完全隐式返回映射算法对岩土地基问题的求解
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摘要
针对岩土工程中的复杂力学问题,在弹塑性力学理论框架和非线性有限元理论基础上,采用非关联等向硬化Drucker-Prager模型的完全隐式积分算法—返回映射算法(Return MappingAlgorithm)编制了有限元求解程序。该算法可以避免预测应力漂移屈服面的现象,对准静态变形条件下的本构方程可以获得准确解,在迭代中使用Newton-Raphson法获得近似平方的收敛速率,具有较高的精确性和稳定性。对岩土工程中的地基问题进行求解,计算得出位移、应力等结果,模拟了塑性区随载荷步增加的演化过程,对地基极限承载力进行了解析解和数值解的对比。结果表明了算法的优越性、程序的正确性和实用性。
To solve complicated mechanics problems in geotechnical engineering,the finite element program is compiled with fully implicit integration algorithm which is return mapping algorithm of non-associative isotropic hardening Drucker-Prager criterion based on the elastic-plastic mechanics theory frame and the nonlinear finite element theory.Return mapping algorithm can avoid the drift phenomenon of the trial stress,and can achieve the accurate solution of the constitutive equation on the condition of the quasi-static deformation,a quadratic convergence rate when using the Newton-Raphson iteration scheme,higher accuracy and stability.To solve the problems of foundations in geotechnical engineering,displacements and stresses are calculated,and the evolution process of plastic zones is simulated.The ultimate bearing capacity of the analytical solution is compared with that from the numerical solution.The results demonstrate the superiority of the algorithm,and the correctness and the practicality of the program.
To solve complicated mechanics problems in geotechnical engineering,the finite element program is compiled with fully implicit integration algorithm which is return mapping algorithm of non-associative isotropic hardening Drucker-Prager criterion based on the elastic-plastic mechanics theory frame and the nonlinear finite element theory.Return mapping algorithm can avoid the drift phenomenon of the trial stress,and can achieve the accurate solution of the constitutive equation on the condition of the quasi-static deformation,a quadratic convergence rate when using the Newton-Raphson iteration scheme,higher accuracy and stability.To solve the problems of foundations in geotechnical engineering,displacements and stresses are calculated,and the evolution process of plastic zones is simulated.The ultimate bearing capacity of the analytical solution is compared with that from the numerical solution.The results demonstrate the superiority of the algorithm,and the correctness and the practicality of the program.
引文
[1]Ted Belytschko,Wing Kam Liu,Brian Moran.连续体和结构的非线性有限元[M].庄茁,译.北京:清华大学出版社,2002:241―250.Ted Belytschko,Wing Kam Liu,Brian Moran.Nonlinearfinite elements for continua and structures[M].Translated by Zhuang Zhuo.Beijing:Tsinghua UniversityPress,2002:241―250.(in Chinese)
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[6]Wang X,Wang L B,Xu L M.Formulation of the returnmapping algorithm for elastoplastic soil models[J].Computers and Geotechnics,2004,31(4):315―338.
[7]杨强,杨晓君,陈新.基于D-P准则的理想弹塑性本构关系积分研究[J].工程力学,2005,22(4):15―19,47.Yang Qiang,Yang Xiaojun,Chen Xin.On integrationalgorithms for perfect plasticity based on Drucker-Pragercriterion[J].Engineering Mechanics,2005,22(4):15―19,47.(in Chinese)
[8]Nukala P K V V.A return mapping algorithm for cyclicviscoplastic constitutive models[J].Comput.MethodsAppl.Mech.Engrg,2006,195(1/2/3):148―178.
[9]Clausen J,Damkilde L,Andersen L.An efficient returnalgorithm for non-associated plasticity with linear yieldcriteria in principal stress space[J].Computers andStructures,2007,85:1795―1807.
[10]陈明祥.关于返回映射算法中应力的四阶张量值函数[J].力学学报,2010,42(2):228―237.Cheng Mingxiang.On the fourth order tensor valuedfunction of the stress in return map algorithm[J].ChineseJournal of Theoretical and Applied Mechanics,2010,42(2):228―237.(in Chinese)
[11]Huang Jinsong,Griffiths D V.Return mapping algorithmsand stress predictors for failure analysis in geomechanics[J].Journal of Engineering Mechanics,2009,135(4):276―284.
[12]Neto E A de S,Peri′c D,Owen D R J.Computationalmethods for plasticity theory and applications[M].Printed in Singapore by Markono,2008:194―343.
[13]茹忠亮.三维进化弹塑性并行有限元反分析系统研究[D].沈阳:东北大学,2004:50―53.Ru Zhongliang.Research on the 3D evolutionelasto-plastic parallel finite element method back analysissystem[D].Shenyang:Northeastern University,2004:50―53.(in Chinese)
[2]Krieg R D,Krieg D B.Accuracies of numerical solutionmethods for the elastic-perfectly plastic model[J].ASMEJ.Pressure Vessel Technol,1977,99(4):510―515.
[3]Simo J C,Taylor R L.Consistent tangent operators forrate independent elastoplasticity[J].Computer Methodsin Applied Mechanics and Engineering,1985,48(1):101―118.
[4]Simo J C,Hughes T J R.Computational inelasticity[M].New York:Springer-Verlag,1998:32―43.
[5]Asensio G,Moreno C.Linearization and return mappingalgorithms for elastoplasticity models[J].InternationalJournal for Numerical Methods in Engineering,2003,57(7):991―1014.
[6]Wang X,Wang L B,Xu L M.Formulation of the returnmapping algorithm for elastoplastic soil models[J].Computers and Geotechnics,2004,31(4):315―338.
[7]杨强,杨晓君,陈新.基于D-P准则的理想弹塑性本构关系积分研究[J].工程力学,2005,22(4):15―19,47.Yang Qiang,Yang Xiaojun,Chen Xin.On integrationalgorithms for perfect plasticity based on Drucker-Pragercriterion[J].Engineering Mechanics,2005,22(4):15―19,47.(in Chinese)
[8]Nukala P K V V.A return mapping algorithm for cyclicviscoplastic constitutive models[J].Comput.MethodsAppl.Mech.Engrg,2006,195(1/2/3):148―178.
[9]Clausen J,Damkilde L,Andersen L.An efficient returnalgorithm for non-associated plasticity with linear yieldcriteria in principal stress space[J].Computers andStructures,2007,85:1795―1807.
[10]陈明祥.关于返回映射算法中应力的四阶张量值函数[J].力学学报,2010,42(2):228―237.Cheng Mingxiang.On the fourth order tensor valuedfunction of the stress in return map algorithm[J].ChineseJournal of Theoretical and Applied Mechanics,2010,42(2):228―237.(in Chinese)
[11]Huang Jinsong,Griffiths D V.Return mapping algorithmsand stress predictors for failure analysis in geomechanics[J].Journal of Engineering Mechanics,2009,135(4):276―284.
[12]Neto E A de S,Peri′c D,Owen D R J.Computationalmethods for plasticity theory and applications[M].Printed in Singapore by Markono,2008:194―343.
[13]茹忠亮.三维进化弹塑性并行有限元反分析系统研究[D].沈阳:东北大学,2004:50―53.Ru Zhongliang.Research on the 3D evolutionelasto-plastic parallel finite element method back analysissystem[D].Shenyang:Northeastern University,2004:50―53.(in Chinese)
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