基于变阶分数阶导数的岩石蠕变模型
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摘要
蠕变模型是描述岩石流变行为的主要形式。建立一个参数少、模拟性能好的岩石蠕变模型是岩石蠕变研究的一个重要方向。为此,从分数阶蠕变元件的物理意义出发,将材料的蠕变过程划分为弹性、弹性、黏塑性3个阶段,并通过引入变阶分数阶导数来描述这3个阶段。然而当载荷应力超过屈服应力时,岩石中微观裂纹会萌生、扩展和演化,导致蠕变损伤的积累并在黏塑性蠕变后期引起加速蠕变的发生。因此,考虑到损伤演化对岩石蠕变的影响,在加速蠕变阶段引入损伤系数来描述这一阶段应变的非线性增长。基于以上分析,在Scott-Blair分数阶元件和变系数分数阶元件的基础上,提出一种变阶分数阶非线性黏弹塑性蠕变模型,并将模型拓展到三维情形。平顶山深部煤体三轴蠕变实验的分段拟合结果表明,基于变阶分数阶导数的蠕变模型与实验数据吻合较好。这也验证了将分数阶导数的变阶看作是一个阶跃函数是合理的、可靠的。此外,通过进一步的参数拟合,在现有实验结果的基础上确定模型中的参数。结果表明,所提出的理论模型能较好地描述材料的蠕变特性,与实验数据吻合较好。
The creep model is a main form to describe the rheological behavior of rocks. An important focus of research on rock creep is to develop a model with fewer parameters and better simulation performance. From the aspect of the physical meaning of fractional creep element,in this study,the creep process of materials was divided into elastic,viscoelastic and viscoplastic stages,and the variable-order fractional derivative was introduced to describe these three segments. In the case of the loading stress exceeding the yield stress,nevertheless,the creep damage accumulates and the accelerating creep occurs due to that microscopic cracks initiate,expand and evolve.Therefore,a damage coefficient was introduced to describe the non-linear strain at the accelerating creep stage considering the influence of damage evolution. A variable order non-linear visco-elastic-plastic creep model was proposed based on Scott-Blair fractional element and time-dependent fractional element and further generalized to three-dimension situations. A series of three-dimensional creep experiments of deep coal from Pingdingshan were analyzed by segment treatment,showing that the creep model based on the variable order fractional derivative is in good agreement with the experimental data. It is also proved that it is reasonable and reliable to regard the variable order of fractional derivative as a step function. In addition,the parameters of the model were determined on the basis of fitting the existing experimental results. The results show that the theoretical model proposed in this paper can well describe the creep properties of the material.
The creep model is a main form to describe the rheological behavior of rocks. An important focus of research on rock creep is to develop a model with fewer parameters and better simulation performance. From the aspect of the physical meaning of fractional creep element,in this study,the creep process of materials was divided into elastic,viscoelastic and viscoplastic stages,and the variable-order fractional derivative was introduced to describe these three segments. In the case of the loading stress exceeding the yield stress,nevertheless,the creep damage accumulates and the accelerating creep occurs due to that microscopic cracks initiate,expand and evolve.Therefore,a damage coefficient was introduced to describe the non-linear strain at the accelerating creep stage considering the influence of damage evolution. A variable order non-linear visco-elastic-plastic creep model was proposed based on Scott-Blair fractional element and time-dependent fractional element and further generalized to three-dimension situations. A series of three-dimensional creep experiments of deep coal from Pingdingshan were analyzed by segment treatment,showing that the creep model based on the variable order fractional derivative is in good agreement with the experimental data. It is also proved that it is reasonable and reliable to regard the variable order of fractional derivative as a step function. In addition,the parameters of the model were determined on the basis of fitting the existing experimental results. The results show that the theoretical model proposed in this paper can well describe the creep properties of the material.
引文
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[9]MARKELOVA A,TRIFONOV A,OLKHOVSKAYA V.Two-phase Filtration model for nonlinear viscoplastic oil and hard water drive[J].Applied Mechanics and Materials,2014,698:679-682.
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[11]ZHOU H W,WANG C P,HAN B B,et al.A creep constitutive model for salt rock based on fractional derivatives[J].International Journal of Rock Mechanics and Mining Sciences,2011,48(1):116-121.
[12]MINO G D,AIREY G,PAOLA M D,et al.Linear and nonlinear fractional hereditary constitutive laws of asphalt mixtures[J].Journal of Civil Engineering and Management,2016,22(7):882-889.
[13]ZHOU H W,WANG C P,MISHNAEVSKY L J,et al.A fractional derivative approach to full creep regions in salt rock[J].Mech Time Depend Mater,2013,17(3):413-425.
[14]CHEN L,WANG C P,LIU J F,et al.A damage-mechanism-based creep model considering temperature effect in granite[J].Mechanics Research Communications,2014,56:76-82.
[15]WU F,CHEN J,ZOU Q L.A nonlinear creep damage model for salt rock[J].International Journal of Damage Mechanics,2019,28(5):758-771.
[16]RAMIREZ L,COIMBRA C.A variable order constitutive relation for viscoelasticity[J].Ann Phys,2007,16(8):543-552.
[17]WU F,LIU J F,WANG Jun.An improved Maxwell creep model for rock based on variable-order fractional derivatives[J].Environmental Earth Sciences,2015,73(11):6 965-6 971.
[18]BLAIR G W S.The role of psychophysics in rheology[J].Journal of Colloid Science,1947,2(1):21-32.
[19]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of fractional differential equations[M].Amsterdam:Elsevier,2006:204.
[20]ZHOU H W,LIU D,LEI G,et al.The creep-damage model of salt rock based on fractional derivative[J].Energies,2018,11(9):2 349.
[21]COLOMBARO I,GARRA R,GIUSTI A,et al.Scott-Blair models with time-varying viscosity[J].Applied Mathematics Letters,2018,86:57-63.
[22]张龙云.硬脆性岩体卸荷非线性流变模型及工程应用[博士学位论文][D].济南:山东大学,2016.(ZHANG Longyun.Nonlinear unloading rheological model for hard brittle rock mass and its engineering application[Ph.D.Thesis][D].Jinan:Shandong University,2016.(in Chinese))
[2]张玉军,张维庆.锚固的双重孔隙-裂隙岩体流变模型及其地下洞室二维有限元分析[J].岩石力学与工程学报,2018,37(增1):3 300-3 309.(ZHANG Yujun,ZHANG Weiqing.A bolted rheological model for dual-pore-fracture rock mass and 2D FEM analyses for underground cavern[J].Chinese Journal of Rock Mechanics and Engineering,2018,37(Supp.1):3 300-3 309.(in Chinese))
[3]OLDHAM K B,SPANIER J.The fractional calculus[J].Mathematical Gazette,1974,56:396-400.
[4]SABATIER J,AGRAWAL O P,MACHADO J A T.Advances in Fractional Calculus[M].The Netherlands:Springer,2007:115-116.
[5]MAINARDI F.Fractional calculus and waves in linear viscoelasticity:an introduction to mathematical models[M].London:Imperial College Press,2010:368.
[6]WILKIE K P,DRAPACA C S,SIVALOGANATHAN S.A nonlinear viscoelastic fractional derivative model of infant hydrocephalus[J].Applied Mathematics and Computation,2011,217(21):8 693-8 704.
[7]MüLLER S,K?STNER M,BRUMMUND J,et al.A nonlinear fractional viscoelastic material model for polymers[J].Computational Materials Science,2011,50(10):2 938-2 949.
[8]XU H,JIANG X.Creep constitutive models for viscoelastic materials based on fractional derivatives[J].Computers and Mathematics with Applications,2017,73(6):1 377-1 384.
[9]MARKELOVA A,TRIFONOV A,OLKHOVSKAYA V.Two-phase Filtration model for nonlinear viscoplastic oil and hard water drive[J].Applied Mechanics and Materials,2014,698:679-682.
[10]YIN D,ZHANG W,CHENG C,et al.Fractional time dependent Bingham model for muddy clay[J].Journal of Non-Newtonian Fluid Mechanics,2012,187-188:32-35.
[11]ZHOU H W,WANG C P,HAN B B,et al.A creep constitutive model for salt rock based on fractional derivatives[J].International Journal of Rock Mechanics and Mining Sciences,2011,48(1):116-121.
[12]MINO G D,AIREY G,PAOLA M D,et al.Linear and nonlinear fractional hereditary constitutive laws of asphalt mixtures[J].Journal of Civil Engineering and Management,2016,22(7):882-889.
[13]ZHOU H W,WANG C P,MISHNAEVSKY L J,et al.A fractional derivative approach to full creep regions in salt rock[J].Mech Time Depend Mater,2013,17(3):413-425.
[14]CHEN L,WANG C P,LIU J F,et al.A damage-mechanism-based creep model considering temperature effect in granite[J].Mechanics Research Communications,2014,56:76-82.
[15]WU F,CHEN J,ZOU Q L.A nonlinear creep damage model for salt rock[J].International Journal of Damage Mechanics,2019,28(5):758-771.
[16]RAMIREZ L,COIMBRA C.A variable order constitutive relation for viscoelasticity[J].Ann Phys,2007,16(8):543-552.
[17]WU F,LIU J F,WANG Jun.An improved Maxwell creep model for rock based on variable-order fractional derivatives[J].Environmental Earth Sciences,2015,73(11):6 965-6 971.
[18]BLAIR G W S.The role of psychophysics in rheology[J].Journal of Colloid Science,1947,2(1):21-32.
[19]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of fractional differential equations[M].Amsterdam:Elsevier,2006:204.
[20]ZHOU H W,LIU D,LEI G,et al.The creep-damage model of salt rock based on fractional derivative[J].Energies,2018,11(9):2 349.
[21]COLOMBARO I,GARRA R,GIUSTI A,et al.Scott-Blair models with time-varying viscosity[J].Applied Mathematics Letters,2018,86:57-63.
[22]张龙云.硬脆性岩体卸荷非线性流变模型及工程应用[博士学位论文][D].济南:山东大学,2016.(ZHANG Longyun.Nonlinear unloading rheological model for hard brittle rock mass and its engineering application[Ph.D.Thesis][D].Jinan:Shandong University,2016.(in Chinese))
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