ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ─ DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

Abstract

Abstract: In this two-part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients 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 2- and 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.

2010 MSC Number:

O331

ZHENG Quan-shui;ZOU Wen-nan. ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ─ DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS. Applied Mathematics and Mechanics (English Edition), 2001, 22(8): 865-884.

TrendMD

References

[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. Aself 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 microstructural and physical properties[J]. Proc Roy Soc Lond A, 1998,454(1973):1469- 1498.
[18] Coleman B D, Gurtin M E. Thermodynamics with internal state variables[J]. J Chem Phys, 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: International Tables for Crystallography , 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 Phys 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. Mathematical Handbookfor Scientists and Engineers[M]. 2nd Ed. New York: McGraw-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 . Theory of Group Representations and Applications [M]. 2 nd Ed.Warszawa: Polish Scientific Publishers, 1980.
[31] Broker T, Tom Dieck T. Representations of Compact Lie Groups [M]. New York: SpringerVerlag, 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)