基于中间构形的大变形弹塑性模型

孟春宇; 汤正俊; 陈明祥

doi:10.6052/0459-1879-18-138

基于中间构形的大变形弹塑性模型

武汉大学土木建筑工程学院,武汉430072

详细信息

作者简介:
作者简介: 1) 陈明祥，教授，主要研究方向：材料宏观本构关系.E-mail: mxchen@whu.edu.cn

中图分类号: O33;
计量
- 文章访问数: 1487
- HTML全文浏览量: 233
- PDF下载量: 280
出版历程
- 刊出日期: 2019-01-17

A LARGE DEFORMATION ELASTOPLASTIC MODEL BASED ON THE INTERMEDIATE CONFIGURATION

School of Civil Engineering, Wuhan University, Wuhan 430072, China

摘要

摘要: 在大变形弹塑性本构理论中,一个基本的问题是弹性变形和塑性变形的分解.通常采用两种分解方式,一是将变形率(或应变率)加法分解为弹性和塑性两部分,其中,弹性变形率与Kirchhoff应力的客观率通过弹性张量联系起来构成所谓的次弹性模型,而塑性变形率与Kirchhoff应力使用流动法则建立联系;另一种是基于中间构形将变形梯度进行乘法分解,它假定通过虚拟的卸载过程得到一个无应力的中间构形,建立所谓超弹性-塑性模型.研究了基于变形梯度乘法分解并且基于中间构形的大变形弹塑性模型所具有的若干性质,包括：在不同的构形上,塑性旋率的存在性、背应力的对称性、塑性变形率与屈服面的正交性以及它们之间的关系.首先,使用张量函数表示理论,建立了各向同性函数的若干特殊性质,并导出了张量的张量值函数在中间构形到当前构形之间进行前推后拉的简单关系式.然后,基于这些特殊性质和关系式,从热力学定律出发,建立模型在不同构形上的数学表达,包括客观率表示的率形式和连续切向刚度等,从而获得模型所具有的若干性质.最后,将模型与4种其他模型进行了比较分析.
- 大变形弹塑性模型 /
- 变形梯度乘法分解 /
- 张量函数表示理论 /
- 本构关系 /
- 基于中间构形
Abstract: In the large deformation elastoplastic constitutive theory, a basic problem is the decomposition of elastic deformation and plastic deformation. In the usual case, two decomposition methods are adopted. One method is to decompose deformation rate (or strain rate) into elastic and plastic parts. Among them, the elastic deformation rate and the objective rate of Kirchhoff stress are linked by elastic tensors, and by using this way, a sub-elastic model can be established. In the mean time,the plastic deformation rate is related to Kirchhoff stress by using flow law. Another method is to decompose the deformation gradient tensor based in the intermediate configuration. It is supposed that by considering the virtual unloading process, an unstressed intermediate configuration can be obtained and the so-called hyperelastic-plastic model can be established. In this paper, a large number of properties of a large deformation elastoplastic model which is based on the deformation gradient multiplicative decomposition are studied, and the model is built in the intermediate configuration. These properties include: in different configurations, the existence of the plastic spin rate; the symmetry of back stress; the orthogonality of plastic deformation rate and yield surface; and the relationships between plastic spin rate, back stress, plastic deformation rate and yield surface. First of all, by using the tensor function representation theorem which is in the appendix, some special properties of the isotropic function are obtained and some formula relationships are established, and some simple relationships of tensor value function between the intermediate configuration and the current configuration are derived. Secondly, based on these properties and relationships, and in combination with the laws of thermodynamics, the mathematical expression of the model in different configurations is established, which consists of objective rate representation and continuous tangential stiffness. Thus, some properties of the large deformation elastoplastic model based on the intermediate configuration are obtained. Finally, the model is compared and analyzed with the four models.

HTML全文

参考文献(1)

[1]	谢多夫. 连续介质力学. 李植译. 北京: 高等教育出版社, 2007
[1]	(Aleksandr Aleksandroviq Andronov.Continuum Mechanics, fransl. Li Zhi. Beijing: Higher Education Press, 2007 (in Chinese))
[2]	黄克智, 程莉. 大变形弹塑性本构理论的几个基本问题. 力学学报, 1989, 21(s1): 7-17
[2]	(Huang Kezhi, Cheng Li.Some fundamental problems in elastic-plastic constitutive theory of finite deformation. Chin J Theor Appl Mech, 1989, 21(s1): 7-17 (in Chinese))
[3]	Bertram A.Elasticity and Plasticity of Large Deformations. Berlin,Heidelberg: Springer 2005
[4]	Lee EH.Elasto-plastic deformation at finite strain. Journal of Applied Mechanics, 1969, 36: 1-23
[5]	Lee EH, Mcmeeking RM.Concerning elastic and plastic components of deformation. International Journal of Solids & Structures, 1980, 16(8): 715-721
[6]	Brannon RM.Caveats concerning conjugate stress and strain measures for frame indifferent anisotropic elasticity. Acta Mechanica, 1998, 129(1-2): 107-116
[7]	Nemat-Nasser S.Decomposition of strain measures and their rates in finite deformation elastoplasticity. International Journal of Solids & Structures 1979, 15(2): 155-166
[8]	Nemat-Nasser S.On finite deformation elasto-plasticity. International Journal of Solids & Structures, 1982, 18(10): 857-872
[9]	Nemat-Nasser S.Certain basic issues in finite-deformation continuum plasticity. Meccanica, 1990, 24: 223-229
[10]	Schieck B, Stumpf H.Deformation analysis for finite elastic-plastic strains in a lagrangean-type description. International Journal of Solids & Structures, 1993, 30(19): 2639-2660
[11]	Ghavam K, Naghdabadi R.Hardening materials modeling in finite elastic--plastic deformations based on the stretch tensor decomposition. Materials & Design, 2008, 29(1): 161-172
[12]	Heidari M, Vafai A, Desai C.An Eulerian multiplicative constitutive model of finite elastoplasticity. European Journal of Mechanics-A Solids, 2009, 28(6): 1088-1097
[13]	Moran B, Ortiz M, Shih CF.Formulation of implicit finite element methods for multiplicative finite deformation plasticity. International Journal for Numerical Methods in Engineering, 2010, 29(3): 483-514
[14]	王自强. 理性力学基础. 北京: 科学出版社, 2000
[14]	(Wang Ziqiang.Foundation of Rational Mechanics. Beijing: Science Press, 2000 (in Chinese))
[15]	Mandel J.Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. International Journal of Solids & Structures, 1973, 9(6): 725-740
[16]	Asaro RJ.Micromechanics of crystals and polycrystals. Advances in Appl Mech, 1983, 23(8): 1-115
[17]	Mandel J.Plasticité Elassique et Viscoplasticité. Vienna-New York: Springer-Verlag, 1972
[18]	Coleman BD, Noll W.The Thermodynamics of Elastic Materials with Heat Conduction and Viscosity. The Foundations of Mechanics and Thermodynamics. Berlin, Heidelberg: Springer 1974: 167-178
[19]	Coleman BD, Gurtin ME.Thermodynamics with internal state variables. Journal of Chemical Physics, 2004, 47(2): 597-613
[20]	Lubliner J.Normality rules in large-deformation plasticity. Mechanics of Materials, 1986, 5(1): 29-34
[21]	Xiao H, Bruhns QT, Meyers A.A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient. International Journal of Plasticity, 2000, 16(2): 143-177
[22]	Xiao H, Bruhns QT, Meyers A.The integrability criterion in finite elastoplasticity and its constitutive implications. Acta Mechanica, 2007, 188(3-4): 227-244
[23]	Eidel B, Gruttmann F, Elastoplastic orthotropy at finite strains: Multiplicative formulation and numerical implementation. Computational Materials Science, 2003, 28: 732-742
[24]	Xiao H, Bruhns OT, Meyers A,Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica. 1997, 124: 89-105
[25]	Xiao H, Bruhns OT, Meyers A.Hypoelasticity model based upon the logarithmic stress rate. Journal of Elasticity, 1997, 47: 51-68
[26]	Xiao H, Bruhns OT, Meyers A.Elastoplasticity beyond small deformations. Acta Mechanica, 2006, 182: 31-111
[27]	Xiao H, Chen LS.Hencky's logarithmic strain and dual stress--strain and strain--stress relations in isotropic finite hyperelasticity. International Journal of Solids & Structures, 2003, 40(6): 1455-1463
[28]	Charlton DJ, Yang J, Teh KK.A review of methods to characterize rubber elastic behavior for use in finite element analysis. Rubber Chemistry & Technology, 67.3(1994): 481-503
[29]	Frederick CO, Armstrong PJ.A mathematical representation of the multiaxial Bauschinger effect. High Temperature Technology, 2007, 24(1): 1-26
[30]	Dettmer W, Reese S.On the theoretical and numerical modelling of Armstrong--Frederick kinematic hardening in the finite strain regime. Computer Methods in Applied Mechanics & Engineering, 2004, 193(1): 87-116
[31]	Zheng QS.Theory of representations for tensor functions-A unified invariant approach to constitutive equations. Agronomy Journal, 1994, 47(11): 545
[32]	兑关锁, 王正道, 金明. 各向同性张量函数的导数. 中国科学:物理学力学天文学, 2006, 36(1): 89-102
[32]	(Dui Guansuo, Wang Zhengdao, Jin Ming. The derivative of an isotropic tensor. Scientia Sinica Physica,Mechanica & Astronomica, 2006, 36(1): 89-102(in Chinese))
[33]	王足. 连续介质力学中某些物理量的近似和大变形弹塑性定义的比较. [博士论文]. 北京:北京交通大学, 2010
[33]	(Wang Zu.The approximation of some variables and the comparison of elasto-plastic large deformation definitions in continuum mechanics. [PhD Thesis]. Beijing: Beijing Jiaotong University, 2010 (in Chinese))