ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)──CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

Applied Mathematics and Mechanics (English Edition) ›› 2001, Vol. 22 ›› Issue (8): 885-903.

• Articles • Previous Articles Next Articles

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:2000-10-09 Revised:2001-03-20 Online:2001-08-18 Published:2001-08-18
Supported by:
the National Natural Science Foundation of China(19525207, 19891180);the Y-D Huo Education Foundation

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 mth term of the Fourier expansion of a 3-D ODF make up just a single irreducible mth-order tensor, the coefficients in the mth term of the Fourier expansion of a 3-D CODF constitute generally so many as 2m+1 irreducible mth-order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3-D CODFs imposed by various micro- and 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 mth-order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point-group symmetries, which are also thoroughly investigated in the present part in both 2- and 3-D spaces.

2010 MSC Number:

O331

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. Applied Mathematics and Mechanics (English Edition), 2001, 22(8): 885-903.

TrendMD

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] Hahn T. Space- Group Symmetry [M]. In: International Tables for Crystallography, Vol. A, 2nd Ed. Dordrecht: D Reidel.1987.
[6] 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.
[7] Barut A O, Raczka R. Theory of Group Representations and Applications[M]. 2nd Ed. Warszawa: Polish Scientific Publishers, 1980.
[8] Brocker T, Tom Dieck T. Representations of Compact Lie Groups [M]. New York: SpringerVerlag, 1985.
[9] Murnagham F D. The Theory of Group Representations [M]. Baltimore, MD: Johns Hopkins Univ Press, 1938.
[10] 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.
[11] Korn G A, Korn T M. Mathematical Handbook for Scientists and Engineers[M]. 2nd Ed. New York: McGraw-Hill, 1968.
[12] Eringen A C. Mechanics of Continua[M]. 2nd Ed. New York: Wiley,1980.
[13] Zhang J M, Rychlewski J. Structural tensors for anisotropic solids[J]. Arch Mech, 1990,42:267-277.
[14] 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.
[15] Smith G F, Smith M M, Rivlin R S. Integrity bases for a symmetric tensor and a vector- the crystal classes[J]. Arch Ratl Mech Anal, 1963,12(1):93-133.
[16] Smith G F. Constitutive Equations for Anisotropic and Isotropic Materials [M]. Amsterdam:North- Holland, 1994.
[17] 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.
[18] Hannabuss K C. The irreducible components of homogeneous functions and symmetric tensors[J].J Inst Maths Applics, 1974,14(11):83-88.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)──CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 3

Recommended Articles

Metrics

Comments

[1]	B. AMIRIAN;R. HOSSEINI-ARA;H. MOOSAVI. Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model [J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(7): 875-886.
[2]	WANG Zhi-qiao;DUI Guan-suo. Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions [J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(9): 1249-1257 .
[3]	ZHENG Quan-shui;ZOU Wen-nan. ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ─ DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS [J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(8): 865-884.