岩石应变软化本构模型建立及NR-AL法求解研究
|
摘要
针对岩土工程材料应变软化问题及有限元对其数值计算时切线刚度矩阵负定造成求解困难的问题进行研究。建立了基于Drucker-Prager(D-P)强度准则的岩石弹塑性应变软化本构模型,本构积分算法采用一种完全隐式返回映射算法,它具有无条件稳定和精确的特点,详细论述了如何进行本构模型的程序化求解;考虑弧长法在判断切线刚度矩阵正定性导致效率低的缺点,在弹塑性增量有限元方程的迭代计算中尝试采用Newton-Raphson法和arc-length法(NR-AL法)联合迭代求解的思路,即在结构未达到极限荷载前采用NR迭代法,而当结构接近极限荷载时转换为AL法控制迭代,从而使结构越过峰值点进入软化区直至破坏,NR-AL法汲取了2者迭代求解中具有的优势;利用C++语言对所建应变软化模型的本构求解和弹塑性增量有限元方程迭代求解过程给予程序实现,应用所编程序进行数值计算,分析了D-P理想弹塑性模型、应变软化模型、应变硬化模型计算的应力-应变曲线的区别,同时将应变软化模型计算结果与试验数据进行了对比。研究结果表明:所建应变软化本构模型可以较好地模拟岩石材料的峰后软化特性,能够揭示峰后应变软化特性和破坏机制,同时NR-AL法能够求解由于应变软化造成的负刚度问题,也克服了单独使用弧长法时判断切线刚度矩阵正定性效率低的缺点。
Strain softening problem in geotechnical engineering and the difficult solution problem of the finite element numerical calculation due to the negative tangent stiffness of strain softening model are studied. An elastoplastic strain softening constitutive model of rock is established based on the Drucker-Prager strength criteria. A fully implicit return mapping algorithm which has characteristics of the unconditional stability and precision is used to solve the constitutive equation, and how the programmed constitutive model to be solved is discussed in detail. Then, the shortcomings of low efficiency of the arc-length method in judging stiffness matrix is considered, Newton-Raphson scheme and arc-length method(NR-AL method) are combined to iteratively solve the calculation of elastoplastic incremental finite element equations. Namely Newton-Raphson scheme is used before the structure reaching the limit load, and when the structure is close to the limit load, turning to the arc-length method, so that the structure can go over the peak point into the softening phase until destruction. NR-AL method has the advantages in the iterative solution. A program of the built strain softening model and elastoplastic incremental finite element to solve the constitutive equation for the iterative process is compiled using C++ language. The program is applied to numerical calculation, and the stress-strain curves of the idealized elastoplastic model, strain softening and strain hardening model based on the Drucker-Prager strength criteria are comparatively analyzed. The results show that the strain softening constitutive model can simulate the characteristics the post-peak softening of rock material well, and it can reveal the features of the post-peak strain softening and failure mechanism. NR-AL method can solve the negative stiffness problem caused by strain softening and also overcome the shortcomings of low efficiency in judging stiffness matrix using the arc-length only.
Strain softening problem in geotechnical engineering and the difficult solution problem of the finite element numerical calculation due to the negative tangent stiffness of strain softening model are studied. An elastoplastic strain softening constitutive model of rock is established based on the Drucker-Prager strength criteria. A fully implicit return mapping algorithm which has characteristics of the unconditional stability and precision is used to solve the constitutive equation, and how the programmed constitutive model to be solved is discussed in detail. Then, the shortcomings of low efficiency of the arc-length method in judging stiffness matrix is considered, Newton-Raphson scheme and arc-length method(NR-AL method) are combined to iteratively solve the calculation of elastoplastic incremental finite element equations. Namely Newton-Raphson scheme is used before the structure reaching the limit load, and when the structure is close to the limit load, turning to the arc-length method, so that the structure can go over the peak point into the softening phase until destruction. NR-AL method has the advantages in the iterative solution. A program of the built strain softening model and elastoplastic incremental finite element to solve the constitutive equation for the iterative process is compiled using C++ language. The program is applied to numerical calculation, and the stress-strain curves of the idealized elastoplastic model, strain softening and strain hardening model based on the Drucker-Prager strength criteria are comparatively analyzed. The results show that the strain softening constitutive model can simulate the characteristics the post-peak softening of rock material well, and it can reveal the features of the post-peak strain softening and failure mechanism. NR-AL method can solve the negative stiffness problem caused by strain softening and also overcome the shortcomings of low efficiency in judging stiffness matrix using the arc-length only.
引文
[1]韩建新,李术才,李树忱,等.基于强度参数演化行为的岩石峰后应力-应变关系研究[J].岩土力学,2013,34(2):342-346.HAN Jian-xin,LI Shu-cai,LI Shu-chen,et al.Study of post-peak stress-strain relationship of rock material based on evolution of strength parameters[J].Rock and Soil Mechanics,2013,34(2):342-346.
[2]NAGHDI P M,TRAPP J A.The significance of formulating plasticity theory with reference to loading surfaces in strain space[J].International Journal of Engineering Science,1975,13(9):785-797.
[3]阎金安,张宪宏.岩石材料应变软化模型及有限元分析[J].岩土力学,1990,11(1):19-27.YAN Jin-an,ZHANG Xian-hong.Strain softening model of rock material and associated finite element analysis[J].Rock and Soil Mechanics,1990,11(1):19-27.
[4]吴玉忠.岩石应变软化弹塑性有限元分析[J].有色金属,1990,42(1):1-8.WU Yu-zhong.Finite element analysis of elastoplastic rock with strain-softening[J].Nonferrous Metals,1990,42(1):1-8.
[5]王兵,陈炽昭,张金荣.考虑岩石应变软化特性隧道的弹-塑性分析[J].铁道学报,1992,14(2):86-95.WANG Bing,CHEN Chi-zhao,ZHANG Jin-rong.An elasto-plastic analysis of a tunnel in consideration of the strain softening behaviour of rock[J].Journal of the China Railway Society,1992,14(2):86-95.
[6]沈新普,岑章志,徐秉业.弹脆塑性软化本构理论的特点及其数值计算[J].清华大学学报(自然科学版),1995,35(2):22-27.SHEN Xin-pu,CEN Zhang-zhi,XU Bing-ye.The characteristics of elasto-brittle-plastic softening constitutive theory and its numerical calculation[J].Journal of Tsinghua University(Sci.&Tech.),1995,35(2):22-27.
[7]杨超,崔新明,徐水平.软岩应变软化数值模型的建立与研究[J].岩土力学,2002,23(6):695-697,701.YANG Chao,CHUI Xin-ming,XU Shui-ping.Establishment and study of strain-softening numerical constitutive model for soft rock[J].Rock and Soil Mechanics,2002,23(6):695-697,701.
[8]ALONSO E,ALEJANO L R,VARAS F,et al.Ground response curves for rock masses exhibiting strainsoftening behaviour[J].International Journal for Numerical and Analytical Methods in Geomechanics,2003,27(13):1153-1185.
[9]张帆,盛谦,朱泽奇,等.三峡花岗岩峰后力学特性及应变软化模型研究[J].岩石力学与工程学报,2008,27(增刊1):2651-2655.ZHANG Fan,SHENG Qian,ZHU ZE-qi,et al.Study on post-peak mechanical behaviour and strain-softening model of Three Gorges granite[J].Chinese Journal of Rock Mechanics and Engineering,2008,27(Supp.1):2651-2655.
[10]周家文,徐卫亚,李明卫,等.岩石应变软化模型在深埋隧洞数值分析中的应用[J].岩石力学与工程学报,2009,28(6):1116-1127.ZHOU Jia-wen,XU Wei-ya,LI Ming-wei,et al.Application of rock strain softening model to numerical analysis of deep tunnel[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(6):1116-1127.
[11]王水林,王威,吴振君.岩土材料峰值后区强度参数演化与应力-应变曲线关系研究[J].岩石力学与工程学报,2010,29(8):1524-1529.WANG Shui-lin,WANG Wei,WU Zhen-jun.Study of relationship between evolution of post-peak strength parameters and stress-strain curves of geomaterial[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(8):1524-1529.
[12]李文婷,李树忱,冯现大,等.基于摩尔-库仑准则的岩石峰后应变软化力学行为研究[J].岩石力学与工程学报,2011,30(7):1460-1466.LI Wen-ting,LI Shu-chen,FENG Xian-da,et al.Study of post-peak strain softening mechanical properties of rock based on Mohr-Coulomb criterion[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(7):1460-1466.
[13]李英杰,张顶立,刘保国.考虑变形模量劣化的应变软化模型在FLAC3D中的开发与验证[J].岩土力学,2011,32(增刊2):647-652,659.LI Ying-jie,ZHANG Ding-li,LIU Bao-guo.Development and verification of strain-softening model considering deformation modulus degradation in FLAC3D[J].Rock and Soil Mechanics,2011,32(Supp.2):647-652,659.
[14]赵启林,牛海清,卓家寿.应变软化材料的几个基本问题研究进展[J].水利水运工程学报,2001,3:73-77.ZHAO Qi-lin,NIU Hai-qing,ZHUO Jia-shou.Basic problems of strain-softening material[J].Hydro-Science and Engineering,2001,3:73-77.
[15]江见鲸,陆新征.混凝土结构有限元分析(第2版)[M].北京:清华大学出版社,2013:239-247.
[16]郑宏,葛修润,李焯芬.脆塑性岩体的分析原理及其应用[J].岩石力学与工程学报,1997,16(1):8-12.ZHENG Hong,GE Xiu-run,LEE C.F.Analysis principle for rock mass with brittle-plasticity and its applications[J].Chinese Journal of Rock Mechanics and Engineering,1997,16(1):8-12.
[17]WEMPNER G A.Discrete approximation related to nonlinear theories of solids[J].International Journal of Solids and Structures,1971,7(11):1581-1599.
[18]RIKS E.The application of Newton’s method to the problem of elastic stability[J].Journal of Applied Mechanics,1972,39(4):1060-1065.
[19]RIKS E.An incremental approach to the solution of snapping and bucking problems[J].International Journal of Solids and Structures,1979,15:529-551.
[20]CRISFIELD M A.A fast incremental iterative solution procedure that handles“snap-through”[J].Computers and Structures,1981,13:55-62.
[21]CRISFIELD M A.Non-linear finite element analysis of solids and structures.Volume 1:Essential[M].New York:John Wiley&Sons,1991:266-286.
[22]RAMM E.Strategies for tracing the nonlinear response near limit points[M].New York:Springer Berlin Heidelberg,1981:63-89.
[23]竺润祥,派列希.解非线性有限元问题的组合弧长法[J].西北工业大学学报,1984,2(3):291-303.ZHU Run-xiang,PARISCH H.A modified arc-length method for solving nonlinear problems[J].Journal of Northwestern Polytechnical University.1984,2(3):291-303.
[24]刘国明,卓家寿,夏颂佑.求解非线性有限元方程的弧长法及在工程稳定分析中的应用[J].岩土力学,1993,14(4):57-67.LIU Guo-ming,ZHUO Jia-shou,XIA Song-you.Arc length method for solving non-linear FEM equations and its applications in engineering stability analysis[J].Rock and Soil Mechanics,1993,14(4):57-67.
[25]李元齐,沈祖炎.弧长法中初始荷载增量参数符号确定准则的改进[J].工程力学,2001,18(3):24-39.LI Yuan-qi,SHEN Zu-yan.Discussion on the criteria for determination of initial loading parameter in arc-length methods[J].Engineering Mechanics,2001,18(3):24-39.
[26]SIMO J C,HUGHES T J R.Computational Inelasticity[M].New York:Springer-Verlag,1998.
[27]BELYTSCHKO T,LIU W K,MORAN B.连续体和结构的非线性有限元[M].北京:清华大学出版社,2002:241-250.
[28]NETO E A d E SOUZA,PERI′C D,OWEN D R J.Computational methods for plasticity theory and applications[M].West Sussex:John Wiley&Sons Ltd,2008:194-343.
[29]王军祥,姜谙男.完全隐式返回映射算法对岩土地基问题的求解[J].工程力学,2013,30(8):83-89.WANG Jun-xiang,JIANG An-nan.Fully implicit return mapping algorithm for solving the problems of geotechnical foundation[J].Engineering Mechanics,2013,30(8):83-89.
[30]周应华,周德培,封志军.三种红层岩石常规三轴压缩下的强度与变形特性研究[J].工程地质学报,2005,13(4):477-480.ZHOU Ying-hua,ZHOU De-pei,FENG Zhi-jun.Strength and deformation of three types of red beds under conventional triaxial compression[J].Journal of Engineering Geology,2005,13(4):477-480.
[2]NAGHDI P M,TRAPP J A.The significance of formulating plasticity theory with reference to loading surfaces in strain space[J].International Journal of Engineering Science,1975,13(9):785-797.
[3]阎金安,张宪宏.岩石材料应变软化模型及有限元分析[J].岩土力学,1990,11(1):19-27.YAN Jin-an,ZHANG Xian-hong.Strain softening model of rock material and associated finite element analysis[J].Rock and Soil Mechanics,1990,11(1):19-27.
[4]吴玉忠.岩石应变软化弹塑性有限元分析[J].有色金属,1990,42(1):1-8.WU Yu-zhong.Finite element analysis of elastoplastic rock with strain-softening[J].Nonferrous Metals,1990,42(1):1-8.
[5]王兵,陈炽昭,张金荣.考虑岩石应变软化特性隧道的弹-塑性分析[J].铁道学报,1992,14(2):86-95.WANG Bing,CHEN Chi-zhao,ZHANG Jin-rong.An elasto-plastic analysis of a tunnel in consideration of the strain softening behaviour of rock[J].Journal of the China Railway Society,1992,14(2):86-95.
[6]沈新普,岑章志,徐秉业.弹脆塑性软化本构理论的特点及其数值计算[J].清华大学学报(自然科学版),1995,35(2):22-27.SHEN Xin-pu,CEN Zhang-zhi,XU Bing-ye.The characteristics of elasto-brittle-plastic softening constitutive theory and its numerical calculation[J].Journal of Tsinghua University(Sci.&Tech.),1995,35(2):22-27.
[7]杨超,崔新明,徐水平.软岩应变软化数值模型的建立与研究[J].岩土力学,2002,23(6):695-697,701.YANG Chao,CHUI Xin-ming,XU Shui-ping.Establishment and study of strain-softening numerical constitutive model for soft rock[J].Rock and Soil Mechanics,2002,23(6):695-697,701.
[8]ALONSO E,ALEJANO L R,VARAS F,et al.Ground response curves for rock masses exhibiting strainsoftening behaviour[J].International Journal for Numerical and Analytical Methods in Geomechanics,2003,27(13):1153-1185.
[9]张帆,盛谦,朱泽奇,等.三峡花岗岩峰后力学特性及应变软化模型研究[J].岩石力学与工程学报,2008,27(增刊1):2651-2655.ZHANG Fan,SHENG Qian,ZHU ZE-qi,et al.Study on post-peak mechanical behaviour and strain-softening model of Three Gorges granite[J].Chinese Journal of Rock Mechanics and Engineering,2008,27(Supp.1):2651-2655.
[10]周家文,徐卫亚,李明卫,等.岩石应变软化模型在深埋隧洞数值分析中的应用[J].岩石力学与工程学报,2009,28(6):1116-1127.ZHOU Jia-wen,XU Wei-ya,LI Ming-wei,et al.Application of rock strain softening model to numerical analysis of deep tunnel[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(6):1116-1127.
[11]王水林,王威,吴振君.岩土材料峰值后区强度参数演化与应力-应变曲线关系研究[J].岩石力学与工程学报,2010,29(8):1524-1529.WANG Shui-lin,WANG Wei,WU Zhen-jun.Study of relationship between evolution of post-peak strength parameters and stress-strain curves of geomaterial[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(8):1524-1529.
[12]李文婷,李树忱,冯现大,等.基于摩尔-库仑准则的岩石峰后应变软化力学行为研究[J].岩石力学与工程学报,2011,30(7):1460-1466.LI Wen-ting,LI Shu-chen,FENG Xian-da,et al.Study of post-peak strain softening mechanical properties of rock based on Mohr-Coulomb criterion[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(7):1460-1466.
[13]李英杰,张顶立,刘保国.考虑变形模量劣化的应变软化模型在FLAC3D中的开发与验证[J].岩土力学,2011,32(增刊2):647-652,659.LI Ying-jie,ZHANG Ding-li,LIU Bao-guo.Development and verification of strain-softening model considering deformation modulus degradation in FLAC3D[J].Rock and Soil Mechanics,2011,32(Supp.2):647-652,659.
[14]赵启林,牛海清,卓家寿.应变软化材料的几个基本问题研究进展[J].水利水运工程学报,2001,3:73-77.ZHAO Qi-lin,NIU Hai-qing,ZHUO Jia-shou.Basic problems of strain-softening material[J].Hydro-Science and Engineering,2001,3:73-77.
[15]江见鲸,陆新征.混凝土结构有限元分析(第2版)[M].北京:清华大学出版社,2013:239-247.
[16]郑宏,葛修润,李焯芬.脆塑性岩体的分析原理及其应用[J].岩石力学与工程学报,1997,16(1):8-12.ZHENG Hong,GE Xiu-run,LEE C.F.Analysis principle for rock mass with brittle-plasticity and its applications[J].Chinese Journal of Rock Mechanics and Engineering,1997,16(1):8-12.
[17]WEMPNER G A.Discrete approximation related to nonlinear theories of solids[J].International Journal of Solids and Structures,1971,7(11):1581-1599.
[18]RIKS E.The application of Newton’s method to the problem of elastic stability[J].Journal of Applied Mechanics,1972,39(4):1060-1065.
[19]RIKS E.An incremental approach to the solution of snapping and bucking problems[J].International Journal of Solids and Structures,1979,15:529-551.
[20]CRISFIELD M A.A fast incremental iterative solution procedure that handles“snap-through”[J].Computers and Structures,1981,13:55-62.
[21]CRISFIELD M A.Non-linear finite element analysis of solids and structures.Volume 1:Essential[M].New York:John Wiley&Sons,1991:266-286.
[22]RAMM E.Strategies for tracing the nonlinear response near limit points[M].New York:Springer Berlin Heidelberg,1981:63-89.
[23]竺润祥,派列希.解非线性有限元问题的组合弧长法[J].西北工业大学学报,1984,2(3):291-303.ZHU Run-xiang,PARISCH H.A modified arc-length method for solving nonlinear problems[J].Journal of Northwestern Polytechnical University.1984,2(3):291-303.
[24]刘国明,卓家寿,夏颂佑.求解非线性有限元方程的弧长法及在工程稳定分析中的应用[J].岩土力学,1993,14(4):57-67.LIU Guo-ming,ZHUO Jia-shou,XIA Song-you.Arc length method for solving non-linear FEM equations and its applications in engineering stability analysis[J].Rock and Soil Mechanics,1993,14(4):57-67.
[25]李元齐,沈祖炎.弧长法中初始荷载增量参数符号确定准则的改进[J].工程力学,2001,18(3):24-39.LI Yuan-qi,SHEN Zu-yan.Discussion on the criteria for determination of initial loading parameter in arc-length methods[J].Engineering Mechanics,2001,18(3):24-39.
[26]SIMO J C,HUGHES T J R.Computational Inelasticity[M].New York:Springer-Verlag,1998.
[27]BELYTSCHKO T,LIU W K,MORAN B.连续体和结构的非线性有限元[M].北京:清华大学出版社,2002:241-250.
[28]NETO E A d E SOUZA,PERI′C D,OWEN D R J.Computational methods for plasticity theory and applications[M].West Sussex:John Wiley&Sons Ltd,2008:194-343.
[29]王军祥,姜谙男.完全隐式返回映射算法对岩土地基问题的求解[J].工程力学,2013,30(8):83-89.WANG Jun-xiang,JIANG An-nan.Fully implicit return mapping algorithm for solving the problems of geotechnical foundation[J].Engineering Mechanics,2013,30(8):83-89.
[30]周应华,周德培,封志军.三种红层岩石常规三轴压缩下的强度与变形特性研究[J].工程地质学报,2005,13(4):477-480.ZHOU Ying-hua,ZHOU De-pei,FENG Zhi-jun.Strength and deformation of three types of red beds under conventional triaxial compression[J].Journal of Engineering Geology,2005,13(4):477-480.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |