed.A modified Maxwell model(kHP variable order fractional Maxwell model)is proposed,which combines the memory and heredity of k-Hilfer-Prabhakar fractional derivative with the stress-dependent order.The kHP variable order fractional Maxwell model contains nine parameters,two of them inherit the genes of the classical Maxwell model,three of them reflect the time-varying character of the order of the fractional derivative,and the other parameters reflect the special features of the k-HilferPrabhakar fractional derivative.Its advantage lies in the unnecessary piecewise of constitutive equation according to the magnitude of stress.By using Laplace transform,the general solution of strain deviation and volumetric strain in the process of conventional triaxial compression rheology(creep and/or relaxation)and the special solution of the creep problem under graded loading are derived.(3)According to the definition of k-Hilfer-Prabhakar fractional derivative,the difference scheme of initial value of strain deviation and volumetric strain of kHP variable order fractional Maxwell model is constructed.The methods to overcome or weaken the singularity of the kernel function are explored,and the truncation error of the kernel function expressed in series is analyzed.(4)A particle swarm optimization algorithm for parameter optimization of the kHP variable order fractional Maxwell model is designed based on the fitness of the relative error between the difference solution of strain deviation and volumetric strain and the test value.How to define fitness function,how to allocate particle swarm size and state update times,and how to adjust the value range of learning factor and inertia weight are considered seriously.The aggregation effect of particle swarm is analyzed through examples.The particle swarm optimization algorithm is used to invert the nine parameters of the kHP variable order fractional Maxwell model.The influence of the fractal dimension of aggregate size distribution and confining pressure on the parameters of the kHP variable fractional Maxwell model is obtained through inversion.(5)The mechanical model of random mixed cemented backfill materials is established,which is composed of four units(ED1～ED4)with the fractal dimensions of aggregate size distribution of 2.2106,2.4150,2.6084 and 2.7824,respectively.The kHP variable order fractional Maxwell model is input into the constitutive relationship of the random mixed backfill through the VUMAT interface.Using Abaqus/Explicit solver,Mises equivalent stress nephogram and strain deviation time-history curve of random mixed cemented backfill materials are obtained,and the influence of units ratio on the creep of random mixed cemented backfill materials is analyzed.There are 95 figures,36 tables and 196 references in this dissertation.