Anisotropy of Sheet Metal

  • D. Banabic
Part of the Engineering Materials book series (ENG.MAT.)


Due to their crystallographic structure and the characteristics of the rolling process, sheet metals generally exhibit a significant anisotropy of mechanical properties. The variation of their plastic behavior with direction is assessed by a quantity called Lankford parameter or anisotropy coefficient [4.1]. This coefficient is determined by uniaxial tensile tests on sheet specimens in the form of a strip. The anisotropy coefficient (r) is defined by
$$r = \frac{{{\varepsilon _2}}}{{{\varepsilon _3}}} $$
where ε 2; ε 3 are the strains in the width and thickness directions, respectively. Eq. 4.1 can be written in the form
$$r = \frac{{In\frac{b}{{{b_0}}}}}{{In\frac{t}{{{t_0}}}}} $$
where b0 and b are the initial and final width, while t0 and t are the initial and final thickness of the specimen, respectively.


Principal Stress Yield Function Yield Surface Yield Criterion Anisotropic Material 
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

a, b

coefficients in the Hill 1990 yield criterion

a, b, c, f, g, h

material parameters in the Barlat 1991 yield criterion

a, b, c, h, p

coefficients in the Barlat 1989 yield criterion

a, b, m, n, p, q

parameters describing the planar anisotropy of the material in the Ferron yield criterion

A0, ..., A9

coefficients in the Gotoh yield criterion


final width of the specimen

B, C, D, H

coefficients in the Chu yield criterion

b, c, h, α

coefficients in the Zhou 1994 yield criterion


initial width of the specimen


weighting coefficient in the Karafillis-Boyce yield criterion

c, h, n, α1, α2

coefficients in the Montheillet yield criterion

c, p, q

coefficients in the Hill 1993 yield criterion

c1, c2, c3

material coefficients describing the material anisotropy in the Barlat 1994 yield criterion


material constant in the Drucker yield criterion


strain-rate tensor


elastic modulus

f, F, φ

yield function

f, g, h, a, b, c

coefficients in the Hill 1979 yield criterion

F, G, H, L, M, N

coefficients in the Hill 1948 yield criterion


function used to define the Budiansky yield criterion

g(θ, α)

function used to define the Ferron yield criterion


anisotropy coefficients in the von Mises 1928 yield criterion

I2, I3

second and third of the stress tensor

J2, J3

second and third invariants of the stress tensor

K1, K2

invariants of the stress tensor


linear transformation tensor in the Karafillis-Boyce yield criterion


integer exponent used by the yield criteria

m, n

exponents used by the yield criteria

m, n, p, q, r, s

coefficients in the Banabic-Balan yield criterion


material parameter in the Lin-Ding yield criterion

r, R

normal anisotropy coefficient


parameter in the Banabic-Balan yield criterion

R, S, T

shear yield stresses in the principal anisotropie directions (Hill 1948)

r0, r45, r90

anisotropy coefficients at 0°, 45° and 90° from the rolling direction


exponent in the Lin-Ding yield criterion


IPE stress tensor used by the Karafillis-Boyce yield criterion

S1, S2, S3

principal deviatoric stresses

Sx, Sy, Sz, Sxy, Syz, Szx

components of the IPE stress tensor used by the Karafillis-Boyce yield criterion

t0, t

initial and final thickness of the specimen


energy of distortion


elastic potential energy


volumetric change energy

X, Y, Z

tensile yield stresses in principal anisotropic directions (Hill ‘48)


yield stress


angle between principal stress σ 1 and rolling direction

α = σ21

ratio of the principal stresses

αl, α2, α3

coefficients in the Barlat 1994 yield criterion

αl, α2, γl, γ2, γ3, C

parameters defining the anisotropy of the material in the Karafillis-Boyce yield criterion

αx, αy, αz

coefficients in the Barlat 1994 yield criterion

βl, β2, β3

auxiliary coefficients used to define the linear transformation tensor in the Karafillis-Boyce yield criterion


variation of anisotropy coefficients


equivalent (effective) strain

ε1, ε2, ε3

principal (logarithmic) strains


parameter of the Bézier function used in Vegter’s yield criterion


plastic multiplier in the flow rule


Poisson’s ratio


actual stress tensor in the Karafillis-Boyce yield criterion

σ0, σ45, σ90

uniaxial yield stress at 0°, 45° and 90° from the rolling direction

σ1, σ2, σ3

principal stresses


equibiaxial yield stress


equivalent (effective) stress


uniaxial yield stress

σx, σy, σz, σxy, σyz, σzx

components of the actual stress tensor in the Karafillis-Boyce yield criterion

σx, σy, τxy

planar components of the stress tensor


shear yield stress

References to Chapter 4

  1. 4.1
    Lankford, W. I.; Snyder, S. C.; Bauscher, J. A.: New criteria for predicting the press performance of deep-drawing sheets, Trans. ASM. 42 (1950), 1196–1232.Google Scholar
  2. 4.2
    Wech, P. I.; Radtke, L.; Bunge, H. J.: Comparison of plastic anisotropy parameters, Sheet Metal Ind. (1983), 594–597.Google Scholar
  3. 4.3
    Findley, W. N.; Michno, M. J.: A historical perspective of yield surface investigations for Metals, Int. J. Non-Linear Mech. 11 (1976), 59–80.CrossRefGoogle Scholar
  4. 4.4
    Zyckovski, M.: Combined loadings in the theory of plasticity. Polish Scientific Publishers, Warsaw 1981.Google Scholar
  5. 4.5
    Barlat, F.; Lege, D. J.; Brem, J. C.: A six-component yield function for anisotropic materials, Int. J. Plasticity 7 (1991), 693–712.CrossRefGoogle Scholar
  6. 4.6
    Tresca, H.: On the yield of solids at high pressures (in French), Comptes Rendus Academie des Sciences, Paris 59 (1864), 754.Google Scholar
  7. 4.7
    Huber, M. T.: C T 22 (1904), 34–81.Google Scholar
  8. 4.8
    Mises, R.: Mechanics of solids in plastic state 592 (in German), Göttinger Nachrichten Math. Phys. Klasse 1 (1913), 582.Google Scholar
  9. 4.9
    Hencky, H.: On the theory of plastic deformations 592 (in German), Z. Ang. Math. Mech. 4 (1924), 323–334.CrossRefGoogle Scholar
  10. 4.10
    Drucker, D. C.: Relations of experiments to mathematical theories of plasticity, J. Appl. Mech. 16 (1949), 349–357.Google Scholar
  11. 4.11
    Hosford, W. F.: A generalised isotropic yield criterion, J. Appl. Mech. 39 (1972), 607–609.CrossRefGoogle Scholar
  12. 4.12
    von Mises, R.V.: Mechanics of plastic deformation of crystals 592 (in German), Z. Ang. Math. Mech.. 8 (1928), 161–185.CrossRefGoogle Scholar
  13. 4.13
    Olszak, W.; Urbanowski, W.: The orthotropy and the non-homogeneity in the theory of plasticity, Pol. Arch. Mech. Stos. 8 (1956), 85–110.Google Scholar
  14. 4.14
    Hill, R.: A theory of the yielding and plastic flow of anisotropic metals, Proc. Roy. Soc. London A 193 (1948), 281–297.CrossRefGoogle Scholar
  15. 4.15
    Woodthrope, J.; Pearce, R.: The anomalous behaviour of aluminium sheet under balanced biaxial tension, Int. J. Mech. Sci., 12 (1970), 341–347.CrossRefGoogle Scholar
  16. 4.16
    Pearce, R.: Some aspects of anisotropic plasticity in sheet metals, Int. J. Mech. Sci. 10 (1968), 995–1001.CrossRefGoogle Scholar
  17. 4.17
    Hill, R.: Theoretical plasticity of textured aggregates, Math. Proc. Cambridge Philosophical Soc. 85 (1979), 179–191.CrossRefGoogle Scholar
  18. 4.18
    Lian, J.; Zhou, D.; Baudelet, B.: Application of Hill’s new theory to sheet metal forming- Pt. I. Hill’s 1979 criterion and its application to predicting sheet forming limits, Int. J. Mech. Sci. 31 (1989), 237–244.CrossRefGoogle Scholar
  19. 4.19
    Müller, W.: Characterization of sheet metal under multiaxial load (in German). Berichte aus dem Institut für Umformtechnik, Universität Stuttgart, Nr. 123, Berlin, Springer 1996.Google Scholar
  20. 4.20
    Bishop, J. F. W.; Hill, R.: A theory of the plastic distortion of polycrystalline aggregates under combined stress, Phil. Mag. 42 (1951).Google Scholar
  21. 4.21
    Bassani, J. L.: Yield characterisation of metals with transversally isotropic plastic properties, Int. J. Mech. Sci. 19 (1977), 651–654.CrossRefGoogle Scholar
  22. 4.22
    Hosford, W. F.: On yield loci of anisotropic cubic metals. In: Proc. 7“’ North American Metalworking Conf. (NMRC), SME, Dearborn, MI (1979), 191–197.Google Scholar
  23. 4.23
    Hosford, W. F.: The Mechanics of Crystals and Textured Polycrystals. New York, Oxford University Press, (1993).Google Scholar
  24. 4.24
    Logan, R.; Hosford, W. F.: Upper-bound anisotropic yield locus calculations assuming (111)–pencil glide, Int. J. Mech. Sci. 22 (1980), 419–430.CrossRefGoogle Scholar
  25. 4.25
    Hosford, W. F.: On the crystallographic basis of yield criteria, Texture and Micro-Microstructures, 26–27 (1996), 479–493.CrossRefGoogle Scholar
  26. 4.26
    Hosford, W. F.: Comments on anisotropic yield criteria, Int. J. Mech. Sci. 27 (1985), 423–427.CrossRefGoogle Scholar
  27. 4.27
    Barlat, F.; Richmond, O.: Prediction of tricomponent plane stress yield surfaces and associated flow and failure behaviour of strongly textured FCC polycrystalline sheets, Mat. Sci. Eng. 91 (1987), 15–29.Google Scholar
  28. 4.28
    Barlat, F.; Lian, J.: Plastic behaviour and stretchability of sheet metals (Part I): A yield function for orthotropic sheet under plane stress conditions, Int. J. Plasticity 5 (1989), 51–56.CrossRefGoogle Scholar
  29. 4.29
    Chu, E.: Generalization of Hill’s 1979 anisotropic yield criteria, J. Mater. Process. Technol. 50 (1995), 207–215.CrossRefGoogle Scholar
  30. 4.30
    Hill, R.: Constitutive modelling of orthotropic plasticity in sheet metals, J. Mech. Phys. Solids 38 (1990), 405–417.CrossRefGoogle Scholar
  31. 4.31
    Lin, S.B.; Ding, J. L.: A modified form of Hill’s orientation-dependent yield criterion for orthotropic sheet metals, J. Mech. Phys. Solids, 44 (1996), 1739–1764.CrossRefGoogle Scholar
  32. 4.32
    Hill, R.: A user-friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci. 15 (1993), 19–25.CrossRefGoogle Scholar
  33. 4.33
    Stout, M. G.; Hecker, S. S.: Role of geometry in plastic instability and fracture of tubes sheet, Mechanics of Materials 2 (1983), 23–31.CrossRefGoogle Scholar
  34. 4.34
    Banabic, D.; Müller, W.; Pöhlandt, K.: Determination of yield loci from cross tensile tests assuming various kinds of yield criteria. In: Sheet metal forming beyond 2000, Brussels 1998, 343–349.Google Scholar
  35. 4.35
    Banabic, D.; et al.: A new criterion for anisotropic sheet metals, 8th Int. Conf. Achievements in the Mechanical and Materials Engineering, Gliwice, Poland 1999, 33–36.Google Scholar
  36. 4.36
    Chung, K.; Shah, K.: Finite element simulation of sheet metal forming for planar anisotropic metals, Int. J. of Plasticity 8 (1992), 453–476.CrossRefGoogle Scholar
  37. 4.37
    Vegter, D.: On the plastic behaviour of steel during sheet forming. Thesis, Univ. Twente, The Netherlands, 1991.Google Scholar
  38. 4.38
    Choi, S. H.; et al.: Prediction of yield surfaces of textured sheet metals, Metall. Trans. 30A (1999), 377–379.Google Scholar
  39. 4.39
    Karafillis, A. P.; Boyce, M. C.: A general anisotropic yield criterion using bounds and a transformation weighting tensor, J. Mech. Phys. Solids 41 (1993), 1859–1886.Google Scholar
  40. 4.40
    Lian, J.; Chen, J.: Isotropic Polycrystal Yield Surfaces of BCC and FCC Metals: Crystallographic and continuum mechanics approaches, Acta Met. 39 (1991), 2285–2294.CrossRefGoogle Scholar
  41. 4.41
    Barlat, F.; et al.: Yielding description for solution strengthened aluminium alloys, Int. J. Plasticity, 13 (1997), 185–401.CrossRefGoogle Scholar
  42. 4.42
    Hayashida, Y. et al.: FEM analysis of punch stretching and cup drawing tests for aluminium alloys using a planar anisotropic yield function, in: Shen, S. F.; Dawson, P. R. (eds): Simulation of materials processing Theory, methods and applica-cations, Rotterdam, Balkema 1995, 712–722.Google Scholar
  43. 4.43
    Barlat, F.; et al.: Yield function development for aluminium alloy sheets, J. Mech. Phys. Solids, 45 (1997), 1727–1763.CrossRefGoogle Scholar
  44. 4.44
    Yoon, J. W.; et al.: A general elasto-plastic finite element formulation based on incremental deformation theory for planar anisotropy and its application to sheet metal forming, Int. J. Plasticity 15 (1999), 35–67.CrossRefGoogle Scholar
  45. 4.45
    Gotoh, M.: A theory of plastic anisotropy based on a yield function of fourth order, Int. J. Mech. Sci. 19 (1977), 505–520.CrossRefGoogle Scholar
  46. 4.46
    Zhou, W.: A new non-quadratic orthotropic yield criterion, Int. J. Mech. Sci. 32 (1990), 513–520.CrossRefGoogle Scholar
  47. 4.47
    Zhou, W.: A new orthotropic yield function describable anomalous behaviour of materials, Trans. Nonferrous Metals Soc. China 4 (1994), 431–449.Google Scholar
  48. 4.48
    Montheillet, F.; Jonas, J. J.; Benferrah, M.: Development of anisotropy during the cold rolling of aluminium sheet, Int. J. Mech. Sci. 33 (1991), 197–209.CrossRefGoogle Scholar
  49. 4.49
    Banabic, D.; Balan, T.; Pöhlandt, K.: Analytical and experimental investigation on anisotropic yield criteria. In: Geiger, M. (ed): Advanced technology of plasticity 1999, Proc. 6th ICTP, Nürnberg, Germany, 1999, 1739–1764.Google Scholar
  50. 4.50
    Banabic, D.; et al.: Some comments on a new anisotropic yield criterion, 7`11 Natl. Conf. Technology and Machine Tools for Cold Metal Forming (TPR 2000), Cluj-Napoca, Romania, 11.–12. May 2000, 93–100.Google Scholar
  51. 4.51
    Banabic, D.; Kuwabara, T.; Balan, T.: Experimental validation of some anisotropic yield criteria. In: Proc. 7th Natl. Conf. Technology and Machine-Tools for Cold Metal Forming (TPR 2000), Cluj-Napoca, Romania, 11.–12. May 2000, 109–116.Google Scholar
  52. 4.52
    Banabic, D.; Comsa, D. S.; Balan, T.: Yield criterion for anisotropic sheet metals under plane-stress conditions. In: Proc. 76 Natl. Conf. Technology and Machine Tools (TPR 2000), Cluj- Napoca, Romania, 11.–12. May 2000, 215–224.Google Scholar
  53. 4.53
    Budiansky, B.: Anisotropic plasticity of plane-isotropic sheets. In: Dvorak G. J.; Shield, R. T. (eds): Mechanics of material behaviour, Amsterdam, Elsevier 1984, 15–29.Google Scholar
  54. 4.54
    Tourki, Z.; et al.: Orthotropic plasticity in metal sheets. J. Mater. Process. Technol. 45 (1994), 453–458.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • D. Banabic

There are no affiliations available

Personalised recommendations