Crystallographic Texture and Plastic Anisotropy

  • H. J. Bunge
Part of the Engineering Materials book series (ENG.MAT.)


The structure of crystalline materials can be characterized by four structure levels:
  1. 1.

    Crystal Structure specifies the kind and position of atoms in the unit cell of the ideal crystal lattice.

  2. 2.

    Phase Structure specifies the sizes, shapes and mutual arrangement of single-phase volumes (volumes with constant crystal structure).

  3. 3.

    Grain Structure specifies the sizes, shapes, crystal lattice orientation, and mutual arrangement of monocrystal volumes (within the single-phase volumes).

  4. 4.

    Substructure specifies the kind, amount, arrangement, crystallographic orientation of all lattice defects, i.e. all deviations from the ideal crystal lattice such as point defects, dislocations, stacking faults, grain and phase boundaries, the surface, elastic strain, magnetization, electric polarization.



Volume Element Crystal Orientation Crystallographic Texture Orientation Distribution Function Yield Locus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Special Symbols


Relative shear stress of the glide system n


Antisymmetric part of a tensor


Body centered cubic structure


Series expansion coefficients of the orientation distribution function f(g) (texture coefficients)


Substructure function

d, di

Glide direction


Orientation element (volume element of orientation space)


Displacement vector (after the deformation step dη)


Volume of the sample in which the crystal orientation is g


Dissipated deformation work (monocrystal)

\(d\tilde W\)

Dissipated deformation work (polycrystal)

Deformation step

Solid angular element


Orientation density


Orientation distribution function ODF


Orientation distribution function after the average lattice rotation \(\widetilde {\Delta g}\) (after texture spin)


Face centered cubic structure


Glide system tensor


Microstructure function

g = {φ1 ø, φ2}

Crystal orientation (Euler angles)


Orientation matrix g


Orientation-location function (orientation stereology)


Orientation of the principle strain axes with respect to the sample coordinate system KA


Orientation of the principle stress axes with repect to the sample coordinate system KA


Normal direction to the lattice plane (hkl)


Miller indices of a crystal lattice plane


Number of the phase


Sample coordinate system


Crystal coordinate system

L, L0

Series expansion degree (series truncation)


Taylor factor (monocrystal)

\(\tilde M\)

Polycrystal average of the Taylor factor


Number of independent spherical harmonics of the degree λ (crystal symmetry)


Strain-rate sensitivity factor


Series expansion coefficients of monocrystal Taylor factor M(g)


Number of glide systems


Number of independent spherical harmoncis of the degree λ (sample symmetry)


Normal direction to the glide plane


Pole density


Pole density distribution function (pole figure)


Contraction ratio


Contraction ratio requiring minimum deformation work

R, Rij

Lattice rotation rate, rotation rate tensor

\(\tilde R\)

Polycrystal average of the rotation rate


Rotation axis (parameter of g)


Lankford parameter


Numerical coefficients


Symmetrical part of a tensor


Generalized spherical harmonics


Total volume


Volume of the polycrystalline subsample at the location X


Monocrystalline volume element at the location x

X={X1 X2, X3}

Location of a polycrystalline volume element

x={x1, x2, x3}

Location of a monocrystalline volume element


Spherical polar coordinates of a sample direction


Numerical factors characterizing linear independent solutions


Angle in the sheet plane towards the rolling direction


Glide rate in the glide system n


Lattice rotation (after the deformation step dη) (lattice spin)

\(\widetilde {\Delta g}\)

Average lattice rotation after the deformation step dη (texture spin)


Strain, strain tensor

\(\tilde \varepsilon \)

Averaged strain


Total deformation degree

σ, σij

Stress, stress tensor

σ1, σ2

Principle stresses


Stress component falling into the glide system n


Critical resolved shear stress in the glide system n


Reference shear stress (“hardness”)


Texture changing rate


Euler angle (parameter of g)


Euler angle (parameter of g)


Euler angle (parameter of g)


Rotation angle (parameter of g)


Symbol characterizing polycrystal average


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References to Chapter 2

  1. 2.1
    Bunge, H. J.; Schwarzer, R.: Orientation stereology — a new branch of texture studies (in German), TU Contact (Clausthal), 2 (1998), 67–73.Google Scholar
  2. 2.2
    Bunge, H. J.; Schwarzer, R.: Orientation stereology — a new branch in texture research, Advanced Engineering Materials (2000), in print.Google Scholar
  3. 2.3
    Bunge, H. J.: Texture analysis in materials science, mathematical methods, London, Butterworths 1982.Google Scholar
  4. 2.4
    Fischer, A. H.; Schwarzer, R.: X-ray pole figure measurement and texture mapping of selected areas using an X-ray scanning apparatus, Texture and anisotropy of polycrystals, Materials Science Forum 273–275 (1998), 255–262.CrossRefGoogle Scholar
  5. 2.5
    Klein, H.; Bunge, H. J.: Location resolved texture analysis, Z. Metallkde. 90 (1999), 103–110.Google Scholar
  6. 2.6
    Cosserat, E.; Cosserat, F.: Theory of deformable bodies (in French), Paris, Herman et fils 1909.Google Scholar
  7. 2.7
    Wassermann; G.; Grewen, J.: Textures of metallic materials (in German), Berlin, Springer-Verlag 1962.Google Scholar
  8. 2.
    Adams, B. L.. et al.: Orientation imaging microscopy: new possibilities for micro-structural investigations using automated BKD analysis, in: Textures of Materials, Proc. ICOTOM-10, Materials Science Forum 157–162 (1994), 31–43.Google Scholar
  9. 2.9
    Schwarzer, R. A.: Automated crystal lattice orientation mapping using a computer controlled SEM, Micron 28 (1997), 249–265.CrossRefGoogle Scholar
  10. 2.10
    Sachs, G.: On the derivation of a yield condition (in German), Z. VDI 72 (1928), 734.Google Scholar
  11. 2.11
    Canova, G. R.; Kocks, U. F.; Jonas, J. J.: Theory of torsion texture development, Acta Met. 32 (1984), 211–266.CrossRefGoogle Scholar
  12. 2.13
    Taylor, G. I.: Plastic strain in metals, J. Inst. Metals 62 (1938), 307–324.Google Scholar
  13. 2.14
    Bishop, J. F.; Hill, R.: A theoretical deviation of the plastic properties of a polycrystalline face-centered metal, Phil. Mag. Ser. 7 42 (1951), 1298–1307.Google Scholar
  14. 2.15
    Honneff, H.; Mecking, H.: A method for the determination of the active slip systems and orientation changes during single crystal deformation, in (ed) Gottstein, G.; Lücke, K.: Texture of materials, Proc. ICOTOM-5, Vol. 1, Berlin, Springer-Verlag 1978, 265–275.Google Scholar
  15. 2.16
    Kröner, E.: On the plastic deformation of the polycrystal (in German), Acta Met. 9 (1961), 155–161.CrossRefGoogle Scholar
  16. 2.17
    Molinari, A.; Canova, G. R.; Ahzi, S.: A self-consistent approach of the large deformation polycrystal viscoplasticity, Acta Met. 35 (1987), 2983–2994.CrossRefGoogle Scholar
  17. 2.18
    Tome, C. N.; Canova, G. R.: Self-consistent modelling of heterogeneous plasticity, in: Kocks, U. F.; Tome, C. N.; Wenk, R.: Texture and anisotropy, Cambridge University Press 1998.Google Scholar
  18. 2.
    Dawson, P. R.; Beaudoin, A. J.; Mathur, K. K.: Finite element modelling of polycrystalline solids, in: Textures of materials, Proc. ICOTOM-10, Materials Science Forum 157–162 (1994), 1703–1712.Google Scholar
  19. 2.20
    Dawson, P- R.; Beaudoin, A. J.: Finite element simulation of metal forming, in: Kocks, U. F.; Tome, C. N.; Wenk, H. R.: Texture and anisotropy, Cambridge Uni-University Press 1998, 532–559.Google Scholar
  20. 2.21
    Iwakuma, T.; Nemat-Nasser, S.: Finite element elastic-plastic deformation of polycrystalline metals, Proc. Roy. Soc. London A 394 (1984), 87–119.CrossRefGoogle Scholar
  21. 2.22
    Masson, R.; Zaoui, A.: Self-consistent estimates for the rate-dependent elastoplastic behaviour of polycrystalline materials, J. Mechanics and Physics of Solids 47 (1999), 1543.Google Scholar
  22. 2.
    Van Houtte, P.: Some recent developments in the theories for deformation texture Prediction, in: Textures of Materials, Proc. ICOTOM-7, Netherlands Society for Materials Science (1984), 7–13.Google Scholar
  23. 2.24
    Zaoui, A.: Quasi-physical modelling of plastic behaviour of polycrystals, in: (ed) Gittus, J.; Zarka, J.: Modelling small deformations of polycrystals, Amsterdam, Elsevier 1986, 187–225.Google Scholar
  24. 2.25
    Leffers, T.; et al.: Deformation textures: simulation principles, panel report, in (eds) Kallend, J. S.; Gottstein, G.: Textures of materials, Proc. ICOTOM-8, Warrendale, UK, The Metallurgical Society 1988, 265–272.Google Scholar
  25. 2.26
    Lowe, T. C.; et al.: Modelling the deformation of crystalline solids, Warrendale, TMS Publications 1991.Google Scholar
  26. 2.27
    Van Houtte, P.: Microscopic strain heterogeneity and deformation texture prediction, in: (eds) Liang, Z.; Zuo, L.; Chu, Y.: Textures of materials, Proc. ICOTOM11, Beijing, Int. Academic Publishers 1996, 236–247.Google Scholar
  27. 2.28
    Kocks, U. F.; Tome, C. N.; Wenk, H. R.: Texture and anisotropy, Cambridge Uni-University Press 1998.Google Scholar
  28. 2.29
    Raabe, D.: Computational Materials Science, Weinheim, Wiley-VCH 1998.CrossRefGoogle Scholar
  29. 2.30
    Lippmann, H.: A Cosserat theory of plastic flow (in German), Acta Met. 8 (1969), 255–284.Google Scholar
  30. 2.31
    Lippmann, H.: Cosserat plasticity and plastic spin, Appl. Mech. Rev. 48 (1995), 753–762.CrossRefGoogle Scholar
  31. 2.32
    Klein, H.; Bunge, H. J.: Modelling deformation texture formation by orientation flow-field, steel research 62 (1991), 548–559.Google Scholar
  32. 2.33
    Bunge, H. J.; Klein, H.: Model calculations of texture changes by non-unique orientation flow fields, in: (ed) Lee, W. B.: Advances in engineering plasticity and its applications, Amsterdam, Elsevier 1993, 109–117.Google Scholar
  33. 2.34
    Chin, G. Y.: Tension and compression textures, in: (eds) Grewen, J.; Wassermann, G.: Textures in research and practice, Berlin, Springer-Verlag 1969, 51–80.Google Scholar
  34. 2.35
    Bunge, H. J.: Some applications of the Taylor theory of polycrystal plasticity, Kristall und Technik 5 (1970), 145–175.CrossRefGoogle Scholar
  35. 2.36
    Bunge, H. J.; Nielsen, I.: Experimental determination of plastic spin in polycrystalline materials, Int. J. Plasticity 13 (1997), 435–446.CrossRefGoogle Scholar
  36. 2.37
    Clement, A.; Coulomb, P.: Eulerian simulation of deformation textures, Scripta Met. 13 (1979), 899–901.CrossRefGoogle Scholar
  37. 2.38
    Bunge, H. J.; Roberts, W. T.: Orientation distribution, elastic and plastic anisotropy in stabilized steel sheet, J. Appl. Cryst. 2 (1969), 116–128.CrossRefGoogle Scholar
  38. 2.39
    Bunge, H,. J.; Schulze, M.; Grzesik, D.: Calculation of the yield locus of polycrystalline materials according to the Taylor theory, Peine+Salzgitter Berichte, Sonderheft 1980.Google Scholar
  39. 2.40
    Park, N. J.; Klein, H.; Dahlem-Klein, E.: Program system physical properties of textured materials, Göttingen, Cuvillier Verlag 1993.Google Scholar
  40. 2.41
    Hoferlin, E.; van Bael, A.; van Houtte, P.: Influence of texture evolution on finite element simulation of forming processes, in (ed) Szpunar, J.: Textures of Materials, Proc. ICOTOM-12, Ottawa, NRC Research Press 1999, 249–254.Google Scholar
  41. 2.42
    Dahlem-Klein, E.; Klein, H.; Park, N. J.: Program system ODF analysis, Göttingen, Cuvillier Verlag 1993.Google Scholar
  42. 2.43
    Stickels, C. A.; Mould, R. R.: The use of Young’s modulus for predicting the plastic strain ratio of low carbon steels, Met. Trans. 1 (1970), 1303–1312.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • H. J. Bunge

There are no affiliations available

Personalised recommendations