基于分数阶微积分的岩石非线性黏弹性应力松弛模型研究
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摘要
基于Riemann-Liouville分数阶微积分理论,采用Koeller弹壶元件替换整数阶Poynting-Thomson模型中的Newton元件,结合Laplace正逆变换和Mittag-Leffler函数,构建了一种新的岩石非线性黏弹性应力松弛模型-分数阶Poynting-Thomson模型.应用岩石流变仪对三峡库区巴东组粉砂质泥岩进行了单轴压缩应力松弛试验.依据试验结果,分别采用整数阶Poynting-Thomson模型、整数阶五元件模型(H‖M‖M)和分数阶Poynting-Thomson模型对应力松弛试验数据进行拟合分析,比较了各模型的辨识精度.在此基础上,分析了分数阶Poynting-Thomson模型参数的敏感性,揭示了应变水平、分数阶阶数和黏滞系数对岩石应力松弛的影响规律.研究结果表明,分数阶Poynting-Thomson模型能够更准确地描述岩石的应力松弛特性.
According to the Riemann-Liouville fractional calculus theory and Mittag-Leffler function,a nonlinear viscoelastic stress relaxation model of rock was proposed by replacing the Newton dashpot in the classical Poynting-Thomson model with Koeller spring-pot,namely fractional calculus Poynting-Thomson model.The uniaxial compressive stress relaxation tests of silty mudstone from Badong Formation strata in the Three Gorges Reservoir area were performed with the RLJW-2000 servo-controlled rheology testing machine.Then the integer calculus Poynting-Thomson model,the fractional calculus Poynting-Thomson model and the integer calculus five-component model were used to fit the test results of silty mudstone.The parameters of three models were determined,and the fitting accuracy of the models was compared.Furthermore,a sensitivity study for the parameters of fractional calculus Poynting-Thomson model was carried out,showing the effects of strain level,fractional order and viscosity coefficient on stress relaxation behavior of silty mudstone.The results show that the fractional calculus Poynting-Thomson model can more accurately reflect the stress relaxation behavior of silty mudstone than other two models.
According to the Riemann-Liouville fractional calculus theory and Mittag-Leffler function,a nonlinear viscoelastic stress relaxation model of rock was proposed by replacing the Newton dashpot in the classical Poynting-Thomson model with Koeller spring-pot,namely fractional calculus Poynting-Thomson model.The uniaxial compressive stress relaxation tests of silty mudstone from Badong Formation strata in the Three Gorges Reservoir area were performed with the RLJW-2000 servo-controlled rheology testing machine.Then the integer calculus Poynting-Thomson model,the fractional calculus Poynting-Thomson model and the integer calculus five-component model were used to fit the test results of silty mudstone.The parameters of three models were determined,and the fitting accuracy of the models was compared.Furthermore,a sensitivity study for the parameters of fractional calculus Poynting-Thomson model was carried out,showing the effects of strain level,fractional order and viscosity coefficient on stress relaxation behavior of silty mudstone.The results show that the fractional calculus Poynting-Thomson model can more accurately reflect the stress relaxation behavior of silty mudstone than other two models.
引文
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[15]徐国文,何川,胡雄玉,等.基于分数阶微积分的改进西原模型及其参数智能辨识[J].岩土力学,2015,36(S2):132-139Xu Guowen,He Chuan,Hu Xiongyu,et al.A modified Nishihara model based on fractional calculus theory and its parameter intelligent identification[J].Rock and Soil Mechanics,2015,36(S2):132-139
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[17]Koeller R C.Applications of fractional calculus to the theory of viscoelasticity[J].Journal of Applied Mechanics,1984,51(2):299-307
[18]范广勤.岩土工程流变力学[M].北京:煤炭工业出版社,1993:46-47Fan Guangqin.Rheology mechanics of geotechnical engineering[M].Beijing:China Coal Industry Publishing House,1993:46-47
[19]Podlubny L.Fractional differential equations[M].New York:Academic Press,1999,130-132
[20]张龙云.硬脆性岩体卸荷非线性流变模型及工程应用[D].济南:山东大学,2016Zhang Longyun.Nonlinear unloading rheological model for hard brittle rock mass and its engineering application[D].Jinan:Shandong University,2016
[21]于怀昌,周敏,刘汉东,等.粉砂质泥岩三轴压缩应力松弛特性试验研究[J].岩石力学与工程学报,2011,30(4):803-811Yu Huaichang,Zhou Min,Liu Handong,et al.Experimental investigation on stress relaxation properties of silty mudstone under triaxial compression[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(4):803-811
[22]International Society for Rock Mechanics.Suggested methods for determining the strength of rock material in triaxial compression[J].International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts,1978,15(2):47-51
[23]杨挺青,罗文波,徐平,等.黏弹性理论与应用[M].北京:科学出版社,2004:13-19Yang Tingqing,Luo Wenbo,Xu Ping,et al.Theory of viscoelasticity and its applications[M].Beijing:Science Press,2004:13-19
[24]朱杰兵,汪斌,邬爱清.锦屏水电站绿砂岩三轴卸荷流变试验及非线性损伤蠕变本构模型研究[J].岩石力学与工程学报,2010,29(3):528-534Zhu Jiebing,Wang Bin,Wu Aiqing.Study of unloading triaxial rheological tests and its nonlinear damage constitutive model of Jinping hydropower station green sandstone[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(3):528-534
[25]Ma L,Daemen J J K.An experimental study on creep of welded tuff[J].International Journal of Rock Mechanics and Mining Sciences,2006,43(2):282-291
[2]于怀昌,李亚丽,刘汉东.粉砂质泥岩三轴压缩应力松弛本构模型研究[J].煤炭学报,2011,36(8):1258-1263Yu Huaichang,Li Yali,Liu Handong.Study of stress relaxation model of silty mudstone under triaxial compression[J].Journal of China Coal Society,2011,36(8):1258-1263
[3]熊良宵,杨林德,张尧.绿片岩多轴受压应力松弛试验研究[J].岩土工程学报,2010,32(8):1158-1165Xiong Liangxiao,Yang Linde,Zhang Yao.Stress relaxation tests on green schist speciments under multi-axial compression[J].Chinese Journal of Geotechnical Engineering,2010,32(8):1158-1165
[4]李锐铎,乐金朝.基于分数阶导数的软土非线性流变本构模型[J].应用基础与工程科学学报,2014,22(5):856-864Li Ruiduo,Yue Jinchao.Nonlinear rheological constitute of soft soil based on fractional order derivative theory[J].Journal of Basic Science and Engineering,2014,22(5):856-864
[5]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&Mining Sciences,2011,48:116-121
[6]郭佳奇,乔春生,徐冲,等.基于分数阶微积分的Kelvin-Voigt流变模型[J].中国铁道科学,2009,30(4):2-5Guo Jiaqi,Qiao Chunsheng,Xu Chong,et al.The Kelvin-Voigt rheological model based on fractional calculus[J].China Railway Science,2009,30(4):2-5
[7]宋勇军,雷胜友,韩铁林.一种新的岩石非线性黏弹塑性流变模型[J].岩土力学,2012,33(7):2076-2080Song Yongjun,Lei Shengyou,Han Tielin.A new nonlinear viscoelasto-plastic rheological model for rocks[J].Rock and Soil Mechanics,2012,33(7):2076-2080
[8]丁靖洋,周宏伟,刘迪,等.盐岩分数阶三元件本构模型研究[J].岩石力学与工程学报,2014,33(4):672-678Ding Jingyang,Zhou Hongwei,Liu Di,et al.Research on fractional derivative three elements model of salt rock[J].Chinese Journal of Rock Mechanics and Engineering,2014,33(4):672-678
[9]王军保,刘新荣,张倩倩,等.芒硝蠕变特性及本构模型研究[J].四川大学学报(工程科学版),2015,47(5):78-85Wang Junbao,Liu Xinrong,Zhang Qianqian,et al.Study on creep properties and constitutive model of thenardite[J].Journal of Sichuan University(Engineering Science Edition),2015,47(5):78-85
[10]陈家瑞,浦海,肖成.基于分数阶理论的破碎泥岩流变模型试验研究[J].中国矿业大学学报,2015,44(6):996-1001Chen Jiarui,Pu Hai,Xiao Cheng.Research on rheology model of broken mudstone based on the fractional theory[J].Journal of China University of Mining&Technology,2015,44(6):996-1001
[11]Wu Fei,Liu Jianfeng,Wang Jun.An improved Maxwell creep model for rock based on variable-order fractional derivatives[J].Environmental Earth Sciences,2015,73:6965-6971
[12]Kang Jianhong,Zhou Fubao,Liu Chun,et al.A fractional non-linear creep model for coal considering damage effect and experimental validation[J].International Journal of Non-Linear Mechanics,2015,76:20-28
[13]何志磊,朱珍德,朱明礼,等.基于分数阶导数的非定常蠕变本构模型研究[J].岩土力学,2016,37(3):737-744He Zhilei,Zhu Zhende,Zhu Mingli,et al.An unsteady creep constitutive model based on fractional order derivatives[J].Rock and Soil Mechanics,2016,37(3):737-744
[14]陈亮,陈寿根,张恒,等.基于分数阶微积分的非线性黏弹塑性蠕变模型[J].四川大学学报(工程科学版),2013,45(3):7-11Chen Liang,Chen Shougen,Zhang Heng,et al.An onlinear viscoelasto-plastic creep model based on fractional calculus[J].Journal of Sichuan University(Engineering Science Edition),2013,45(3):7-11
[15]徐国文,何川,胡雄玉,等.基于分数阶微积分的改进西原模型及其参数智能辨识[J].岩土力学,2015,36(S2):132-139Xu Guowen,He Chuan,Hu Xiongyu,et al.A modified Nishihara model based on fractional calculus theory and its parameter intelligent identification[J].Rock and Soil Mechanics,2015,36(S2):132-139
[16]Nonnenmacher T F,Metzler R.On the Riemann-Liouville fractional calculus and some recent application[J].Fractals-complex Geometry Patterns&Scaling in Nature&Society,1995,3(3):557-566
[17]Koeller R C.Applications of fractional calculus to the theory of viscoelasticity[J].Journal of Applied Mechanics,1984,51(2):299-307
[18]范广勤.岩土工程流变力学[M].北京:煤炭工业出版社,1993:46-47Fan Guangqin.Rheology mechanics of geotechnical engineering[M].Beijing:China Coal Industry Publishing House,1993:46-47
[19]Podlubny L.Fractional differential equations[M].New York:Academic Press,1999,130-132
[20]张龙云.硬脆性岩体卸荷非线性流变模型及工程应用[D].济南:山东大学,2016Zhang Longyun.Nonlinear unloading rheological model for hard brittle rock mass and its engineering application[D].Jinan:Shandong University,2016
[21]于怀昌,周敏,刘汉东,等.粉砂质泥岩三轴压缩应力松弛特性试验研究[J].岩石力学与工程学报,2011,30(4):803-811Yu Huaichang,Zhou Min,Liu Handong,et al.Experimental investigation on stress relaxation properties of silty mudstone under triaxial compression[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(4):803-811
[22]International Society for Rock Mechanics.Suggested methods for determining the strength of rock material in triaxial compression[J].International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts,1978,15(2):47-51
[23]杨挺青,罗文波,徐平,等.黏弹性理论与应用[M].北京:科学出版社,2004:13-19Yang Tingqing,Luo Wenbo,Xu Ping,et al.Theory of viscoelasticity and its applications[M].Beijing:Science Press,2004:13-19
[24]朱杰兵,汪斌,邬爱清.锦屏水电站绿砂岩三轴卸荷流变试验及非线性损伤蠕变本构模型研究[J].岩石力学与工程学报,2010,29(3):528-534Zhu Jiebing,Wang Bin,Wu Aiqing.Study of unloading triaxial rheological tests and its nonlinear damage constitutive model of Jinping hydropower station green sandstone[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(3):528-534
[25]Ma L,Daemen J J K.An experimental study on creep of welded tuff[J].International Journal of Rock Mechanics and Mining Sciences,2006,43(2):282-291