2021年 02期

Elastoplastic Integral Algorithm for Mohr-Coulomb Criterion


摘要(Abstract):

基于Koiter法则,提出一种适用于Mohr-Coulomb非光滑本构模型的弹塑性积分方法;阐述Mohr-Coulomb准则角点问题产生的原因,采用经典的Kuhn-Tucker互补条件判断可能活跃的屈服面,将Kuhn-Tucker互补方程作为一类特殊的变分不等式,使用投影收缩算法进行求解,并进一步通过迭代确定实际活跃的屈服面;基于主应力特征方程,在主应力空间中计算屈服函数对应力分量的偏导数,同时在一般应力空间中执行应力返回。结果表明,所提出的算法解决了Mohr-Coulomb准则的角点问题,消除了光滑角点带来的误差,既避免了Mohr-Coulomb屈服函数在一般应力空间中角点处的数值奇异性,又不需要进行主应力空间法中所需的应力变换。

关键词(KeyWords): Mohr-Coulomb准则;弹塑性;互补问题

基金项目(Foundation): 国家自然科学基金重点项目(51538001)

作者(Author): 张谭,郑宏,林姗

DOI: 10.13349/j.cnki.jdxbn.20201029.002

参考文献(References):

[1] KOITER W T.Stress-strain relations,uniqueness and variational theorems for elastic-plastic materials with a singular yield surface[J].Quarterly of Applied Mathematics,1953,11(3):350-354.

[2] ZIENKIEWICZ O C,VALLIAPPAN S,KING I P.Elasto-plastic solutions of engineering problems ‘initial stress’,finite element approach[J].International Journal for Numerical Methods in Engineering,1969,1(1):75-100.

[3] OWEN D R J,HINTON E.Finite elements in plasticity:theory and practice[M].Swansea:Pineridge Press,1980:20-45.

[4] MARQUES J M M C.Stress computation in elastoplasticity[J].Engineering Computations,1984,1(1):42-51.

[5] ZIENKIEWICZ O C,PANDE G N.Some useful forms for isotropic yield surfaces for soils and rock mechanics[M]//Finite Elements in Geomechanics.Chichester:Wiley,1977:179-198.

[6] SLOAN S W,BOOKER J R.Removal of singularities in Tresca and Mohr-Coulomb yield functions[J].Communications in Applied Numerical Methods,1986,2(2):173-179.

[7] ABBO A J,LYAMIN A V,SLOAN S W.A C2 continuous approximation to the Mohr-Coulomb yield surface[J].International Journal of Solids and Structures,2011,48(21):3001-3010.

[8] 俞茂宏,昝月稳,李建春.统一强度理论角点奇异性的统一处理[J].岩石力学与工程学报,2000,19(增刊1):849-852.

[9] LARSSON R,RUNESSON K. Implicit integration and consistent linearization for yield criteria of the Mohr-Coulomb type[J].Mechanics of Cohesive-frictional Materials,1996,1(4):367-383.

[10] PERIC D,de SOUZA NETO E A. A new computational model for Tresca plasticity at finite strains with an optimal parametrization in the principal space[J]. Computer Methods in Applied Mechanics and Engineering,1999,171(3/4):463-489.

[11] de SOUZA NETO E A,PERIC D,OWEN D R J. Computational methods for plasticity:theory and applications[M]. Chichester:Wiley,2008:353-386.

[12] de BORST R.Integration of plasticity equations for singular yield functions[J].Computers and Structures,1987,26(5):823-829.

[13] PANKAJ N,BICANIC N. Detection of multiple active yield conditions for Mohr-Coulomb elasto-plasticity[J]. Computers and Structures,1997,62(1):51-61.

[14] CLAUSEN J,DAMKILDE L,ANDERSEN L.An efficient return algorithm for non-associated plasticity with linear yield criteria in principal stress space[J].Computers and Structures,2007,85(23/24):1795-1807.

[15] COOMBS W M,CROUCH R S,AUGARDE C E.Reuleaux plasticity:analytical backward Euler stress integration and consistent tangent[J].Computer Methods in Applied Mechanics and Engineering,2010,199 (25/26/27/28):1733-1743.

[16] 孙林松,郭兴文,王德信.弹塑性问题的互补变分原理与模型[J].河海大学学报(自然科学版),2002,30(2):35-38.

[17] 郑宏,葛修润.弹塑性分析的线性互补问题[J].力学学报,1995,27(1):38-47.

[18] SIMO J C,KENNEDY J G,GOVINDJEE S.Non-smooth multisurface plasticity and viscoplasticity:loading/unloading conditions and numerical algorithms[J].International Journal for Numerical Methods in Engineering,1988,26(10):2161-2185.

[19] KARAOULAINIS F E.Implicit numerical integration of non-smooth multisurface yield criteria in the principal stress space[J].Archives of Computer Methods in Engineering,2013,20(3):263-3087.

[20] 李翠华,姜清辉,周创兵.Mohr-Coulomb准则角点问题的主应力空间互补算法[J].工程力学,2014,31(4):134-140.

[21] ADHIKARY D P,JAYASUNDARA C T,PODGORNEY R K,et al.A robust return-map algorithm for general multisurface plasticity[J].International Journal for Numerical Methods in Engineering,2017,109(2):218-234.

[22] ZIENKIEWICZ O C,HUMPHESON C.Viscoplasticity:a generalized model for description of soil behavior[J].Numerical Methods in Geotechnical Engineering,1977,61(70):116-147.

[23] 戴自航,刘志伟,刘成禹,等.考虑张拉与剪切破坏的土坡稳定数值分析[J].岩石力学与工程学报,2008,27(2):375-382.

[24] DUNCAN J M.State of the art:limit equilibrium and finite-element analysis of slopes[J].Journal of Geotechnical Engineering,1996,122(7):577-596.

[25] ZIENKIEWICZ O C,HUMPHESON C,LEWIS R W.Associated and nonassociated viscoplasticity in soil mechanics[J].Géotechnique,1975,25(4):671-689.

[26] GRIFFITHS D V,LANE P A.Slope stability analysis by finite elements[J].Géotechnique,1999,49(3):387-403.

[27] MATSUI T,SAN K C.Finite element slope stability analysis by shear strength reduction technique[J].Soils and Foundations,1992,32(1):59-70.

[28] 郑颖人,赵尚毅.有限元强度折减法在土坡与岩坡中的应用[J].岩石力学与工程学报,2004,23(19):3381-3388.

[29] 赵尚毅,郑颖人,时卫民,等.用有限元强度折减法求边坡稳定安全系数[J].岩土工程学报,2002,24(3):343-346.

[30] 张爱军,莫海鸿.有限元强度折减法中边坡失稳位移突变判据的改进[J].岩土力学,2013,34(增刊2):332-337.

[31] 陈力华,靳晓光.有限元强度折减法中边坡三种失效判据的适用性研究[J].土木工程学报,2012,45(9):136-146.

[32] 赵尚毅,郑颖人,张玉芳.极限分析有限元法讲座:Ⅱ有限元强度折减法中边坡失稳的判据探讨[J].岩土力学,2005,26(2):332-336.

[33] 郑宏,李春光,李焯芬,等.求解安全系数的有限元法[J].岩土工程学报,2002,24(5):626-628.