Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅱ)-Crystal Distribution Functions and Irreducible Tensors Restricted by Various Material Symmetries

ZHENG Quan-shui; FU Yi-bin

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Article Navigation > Applied Mathematics and Mechanics > 2001 > 22(8): 790-805

ZHENG Quan-shui, FU Yi-bin. Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅱ)-Crystal Distribution Functions and Irreducible Tensors Restricted by Various Material Symmetries[J]. Applied Mathematics and Mechanics, 2001, 22(8): 790-805.

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Orientation Distribution Functions for Microstructures of Heterogeneous Materials(Ⅱ)-Crystal Distribution Functions and Irreducible Tensors Restricted by Various Material Symmetries

ZHENG Quan-shui¹,
FU Yi-bin²

1.
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P R China;
2.
Department of Mathematics, University of Keele, Staffordshire, ST5 586, UK

Received Date: 2000-10-09
Rev Recd Date: 2001-03-20
Publish Date: 2001-08-15

Abstract

Abstract

The explicit representations for tensorial Fourier expansion of 3-D crystal orientation distribution functions(CODFs) are established.In comparison with that the coefficients in the m th-term of the Fourier expansion of a 3-D ODF make up just a single irreducible m th order tensor,the coefficients in the m th term of the Fourier expansion of a 3-D CODF constitute generally so many as 2m+1 irreducible m th order tensors.Therefore,the restricted forms of tensorial Fourier expansions of 3-D CODFs imposed by various microand macro-scopic symmetries are further established,and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3-D CODFs contain remarkably reduced numbers of m th order irreducible tensors than the number 2m+1.These results are based on the restricted forms of irreducible tensors imposed by various pointgroup symmetries,which are also thoroughly investigated in the present part in both 2and 3-D spaces.
- crystal orientation distribution function,
- irreducible tensor,
- Fourier expansion,
- microstructure,
- material symmetry

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References(22)

References

[1]	Molinari A,Canova G R,Ahzi S.A self consistent approach of the large deformation polycrystal viscoplasticity[J].Acta Metall Mater,1987,35(12):2983-2994.
[2]	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.
[3]	Adams B L,Field D P.A statistical theory of creep in polycrystalline materials[J].Acta Metall Mater,1991,39(10):2405-2417.
[4]	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.
[5]	Molinari A,Canova G R,Ahzi S.A self consistent approach of the large deformation polycrystal viscoplasticity[J].Acta Metall Mater,1987,35(12):2983-2994.
[6]	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.
[7]	Adams B L,Field D P.A statistical theory of creep in polycrystalline materials[J].Acta Metall Mater,1991,39(10):2405-2417.
[8]	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.
[9]	Hahn T.Space-Group Symmetry[M].In:International Tables for Cry stallography,Vol.A,2nd Ed.Dordrecht:D Reidel.1987.
[10]	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.
[11]	Barut A O,Raczka R.Theory of Group Repr esentations and Applications[M].2nd Ed.Warsza-wa:Polish Scientific Publishers,1980.
[12]	Br^Lcker T,Tom Dieck T.Representations of Compact Lie Groups[M].New York:Springer-Ver-lag,1985.
[13]	Murnagham F D.The Theory of Group Representations[M].Baltimore,MD:Johns Hopkins Univ Press,1938.
[14]	Zheng Q S,Spencer A J M.On the canonical representations for Kronecker powers of orthogonal tensors with applications to material symmetry problems[J].Int J Engng Sci,1993,31(4):617-635.
[15]	Korn G A,Korn T M.Ma them atical Handbook for Scientists and Engineers[M].2nd Ed.NewYork:McGraw-Hill,1968.
[16]	Eringen A C.Mechanics of Continua[M].2nd Ed.New York:Wiley,1980.
[17]	Zhang J M,Rychlewski J.Structural tensors for anisotropic solids[J].Arch Mech,1990,42:267-277.
[18]	Zheng Q S,Boehler J P.The description,classification,and reality of material and physical symme-tries[J].Acta Mech,1994,102(1-4):73-89.
[19]	Smith G F,Smith M M,Rivlin R S.Integrity bases for a symmetric tensor and avector-the crystalclasses[J].Arch Ratl Mech Anal,1963,12(1):93-133.
[20]	Smith G F.Con stitutive Equations for Anisotropic and Isotropic Materials[M].Amsterdam:North-Holland,1994.
[21]	Spencer A J M.A note on the decomposition of tensors into traceless symmetric tensors[J].Int J Engng Sci,1970,8(8):475-481.
[22]	Hannabuss K C.The irreducible components of homogeneous functions and symmetric tensors[J].J Inst Maths Applics,1974,14(11):83-88.