一种显式子步应力点积分算法及其在SMA数值模拟中的应用
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摘要
形状记忆合金(shape memory alloys,简称SMA)具有复杂的热力本构关系,为了模拟SMA及其组合结构复杂的受力和变形行为,在数值模拟中需要采用可靠且高效的应力点积分算法。隐式应力点回映算法已经成功应用于形状记忆合金的数值模拟,但在复杂加载条件下,荷载增量较大时有可能导致整体非线性迭代求解不收敛。推广了局部误差控制的显式子步积分算法,首次将其应用于形状记忆合金及其组合结构这类热力相变问题的应力点积分,并通过数值算例对所提算法和隐式应力点回映算法进行了比较。数值结果表明:对于大规模数值模拟和计算,整体子步步数决定着总体计算时间;所提出的修正Euler自动子步方案可以有效减少整体子步步数,在保证相同计算精度的前提下能够大幅提高有限元计算效率,因而更适合大规模形状记忆合金智能结构的数值模拟。
Shape memory alloy(SMA) has complex thermomechanical constitutive relation,thus its numerical simulations demand reliable and efficient stress integration algorithms.The implicit return-mapping stress point algorithms,which have been successfully applied to such materials,may encounter convergence difficulties when loading conditions are complicated or load steps are large.Hence,an explicit sub-stepping stress integration method with automatic local error control was proposed for the simulation of the thermomechanical constitutive relation of shape memory alloys.By investigating several numerical examples,the efficiency of the proposed method and the implicit return-mapping stress point algorithm were evaluated and compared.Numerical results indicate that the number of global sub-steps dominates the entire analyzing time for large-scale computations. The proposed modified Euler automatic sub-stepping scheme leads to less global sub-steps so that the computing time is significantly reduced.Therefore,the explicit sub-stepping stress integration method has the potential for large-scale SMA simulations and computations.
Shape memory alloy(SMA) has complex thermomechanical constitutive relation,thus its numerical simulations demand reliable and efficient stress integration algorithms.The implicit return-mapping stress point algorithms,which have been successfully applied to such materials,may encounter convergence difficulties when loading conditions are complicated or load steps are large.Hence,an explicit sub-stepping stress integration method with automatic local error control was proposed for the simulation of the thermomechanical constitutive relation of shape memory alloys.By investigating several numerical examples,the efficiency of the proposed method and the implicit return-mapping stress point algorithm were evaluated and compared.Numerical results indicate that the number of global sub-steps dominates the entire analyzing time for large-scale computations. The proposed modified Euler automatic sub-stepping scheme leads to less global sub-steps so that the computing time is significantly reduced.Therefore,the explicit sub-stepping stress integration method has the potential for large-scale SMA simulations and computations.
引文
[1]Brinson L C,Lammering R.Finite element analysis of the behavior of shape memory alloys and theirapplications[J].International Journal of Solids and Structures,1993,30(23):3261-3280.
[2]Burton D S,Gao X,Brinson L C.Finite element simulation of a self-healing shape memory alloy com-posite[J].Mechanics of Materials,2007,38(5/6):525-537.
[3]王社良,苏三庆,沈亚鹏.形状记忆合金拉索被动控制结构地震响应分析[J].土木工程学报,2000,33(1):56-62.(WANG She-liang,SU San-qing,SHEN Ya-peng.Seismic response analysis forpassive structural control with shape memory alloy tendons[J].China Civil Engineering Journal,2000,33(1):56-62.(in Chinese))
[4]刘爱荣,潘亦苏,周本宽.形状记忆合金热力学行为的模拟[J].计算力学学报,2002,19(1):48-52.(LIU Ai-rong,PAN Yi-su,ZHOU Ben-kuan.Simulation to thermo-mechanial behavior ofshape memory alloys[J].Chinese Journal of Computational Mechanics,2002,19(1):48-52.(in Chi-nese))
[5]朱祎国,吕和祥,杨大智.一个新的形状记忆合金的本构模型[J].应用数学和力学,2002,23(9):896-902.(ZHU Yi-guo,LHe-xiang,YANG Da-zhi.A new model of shape memory alloys[J].Applied Mathematics and Mechanics,2002,23(9):896-902.(in Chinese))
[6]匡亚川,欧进萍.形状记忆合金智能混凝土梁变形特性的研究[J].中国铁道科学,2008,29(4):41-46.(KUANG Ya-chuan,OU Jin-ping.Research on the deformation characteristics of smartconcrete beam embedded with shape memory alloy wires[J].China Railway Science,2008,29(4):41-46.(in Chinese))
[7]陈海泉,李忠献,李延涛.应用形状记忆合金的高层建筑结构智能隔震[J].天津大学学报(自然科学与工程技术版),2002,35(6):761-765.(CHEN Hai-quan,LI Zong-xian,LI Yan-tao.In-telligent isolation of high-rise building structures applying shape memory alloys[J].Journal of TianjinUniversity(Science and Technology),2002,35(6):761-765.(in Chinese))
[8]夏开明,潘同燕,刘山洪.形状记忆合金相变过程三维大变形有限元模拟[J].应用数学和力学,2010,31(10):1201-1210.(XIA Kai-ming,PAN Tong-yan,LIU Shan-hong.Three dimensional largedeformation analysis of phase transformation in shape memory alloys[J].Applied Mathematics and Me-chanics,2010,31(10):1201-1210.(in Chinese))
[9]Tanaka K.A thermomechanical sketch of shape memory effect:one-dimensional tensile behavior[J].Res Mechanica,1986,18:251-263.
[10]Liang C,Rogers C A.One-dimensional thermomechanical constitutive relations for shape memory mate-rials[J].Journal of Intelligent Material Systems and Structures,1990,1(2):207-234.
[11]Boyd J G,Lagoudas D C.A thermodynamic constitutive model for the shape memory materials—partⅠ:the monolithic shape memory alloys[J].International Journal of Plasticity,1996,12(6):805-842.
[12]Lagoudas D C,Bo Z,Qidwai M A.A unified thermodynamic constitutive model for SMA and finite ele-ment analysis of active metal matrix composites[J].Mechanics of Advanced Materials and Structures,1996,3(2):153-179.
[13]徐小明,张盛,姚伟岸,钟万勰.基于辛弹性力学解析本征函数的有限元应力磨平方法[J].计算力学学报,2012,29(4):511-516.(XU Xiao-ming,ZHANG Sheng,YAO Wei-an,ZHONG Wan-xie.A stress recovery method based on the analytical eigenfunctions of symplectic elasticity[J].Chi-nese Journal of Computational Mechanics,2012,29(4):511-516.(in Chinese))
[14]Qidwai M A,Lagoudas D C.Numerical implementation of a shape memory alloy thermomechanical con-stitutive model using return mapping algorithms[J].International Journal for Numerical Methods in En-gineering,2000,47(6):1123-1168.
[15]Chen X,Cheng Y G.On accelerated symmetric stiffness techniques for non-associated plasticity withapplication to soil problems[J].Engineering Computations,2011,28(8):1044-1063.
[16]Sloan S W,Abbo A J,Sheng D.Refined explicit integration of elastoplastic models with automatic er-ror control[J].Engineering Computations,2001,18(1/2):121-154.
[2]Burton D S,Gao X,Brinson L C.Finite element simulation of a self-healing shape memory alloy com-posite[J].Mechanics of Materials,2007,38(5/6):525-537.
[3]王社良,苏三庆,沈亚鹏.形状记忆合金拉索被动控制结构地震响应分析[J].土木工程学报,2000,33(1):56-62.(WANG She-liang,SU San-qing,SHEN Ya-peng.Seismic response analysis forpassive structural control with shape memory alloy tendons[J].China Civil Engineering Journal,2000,33(1):56-62.(in Chinese))
[4]刘爱荣,潘亦苏,周本宽.形状记忆合金热力学行为的模拟[J].计算力学学报,2002,19(1):48-52.(LIU Ai-rong,PAN Yi-su,ZHOU Ben-kuan.Simulation to thermo-mechanial behavior ofshape memory alloys[J].Chinese Journal of Computational Mechanics,2002,19(1):48-52.(in Chi-nese))
[5]朱祎国,吕和祥,杨大智.一个新的形状记忆合金的本构模型[J].应用数学和力学,2002,23(9):896-902.(ZHU Yi-guo,LHe-xiang,YANG Da-zhi.A new model of shape memory alloys[J].Applied Mathematics and Mechanics,2002,23(9):896-902.(in Chinese))
[6]匡亚川,欧进萍.形状记忆合金智能混凝土梁变形特性的研究[J].中国铁道科学,2008,29(4):41-46.(KUANG Ya-chuan,OU Jin-ping.Research on the deformation characteristics of smartconcrete beam embedded with shape memory alloy wires[J].China Railway Science,2008,29(4):41-46.(in Chinese))
[7]陈海泉,李忠献,李延涛.应用形状记忆合金的高层建筑结构智能隔震[J].天津大学学报(自然科学与工程技术版),2002,35(6):761-765.(CHEN Hai-quan,LI Zong-xian,LI Yan-tao.In-telligent isolation of high-rise building structures applying shape memory alloys[J].Journal of TianjinUniversity(Science and Technology),2002,35(6):761-765.(in Chinese))
[8]夏开明,潘同燕,刘山洪.形状记忆合金相变过程三维大变形有限元模拟[J].应用数学和力学,2010,31(10):1201-1210.(XIA Kai-ming,PAN Tong-yan,LIU Shan-hong.Three dimensional largedeformation analysis of phase transformation in shape memory alloys[J].Applied Mathematics and Me-chanics,2010,31(10):1201-1210.(in Chinese))
[9]Tanaka K.A thermomechanical sketch of shape memory effect:one-dimensional tensile behavior[J].Res Mechanica,1986,18:251-263.
[10]Liang C,Rogers C A.One-dimensional thermomechanical constitutive relations for shape memory mate-rials[J].Journal of Intelligent Material Systems and Structures,1990,1(2):207-234.
[11]Boyd J G,Lagoudas D C.A thermodynamic constitutive model for the shape memory materials—partⅠ:the monolithic shape memory alloys[J].International Journal of Plasticity,1996,12(6):805-842.
[12]Lagoudas D C,Bo Z,Qidwai M A.A unified thermodynamic constitutive model for SMA and finite ele-ment analysis of active metal matrix composites[J].Mechanics of Advanced Materials and Structures,1996,3(2):153-179.
[13]徐小明,张盛,姚伟岸,钟万勰.基于辛弹性力学解析本征函数的有限元应力磨平方法[J].计算力学学报,2012,29(4):511-516.(XU Xiao-ming,ZHANG Sheng,YAO Wei-an,ZHONG Wan-xie.A stress recovery method based on the analytical eigenfunctions of symplectic elasticity[J].Chi-nese Journal of Computational Mechanics,2012,29(4):511-516.(in Chinese))
[14]Qidwai M A,Lagoudas D C.Numerical implementation of a shape memory alloy thermomechanical con-stitutive model using return mapping algorithms[J].International Journal for Numerical Methods in En-gineering,2000,47(6):1123-1168.
[15]Chen X,Cheng Y G.On accelerated symmetric stiffness techniques for non-associated plasticity withapplication to soil problems[J].Engineering Computations,2011,28(8):1044-1063.
[16]Sloan S W,Abbo A J,Sheng D.Refined explicit integration of elastoplastic models with automatic er-ror control[J].Engineering Computations,2001,18(1/2):121-154.