非均匀材料细观结构的定向分布函数(Ⅰ)——定向分布函数和不可约张量

郑泉水; 邹文楠

非均匀材料细观结构的定向分布函数(Ⅰ)——定向分布函数和不可约张量

清华大学工程力学系, 北京 100084

基金项目: 国家自然科学基金资助项目(19525207;19891180);霍英东教育基金会资助项目

详细信息

作者简介:
郑泉水(1961- ),男,江西人,教授,博士,教育部长江特聘教授.

中图分类号: O331
计量
- 文章访问数: 2877
- HTML全文浏览量: 257
- PDF下载量: 1096
- 被引次数: 0
出版历程
- 收稿日期: 2000-10-09
- 修回日期: 2001-03-20
- 刊出日期: 2001-08-15

Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅰ)-Directional Distribution Functions and Irreducible Tensors

Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P R China

摘要

摘要: 在最近研究非均匀材料的物理和力学性质的各种基于细观力学的方法中,定向分布函数(ODF)和晶体定向分布函数(CODF)的概念起着重要的作用,它们分别定义在单位球面和旋转群上.本文通过两部分的内容,用具有不可约张量系数的傅立叶展开对它们分别作了深入的研究.群表示理论指出平方可积的定向分布函数可以展开为球谐函数的绝对收敛的傅立叶级数,而其中的球谐函数又能进一步用不可约张量表示.这样一些不可约张量系数的基本重要性在于它们刻划了材料组元和缺陷的体积、形状、相、位置的宏观或全局影响.第(Ⅰ)部分对定义在N维单位球上的定向分布函数的不可约张量Fourier展开的一般性质进行了研究,其中重点是构造二维和三维不可约张量的简单表示,以便于得到它们在各种点群(完全正交群的子群)对称性的约束形式;第(Ⅱ)部分给出了晶体定向分布函数的不可约张量展开的显式表示,并且给出了不可约张量以及定向分布函数和晶体定向分布函数不可约张量展开在各种点群下的约束形式.
- 定向分布函数 /
- 不可约张量 /
- 张量傅立叶展开 /
- 非均匀材料 /
- 细观结构
Abstract: In this two-part paper,a thorough investigation is made on Fourier expansions with irreducible tensorial coefficents for orientation distribution functions(ODFs) and crystal orientation distribution functions(CODFs),which are scalar functions defined on the unit sphere and the rotation group,respectively.Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically-based approaches to mechanical and physical properties of heterogeneous materials.The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors.The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size,shape,phase,position of the material constitutions and defects.In Part(Ⅰ),the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N-dimensional(N-D) unit sphere is carried out.Attention is particularly paid to constructing simple expressions for 2and 3-D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point-group(the synonym of subgroup of the full orthogonal group) symmetries.In the continued work(Part Ⅱ),the explicit expression for the irreducible tensorial expansions of CODFs is established.The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point-group symmetries are derived.
- orientation distribution function /
- irreducible tensor /
- tensorial Fourier expansion /
- heterogeneous material /
- microstructure

HTML全文

参考文献(35)

[1]	Tamuzs V,Lagzdn'sh A Zh.A scheme of a phenomenological fracture theory[J].Mekhan Polim,1968,(4):638-647; see also:Kuksenko V S,Tamuzs V.Fracture Micromechanics of Polymer Materials[M].Boston:Martinus Nijhoff Publ,1981.
[2]	Lagzdyn.sh A Zh,Tamuzs V.Construction of a phenomenological theory of fracture of anisotropic media[J].Polymer Mechanics,1971,7:563-571.
[3]	Lagzdyn'sh A Zh,Tamuzs V.Orientation Averaging in Mechanics of Solids[M].Longman Scientific & Technical Publ,1992.
[4]	Bunge H J.Texture Analysis in Material Science[M].London:Butterworths,1982.
[5]	Onat E T.Representation of mechanical behaviour in the presence of internal damage[J].Engng Fract Mech,1986,25(5-6):605-614.
[6]	Onat E T,Leckie F A.Representation of mechanical behaviour in the presence of changing internal structure[J].Trans ASME,J Appl Mech,1988,55(1):1-10.
[7]	Adams B L,Boehler J P,Guidi M,et al.Group theory and representation of microstructure and mechanical behaviour of polycrystals[J].J Mech Phys Solids,1992,40(4):723-737.
[8]	Kanatani K I.Distribution of directional data and fabric tensors[J].Int J Engng Sci,1984,22(2):149-164.
[9]	Advani S G,Tucker Ⅲ C L.The use of tensors to describe and predict fiber orientation in short fiber composites[J].J Rheology,1987,31(8):751-784.
[10]	Advani S G,Tucker Ⅲ C L.Closure approximation for three-dimensional structure tensors[J].J Rheology,1990,34(3):367-386.
[11]	Molinari A,Canova G R,Ahzi S.A self consistent approach of the large deformation polycrystal viscoplasticity[J].Acta Metall,1987,35(12):2983-2994.
[12]	Harren S V,Asaro R J.Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model[J].J Mech Phys Solids,1989,37(2):191-232.
[13]	Adams B L,Field D P.A statistical theory of creep in polycrystalline materials[J].Acta Metall Mater,1991,39(10):2405-2417.
[14]	Krajcinovic D,Mastilovic S.Some fundamental issues of damage mechanics[J].Mech Mat,1995,21(3):217-230.
[15]	He Q C,Curnier A.A more fundamental approach to damaged elastic stress-strain relations[J].Int J Solids Struct,1995,32(10):1433-1457.
[16]	Chen M X,Zheng Q S,Yang W.A micromechanical model of texture induced orthotropy in planar crystalline polymers[J].J Mech Phys Solids,1996,44(2):157-178.
[17]	Zheng Q S,Collins I F.The relationship of damage variables and their evolution laws and micro-structural and physical properties[J].Pr oc Roy Soc Lond A,1998,454(1973):1469-1498.
[18]	Coleman B D,Gurtin M E.Thermodynamics with internal state variables[J].J Chem Phy s,1967,47(2):597-613.
[19]	Noll W.A mathematical theory of the mechanical behaviour of continuous media[J].Arch Ratl Mech Anal,1958,2(2):197) 226.
[20]	Noll W.A new mathematical theory of simple materials[J].Arch Ratl Mech Anal,1972,48(1):1-50.
[21]	Hahn T.Space-Group Symmetry[M].In:Inter national Tables for Cry stallo graphy,Vol,A,2nd ed.Dordrecht:D Reidel,1987.
[22]	Chen M X,Yang W,Zheng Q S.Simulation of crack tip superblunting of semi-crystalline polymers[J].J Mech Phy s Solids,1998,46(2):337-356.
[23]	Zheng Q S,Spencer A J M.Tensors which characterize anisotropies[J].Int J Engng Sci,1993,31(5):679-693.
[24]	Zheng Q S.Two-dimensional tensor function representations for all kinds of material symmetry[J].Proc R Soc Lond A,1993,443(1917):127-138.
[25]	Zheng Q S,Boehler J P.The description,classification,and reality of material and physical symmetries[J].Acta Mech,1994,102(1-4):73-89.
[26]	Zheng Q S.Theory of representations for tensor functions:A unified invariant approach to constitutive equations[J].Appl Mech Rew,1994,47(11):554-587.
[27]	Korn G A,Korn T M.Mathem atical Handbook for Scientists a nd Engineers[M].2th Ed.New York:Mc Graw-Hill,1968.
[28]	Ryser H J.Combinatorial Mathematics[M].New York:The Mathematical Association of America,1963.
[29]	Zheng Q S.On the roles of initial and induced anisotropies[A].In:D F Parker,A H England Eds.IUTAM Symposium on Anisotropy,Inhomogeneity and Nonlinearity in Solid Mechanics[C].Dordrecht:Kluwer Academic Publishers,1995,57-62.
[30]	Barut A O,Raczka R.Theor y of Gr oup Repr esentation s and Applications[M].2nd Ed.Warszawa:Polish Scientific Publishers,1980.
[31]	Brêcker T,Tom Dieck T.Repr esentations of Compact Lie Groups[M].New York:Springer-Verlag,1985.
[32]	Spencer A J M.A note on the decomposition of tensors into traceless symmetric tensors[J].Int J Engng Sci,1970,8(6):475-481.
[33]	Hannabuss K C.The irreducible components of homogeneous functions and symmetric tensors[J].J Inst Maths Applics,1974,14(1):83-88.
[34]	Zheng Q S,Zou W N.Irreducible decompositions of physical tensors of high orders[J].J Engrg Math,2000,37(1-3):273-288.
[35]	Zou W N,Zheng Q S,Rychlewski J,et al.Orthogonal irreducible decomposition of tensors of high orders[J].Math Mech Solids,2001.(in Press)