Lemaitre等向硬化弹塑性损伤耦合本构模型积分算法及程序实现
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摘要
涉及复杂材料弹塑性损伤问题数值计算研究时,不仅需要选择恰当预测损伤和破坏的本构模型,还需要有效和稳健的本构积分算法。首先,阐述了在热力学和连续介质力学框架下建立弹塑性损伤本构模型的基本步骤;其次,基于Lemaitre等向硬化弹塑性损伤耦合本构模型、相应的本构积分算法-完全隐式返回映射算法(Fully Return Mapping Algorithm)和一致切线模量,采用C++语言在Visual 6.0环境下编制有限元本构求解程序,在塑性损伤修正步中求解返回映射方程时,选取一种简单的形式,只需迭代求解一个标量非线性方程,计算效率较高。最后,通过缺口圆棒数值算例初步验证了程序的正确性,并编制接口程序对计算结果进行可视化。研究结果表明积分算法的有效性及程序的正确性,Lemaitre等向硬化弹塑性损伤耦合本构模型能够较好地模拟韧性材料的破坏发展过程,可以求解类似的有限元边界值问题,为考虑损伤特性的韧性材料结构研究和设计奠定基础。
Research into numerical computing problems involving coupled elastoplastic damage of complex materials not only needs to select the appropriate constitutive model to forecast damage, but also needs a robust and effective constitutive integration algorithm. Firstly, the basic steps of setting up the elastoplastic damage constitutive model are expounded within the framework of thermodynamics and continuum mechanics. Secondly, a finite element solving program of the constitutive model, which is based on Lemaitre's isotropic hardening coupled elastoplastic damage constitutive model, the corresponding constitutive integration algorithm-fully implicit return mapping algorithm, and the consistent tangent modulus, is compiled using C ++ language in Visual 6.0 environment. When solving the full return mapping algorithm, it is possible to develop a simplified form which can be solved by iterating a non-linear equation, so improving computational efficiency is relatively simple. Finally, the validity of the application program is demonstrated through a numerical example of a cylindrical notched bar, and the calculation results are graphically displayed by a self-developed interface program. The research results show the effectiveness of the algorithm and the correctness of program, proving that the development process of damage of the ductile material can be simulated. Boundary value problems of finite elements can be solved well, which provides a basis for research and design for structures which considers the damage characteristics of ductile materials.
Research into numerical computing problems involving coupled elastoplastic damage of complex materials not only needs to select the appropriate constitutive model to forecast damage, but also needs a robust and effective constitutive integration algorithm. Firstly, the basic steps of setting up the elastoplastic damage constitutive model are expounded within the framework of thermodynamics and continuum mechanics. Secondly, a finite element solving program of the constitutive model, which is based on Lemaitre's isotropic hardening coupled elastoplastic damage constitutive model, the corresponding constitutive integration algorithm-fully implicit return mapping algorithm, and the consistent tangent modulus, is compiled using C ++ language in Visual 6.0 environment. When solving the full return mapping algorithm, it is possible to develop a simplified form which can be solved by iterating a non-linear equation, so improving computational efficiency is relatively simple. Finally, the validity of the application program is demonstrated through a numerical example of a cylindrical notched bar, and the calculation results are graphically displayed by a self-developed interface program. The research results show the effectiveness of the algorithm and the correctness of program, proving that the development process of damage of the ductile material can be simulated. Boundary value problems of finite elements can be solved well, which provides a basis for research and design for structures which considers the damage characteristics of ductile materials.
引文
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[2]Rabotnov Y N.On the equations of state for creep[C]//Proceeding of the Institution of Mechanical Engineers,Los Angeles SAGE Publications,1963,178(1):2―117.
[3]Lemaitre J.A Course on Damage Mechanics[M].2nd ed.New York:Springer,1996:1―3.
[4]Vaz M,Owen D R J.Aspects of ductile fracture and adaptive mesh refinement in damaged elasto-plastic materials[J].International Journal for Numerical Methods in Engineering,2001,50(1):29―54.
[5]Lemaitre J.A continuous damage mechanics model for ductile fracture[J].Journal of Engineering Materials and Technology,1985,107(1):83―89.
[6]Lemaitre J,Chaboche J L.Mechanics of Solid Material[M].Cambridge University Press,1994:1―20.
[7]Steinmann P,Miehe C,Stein E.Comparison of different finite deformation inelastic damage models within multiplicative elastoplasticity for ductile materials[J].Computational Mechanics,1994,13(6):458―474.
[8]Neto E A D S,Peric D,Owen D R J.A model for elastoplastic damage at finite strains:algorithmic issues and applications[J].Engineering Computations,1994,11(3):257―281.
[9]Doghri I.Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage[J].International Journal for Numerical Methods in Engineering,1995,38(20):3403―3431.
[10]Bouchard P O,Bourgeon L,Fayolle S,et al.An enhanced Lemaitre model formulation for materials processing damage computation[J].International Journal of Material Forming,2011,4(3):299―315.
[11]Benallal A,Billardon R,Doghri I.An integration algorithm and the corresponding consistent tangent operator for fully coupled elastoplastic and damage equations[J].Communications in Applied Numerical Methods,1988,4(6):731―740.
[12]Neto E A D S,Peric D.A computational framework for a class of fully coupled models for elastoplastic damage at finite strains with reference to the linearization aspects[J].Computer Methods in Applied Mechanics and Engineering,1996,130(1):179―193.
[13]Singh A K,Pandey P C.An implicit integration algorithm for plane stress damage coupled elastoplasticity[J].Mechanics Research Communications,1999,26(6):693―700.
[14]Neto E A D S.A fast,one-equation integration algorithm for the Lemaitre ductile damage model[J].Communications in Numerical Methods in Engineering,2002,18(8):541―554.
[15]Salari M R,Saeb S,Willam K J,et al.A coupled elastoplastic damage model for geomaterials[J].Computer Methods in Applied Mechanics and Engineering,2004,193(27):2625―2643.
[16]Luccioni B,Oller S,Danesi R.Coupled plastic-damaged model[J].Computer Methods in Applied Mechanics and Engineering,1996,129(1):81―89.
[17]李杰,吴建营.混凝土弹塑性损伤本构模型研究:基本公式[J].土木工程学报,2005,38(9):14―20.Li Jie,Wu Jianying.Elastoplastic damage constitutive model for concrete based on damage engineering release,part I:Basic formulations[J].China Civil Engineering Journal,2005,38(9):14―20.(in Chinese)
[18]Tai W H,Yang B X.A new microvoid-damage model for ductile fracture[J].Engineering Fracture Mechanics,1986,25(3):377―384.
[19]Chandrakanth S,Pandey P C.An isotropic damage model for ductile material[J].Engineering Fracture Mechanics,1995,50(4):457―465.
[20]Bonora N.A nonlinear CDM model for ductile failure[J].Engineering Fracture Mechanics 1997,58(1):11―28.
[21]Simo J C,Hughes T J R.Computational Inelasticity[M].New York:Springer-Verlag,1998:32―43.
[22]Belytschko T,Liu W K,Moran B.Nonlinear Finite Elements for Continua and Structures[M].New York:John Wiley&Sons Ltd,2000:277―294.
[23]Simo J C,Taylor R L.A return mapping algorithm for plane stress elastoplasticity[J].International Journal for Numerical Methods in Engineering,1986,22(3):649―670.
[24]杨强,冷旷代,张小寒,等.Drucker-Prager弹塑性本构关系积分:考虑非关联流动与各向同性硬化[J].工程力学,2012,29(8):165―171.Yang Qiang,Leng Kuangdai,Zhang Xiaohan,et al.An integration algorithm for Drucker-Prager elastic-plastic model with non-associated flow rule and isotropic hardening[J].Engineering Mechanics,2012,29(8):165―171.(in Chinese)
[25]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 Junxiang,Jiang Annan.Fully implicit return mapping algorithm for solving the problems geotechnical foundation[J].Engineering Mechanics,2013,30(8):83―89.(in Chinese)
[26]Neto D S E A,Peri′c D,Owen D R J.Computational Methods for Plasticity Theory and Applications[M].West Sussex:United Kingdom John Wiley&Sons Ltd,2008:486―497.
[2]Rabotnov Y N.On the equations of state for creep[C]//Proceeding of the Institution of Mechanical Engineers,Los Angeles SAGE Publications,1963,178(1):2―117.
[3]Lemaitre J.A Course on Damage Mechanics[M].2nd ed.New York:Springer,1996:1―3.
[4]Vaz M,Owen D R J.Aspects of ductile fracture and adaptive mesh refinement in damaged elasto-plastic materials[J].International Journal for Numerical Methods in Engineering,2001,50(1):29―54.
[5]Lemaitre J.A continuous damage mechanics model for ductile fracture[J].Journal of Engineering Materials and Technology,1985,107(1):83―89.
[6]Lemaitre J,Chaboche J L.Mechanics of Solid Material[M].Cambridge University Press,1994:1―20.
[7]Steinmann P,Miehe C,Stein E.Comparison of different finite deformation inelastic damage models within multiplicative elastoplasticity for ductile materials[J].Computational Mechanics,1994,13(6):458―474.
[8]Neto E A D S,Peric D,Owen D R J.A model for elastoplastic damage at finite strains:algorithmic issues and applications[J].Engineering Computations,1994,11(3):257―281.
[9]Doghri I.Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage[J].International Journal for Numerical Methods in Engineering,1995,38(20):3403―3431.
[10]Bouchard P O,Bourgeon L,Fayolle S,et al.An enhanced Lemaitre model formulation for materials processing damage computation[J].International Journal of Material Forming,2011,4(3):299―315.
[11]Benallal A,Billardon R,Doghri I.An integration algorithm and the corresponding consistent tangent operator for fully coupled elastoplastic and damage equations[J].Communications in Applied Numerical Methods,1988,4(6):731―740.
[12]Neto E A D S,Peric D.A computational framework for a class of fully coupled models for elastoplastic damage at finite strains with reference to the linearization aspects[J].Computer Methods in Applied Mechanics and Engineering,1996,130(1):179―193.
[13]Singh A K,Pandey P C.An implicit integration algorithm for plane stress damage coupled elastoplasticity[J].Mechanics Research Communications,1999,26(6):693―700.
[14]Neto E A D S.A fast,one-equation integration algorithm for the Lemaitre ductile damage model[J].Communications in Numerical Methods in Engineering,2002,18(8):541―554.
[15]Salari M R,Saeb S,Willam K J,et al.A coupled elastoplastic damage model for geomaterials[J].Computer Methods in Applied Mechanics and Engineering,2004,193(27):2625―2643.
[16]Luccioni B,Oller S,Danesi R.Coupled plastic-damaged model[J].Computer Methods in Applied Mechanics and Engineering,1996,129(1):81―89.
[17]李杰,吴建营.混凝土弹塑性损伤本构模型研究:基本公式[J].土木工程学报,2005,38(9):14―20.Li Jie,Wu Jianying.Elastoplastic damage constitutive model for concrete based on damage engineering release,part I:Basic formulations[J].China Civil Engineering Journal,2005,38(9):14―20.(in Chinese)
[18]Tai W H,Yang B X.A new microvoid-damage model for ductile fracture[J].Engineering Fracture Mechanics,1986,25(3):377―384.
[19]Chandrakanth S,Pandey P C.An isotropic damage model for ductile material[J].Engineering Fracture Mechanics,1995,50(4):457―465.
[20]Bonora N.A nonlinear CDM model for ductile failure[J].Engineering Fracture Mechanics 1997,58(1):11―28.
[21]Simo J C,Hughes T J R.Computational Inelasticity[M].New York:Springer-Verlag,1998:32―43.
[22]Belytschko T,Liu W K,Moran B.Nonlinear Finite Elements for Continua and Structures[M].New York:John Wiley&Sons Ltd,2000:277―294.
[23]Simo J C,Taylor R L.A return mapping algorithm for plane stress elastoplasticity[J].International Journal for Numerical Methods in Engineering,1986,22(3):649―670.
[24]杨强,冷旷代,张小寒,等.Drucker-Prager弹塑性本构关系积分:考虑非关联流动与各向同性硬化[J].工程力学,2012,29(8):165―171.Yang Qiang,Leng Kuangdai,Zhang Xiaohan,et al.An integration algorithm for Drucker-Prager elastic-plastic model with non-associated flow rule and isotropic hardening[J].Engineering Mechanics,2012,29(8):165―171.(in Chinese)
[25]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 Junxiang,Jiang Annan.Fully implicit return mapping algorithm for solving the problems geotechnical foundation[J].Engineering Mechanics,2013,30(8):83―89.(in Chinese)
[26]Neto D S E A,Peri′c D,Owen D R J.Computational Methods for Plasticity Theory and Applications[M].West Sussex:United Kingdom John Wiley&Sons Ltd,2008:486―497.
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