Abstract
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
where ε 2; ε 3 are the strains in the width and thickness directions, respectively. Eq. 4.1 can be written in the form
where b0 and b are the initial and final width, while t0 and t are the initial and final thickness of the specimen, respectively.
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Abbreviations
- 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
- b:
-
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
- b0 :
-
initial width of the specimen
- c:
-
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
- CD :
-
material constant in the Drucker yield criterion
- D:
-
strain-rate tensor
- E:
-
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
- g(α):
-
function used to define the Budiansky yield criterion
- g(θ, α):
-
function used to define the Ferron yield criterion
- hij :
-
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
- L:
-
linear transformation tensor in the Karafillis-Boyce yield criterion
- M:
-
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
- R:
-
material parameter in the Lin-Ding yield criterion
- r, R:
-
normal anisotropy coefficient
- r:
-
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
- s:
-
exponent in the Lin-Ding yield criterion
- S:
-
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
- Wf :
-
energy of distortion
- Wp :
-
elastic potential energy
- Wv :
-
volumetric change energy
- X, Y, Z:
-
tensile yield stresses in principal anisotropic directions (Hill ‘48)
- Y:
-
yield stress
- α:
-
angle between principal stress σ 1 and rolling direction
- α = σ2/σ1 :
-
ratio of the principal stresses
- αl, α2, α3 :
-
coefficients in the Barlat 1994 yield criterion
- α l, α 2, γ l, γ 2, γ 3, 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
- Δr:
-
variation of anisotropy coefficients
- εe :
-
equivalent (effective) strain
- ε1, ε2, ε3 :
-
principal (logarithmic) strains
- λ:
-
parameter of the Bézier function used in Vegter’s yield criterion
- λ:
-
plastic multiplier in the flow rule
- µ:
-
Poisson’s ratio
- σ:
-
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
- σb :
-
equibiaxial yield stress
- σe :
-
equivalent (effective) stress
- σu :
-
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
- τ:
-
shear yield stress
References to Chapter 4
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.
Wech, P. I.; Radtke, L.; Bunge, H. J.: Comparison of plastic anisotropy parameters, Sheet Metal Ind. (1983), 594–597.
Findley, W. N.; Michno, M. J.: A historical perspective of yield surface investigations for Metals, Int. J. Non-Linear Mech. 11 (1976), 59–80.
Zyckovski, M.: Combined loadings in the theory of plasticity. Polish Scientific Publishers, Warsaw 1981.
Barlat, F.; Lege, D. J.; Brem, J. C.: A six-component yield function for anisotropic materials, Int. J. Plasticity 7 (1991), 693–712.
Tresca, H.: On the yield of solids at high pressures (in French), Comptes Rendus Academie des Sciences, Paris 59 (1864), 754.
Huber, M. T.: C T 22 (1904), 34–81.
Mises, R.: Mechanics of solids in plastic state 592 (in German), Göttinger Nachrichten Math. Phys. Klasse 1 (1913), 582.
Hencky, H.: On the theory of plastic deformations 592 (in German), Z. Ang. Math. Mech. 4 (1924), 323–334.
Drucker, D. C.: Relations of experiments to mathematical theories of plasticity, J. Appl. Mech. 16 (1949), 349–357.
Hosford, W. F.: A generalised isotropic yield criterion, J. Appl. Mech. 39 (1972), 607–609.
von Mises, R.V.: Mechanics of plastic deformation of crystals 592 (in German), Z. Ang. Math. Mech.. 8 (1928), 161–185.
Olszak, W.; Urbanowski, W.: The orthotropy and the non-homogeneity in the theory of plasticity, Pol. Arch. Mech. Stos. 8 (1956), 85–110.
Hill, R.: A theory of the yielding and plastic flow of anisotropic metals, Proc. Roy. Soc. London A 193 (1948), 281–297.
Woodthrope, J.; Pearce, R.: The anomalous behaviour of aluminium sheet under balanced biaxial tension, Int. J. Mech. Sci., 12 (1970), 341–347.
Pearce, R.: Some aspects of anisotropic plasticity in sheet metals, Int. J. Mech. Sci. 10 (1968), 995–1001.
Hill, R.: Theoretical plasticity of textured aggregates, Math. Proc. Cambridge Philosophical Soc. 85 (1979), 179–191.
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.
Müller, W.: Characterization of sheet metal under multiaxial load (in German). Berichte aus dem Institut für Umformtechnik, Universität Stuttgart, Nr. 123, Berlin, Springer 1996.
Bishop, J. F. W.; Hill, R.: A theory of the plastic distortion of polycrystalline aggregates under combined stress, Phil. Mag. 42 (1951).
Bassani, J. L.: Yield characterisation of metals with transversally isotropic plastic properties, Int. J. Mech. Sci. 19 (1977), 651–654.
Hosford, W. F.: On yield loci of anisotropic cubic metals. In: Proc. 7“’ North American Metalworking Conf. (NMRC), SME, Dearborn, MI (1979), 191–197.
Hosford, W. F.: The Mechanics of Crystals and Textured Polycrystals. New York, Oxford University Press, (1993).
Logan, R.; Hosford, W. F.: Upper-bound anisotropic yield locus calculations assuming (111)–pencil glide, Int. J. Mech. Sci. 22 (1980), 419–430.
Hosford, W. F.: On the crystallographic basis of yield criteria, Texture and Micro-Microstructures, 26–27 (1996), 479–493.
Hosford, W. F.: Comments on anisotropic yield criteria, Int. J. Mech. Sci. 27 (1985), 423–427.
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.
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.
Chu, E.: Generalization of Hill’s 1979 anisotropic yield criteria, J. Mater. Process. Technol. 50 (1995), 207–215.
Hill, R.: Constitutive modelling of orthotropic plasticity in sheet metals, J. Mech. Phys. Solids 38 (1990), 405–417.
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.
Hill, R.: A user-friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci. 15 (1993), 19–25.
Stout, M. G.; Hecker, S. S.: Role of geometry in plastic instability and fracture of tubes sheet, Mechanics of Materials 2 (1983), 23–31.
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.
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.
Chung, K.; Shah, K.: Finite element simulation of sheet metal forming for planar anisotropic metals, Int. J. of Plasticity 8 (1992), 453–476.
Vegter, D.: On the plastic behaviour of steel during sheet forming. Thesis, Univ. Twente, The Netherlands, 1991.
Choi, S. H.; et al.: Prediction of yield surfaces of textured sheet metals, Metall. Trans. 30A (1999), 377–379.
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.
Lian, J.; Chen, J.: Isotropic Polycrystal Yield Surfaces of BCC and FCC Metals: Crystallographic and continuum mechanics approaches, Acta Met. 39 (1991), 2285–2294.
Barlat, F.; et al.: Yielding description for solution strengthened aluminium alloys, Int. J. Plasticity, 13 (1997), 185–401.
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.
Barlat, F.; et al.: Yield function development for aluminium alloy sheets, J. Mech. Phys. Solids, 45 (1997), 1727–1763.
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.
Gotoh, M.: A theory of plastic anisotropy based on a yield function of fourth order, Int. J. Mech. Sci. 19 (1977), 505–520.
Zhou, W.: A new non-quadratic orthotropic yield criterion, Int. J. Mech. Sci. 32 (1990), 513–520.
Zhou, W.: A new orthotropic yield function describable anomalous behaviour of materials, Trans. Nonferrous Metals Soc. China 4 (1994), 431–449.
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.
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.
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.
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.
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.
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.
Tourki, Z.; et al.: Orthotropic plasticity in metal sheets. J. Mater. Process. Technol. 45 (1994), 453–458.
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Banabic, D. (2000). Anisotropy of Sheet Metal. In: Banabic, D. (eds) Formability of Metallic Materials. Engineering Materials. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04013-3_4
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