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Orientation distribution functions for microstructures of heterogeneous materials (II)—Crystal distribution functions and irreducible tensors restricted by various material symmetries

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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 a3-D CODF constitute generally so many as2m+1 irreducible mth-order tensors. Therefore, the restricted forms of tensorial Fourier expansions of3-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 of3-D CODFs contain remarkably reduced numbers of mth-order irreducible tensors than the number2m+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 both2- and3-D spaces.

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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.

    Article  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  3. Adams B L, Field D P. A statistical theory of creep in polycrystalline materials[J].Acta Metal Mater, 1991,39(10):2405–2417.

    Article  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  5. Hahn T.Space-Group Symmetry[M]. InInternational Tables for Crystallography, Vol. A, 2nd Ed. Dordrecht: D Reidel. 1987.

    Google Scholar 

  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.

    Google Scholar 

  7. Barut A O, Raczka R.Theory of Group Representations and Applications[M]. 2nd Ed. Warszawa: Polish Scientific Publishers, 1980.

    Google Scholar 

  8. Bröcker T, Tom Dieck T,Representations of Compact Lie Groups[M]. New York: Springer-Verlag, 1985.

    Google Scholar 

  9. Murnagham F D.The Theory of Group Representations[M] Baltimore, MD: Johns Hopkins Univ Press, 1938.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  11. Korn G A, Korn T M.Mathematical Handbook for Scientists and Engineers [M]. 2nd Ed. New York: McGraw-Hill, 1968.

    Google Scholar 

  12. Eringen A C.Mechanics of Continua[M]. 2nd Ed. New York: Wiley, 1980.

    Google Scholar 

  13. Zhang J M, Rychlewski J. Structural tensors for anisotropic solids[J].Arch Mech, 1990,42: 267–277.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  16. Smith G F.Constitutive Equations for Anisotropic and Isotropic Materials [M]. Amsterdam: North-Holland, 1994.

    Google Scholar 

  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.

    Article  MATH  Google Scholar 

  18. Hannabuss K C. The irreducible components of homogeneous functions and symmetric tensors[J].J Inst Maths Applics, 1974,14(11):83–88.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Zheng Quan-shui Professor, Doctor (Chair Professor of the Ministry of Education, China)

  2. Department of Mathematics, University of Keele, ST5 586, Staffordshire, UK

    Fu Yi-bin

Authors
  1. Zheng Quan-shui Professor, Doctor
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  2. Fu Yi-bin
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Additional information

Paper from Zheng Quan-shui, Member of Editorial Committee, AMM

Foundation item: the National Natural Science Foundation of China (19525207, 19891180); the Y-D Huo Education Foundation

Biography: Zheng Quan-shui (1961≈)

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Quan-shui, Z., Yi-bin, F. Orientation distribution functions for microstructures of heterogeneous materials (II)—Crystal distribution functions and irreducible tensors restricted by various material symmetries. Appl Math Mech 22, 885–903 (2001). https://doi.org/10.1007/BF02436388

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  • DOI: https://doi.org/10.1007/BF02436388

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