Skip to main content
Book cover

Sheet Metal Forming Processes pp 27–140Cite as

Plastic Behaviour of Sheet Metal

Abstract

In the Sect. 2.8 the summation convention over repeated indices is used. Let A denote second-order tensor and B a forth-order tensor. One can define the double contracted tensor product as \({\textbf{A}}:{\textbf{A}} = A_{ij} A_{ij}\) and \(({\textbf{B}}:{\textbf{A}})_{ij} = B_{ijkl} A_{kl}\). The norm of A is \(\left\| {\textbf{A}} \right\| = \sqrt {{\textbf{A}}:{\textbf{A}}}\) and its direction is \({\textbf{n}} = {{\textbf{A}} \mathord{\left/ {\vphantom {{\textbf{A}} {\left\| {\textbf{A}} \right\|}}} \right. \kern-\nulldelimiterspace} {\left\| {\textbf{A}} \right\|}}\). The time derivative is \({\boldsymbol{\dot A}} = {{d{\textbf{A}}} \mathord{\left/ {\vphantom {{d{\textbf{A}}} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}\).

Keywords

  • Sheet Metal
  • Yield Function
  • Yield Surface
  • Yield Criterion
  • Equivalent Stress

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.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-540-88113-1_2
  • Chapter length: 114 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   219.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-88113-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   279.99
Price excludes VAT (USA)
Hardcover Book
USD   279.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 2.4
Fig. 2.5
Fig. 2.6
Fig. 2.7
Fig. 2.8
Fig. 2.9
Fig. 2.10
Fig. 2.11
Fig. 2.12
Fig. 2.13
Fig 2.14
Fig 2.15
Fig. 2.16
Fig. 2.17
Fig. 2.18
Fig. 2.19
Fig. 2.20
Fig. 2.21
Fig. 2.22
Fig. 2.23
Fig. 2.24
Fig. 2.25
Fig. 2.26
Fig. 2.27
Fig. 2.28
Fig. 2.29
Fig. 2.30
Fig. 2.31
Fig. 2.32
Fig. 2.33
Fig. 2.34

Notes

  1. 1.

    Research Centre in Sheet Metal Forming Technology belong the Technical University of Cluj Napoca, Romania (http://www.certeta.utcluj.ro).

Abbreviations

a :

exponent in the Hershey and Hosford yield criteria

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

a, b, c, d, e, f, g :

coefficients in the BBC 2000 yield criterion

a 1… a 4 :

coefficients in the Cazacu–Barlat yield criteria

a 1… a 25 :

coefficients in the Soare yield criteria

A 0, …, A 9 :

coefficients in the Gotoh yield criterion

b 1 …b 11 :

coefficients in the Cazacu–Barlat yield criteria

c :

weighting coefficient in the Karafillis–Boyce yield criterion

c, p, q :

coefficients in the Hill 1993 yield criterion

c 1, c 2, c 3 :

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

c 12, c 13 …c 66 :

coefficients in the linear transformation in Barlat 2000 yield criterion

C :

elasticity tensor

C ′, C ″ :

linear transformations in Barlat 2000 yield criterion

C D :

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

F 1, F 2 :

functions in the expresion of the uniaxial yield stress (Barlat 1989 yield criterion)

F b :

function used to define the biaxial yield stress and the biaxial anisotropy coefficient

F θ :

function used to define the uniaxial yield stress and the anisotropy coefficient

g(α):

function used to define the Budiansky yield criterion

g(θ, α):

function used to define the Ferron yield criterion

h :

scalar parameter which defines the plastic deformation accumulated in the material

h ij :

anisotropy coefficients in the von Mises 1928 yield criterion

I 2, I 3 :

second and third invariants of the stress tensor

J 2, J 3 :

second and third invariants of the stress tensor

k :

exponent in the the Karafillis–Boyce and BBC yield criteria

k 1, k 2 :

invariants of the stress tensor

l :

final gage length

l 0 :

initial gage length

L :

linear transformation tensor in the Karafillis–Boyce yield criterion

L, M, N :

function in the Comsa yield criterion

M :

integer exponent used by the yield criteria

M, N, P, Q, R, S, T :

coefficients in the BBC yield criteria

m, n :

exponents used by the yield criteria

p :

exponent in the generalized Drucker yield criterion

p :

accumulated equivalent plastic strain

p 1 …p 8 :

coefficients in the Comsa yield criterion

R :

material parameter in the Lin–Ding yield criterion

R :

isotropic hardening variable

R, S, T :

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

r, R :

normal anisotropy coefficient

r b :

biaxial anisotropy coefficient

r θ :

anisotropy coefficient associated to the direction θ

r 0, r 45, r 90 :

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

s :

exponent in the Lin–Ding yield criterion

s :

deviatoric stress tensor in Barlat 2000 yield criterion

S :

IPE stress tensor used by the Karafillis–Boyce yield criterion

S :

stress deviatoric tensor

S 1, S 2, S 3 :

principal deviatoric stresses

S 11, S 22, S 33,:

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

S 12, S 23, S 31 :

-----

t 0, t :

initial and final thickness of the specimen

t 1, t 2 :

functions in the expresion of the uniaxial anisotropy coefficient (Barlat 1989 yield criterion)

T :

transformation matrix in Barlat 2000 yield criterion

w :

final width of the specimen

w 0 :

initial width of the specimen

W f :

energy of distortion

W p :

elastic potential energy

W v :

volumetric change energy

X :

linear transformation stress tensor in Barlat 2000 yield criterion

X, Y, Z :

tensile yield stresses in principal anisotropic directions (Hill 1948)

Y :

yield stress

Y θ :

uniaxial yield stress in a sample inclined by θ with respect to the rolling direction

Y b :

theoretical biaxial yield stress

α :

angle between principal stress σ1 and rolling direction

α, β, γ :

coefficients in the Wang yield criterion

α = σ 2/σ 1 :

ratio of the principal stresses

α :

back-stress tensor

α 1, α 2, α 3 :

coefficients in the Barlat 1994 yield criterion

α 1, ... α 8 :

coefficients in the Barlat 2000 yield criterion

α 1, α 2, γ 1,γ 2,γ 3, C :

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

α x , α y , α z :

coefficients in the Barlat 1994 yield criterion

β, ϕ, δ, γ :

accuracy index of the yield criteria

β 1, β 2, β 3 :

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

Δr :

variation of anisotropy coefficients

ɛ e :

equivalent (effective) strain

ɛ 1, ɛ 2, ɛ 3 :

principal (logarithmic) strains

ɛ, ɛ e, ɛ p :

tensors of total, elastic and plastic strain respectively

Φ:

plastic potential

φ :

invariant homogeneous function

Γ, Ψ, Λ :

function in the BBC yield criteria angle between the specimen longitudinal axis and the rolling direction

λ :

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

λ :

plastic multiplier in the flow rule

μ :

Poisson’s ratio

σ :

stress tensor

σ 0, σ 45, σ 90 :

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

σ 0 :

initial yield stress

σ 1, σ 2, σ 3 :

principal stresses

σ b :

equibiaxial yield stress

σ e :

equivalent (effective) stress

σ k :

hardening stress

σ u :

uniaxial yield stress

σ 11, σ 22, σ 33,:

components of the actual stress tensor

σ 12, σ 23, σ 31 :

-----

τ :

shear yield stress

References

  1. Lankford WI, Snyder SC, Bauscher JA (1950) New criteria for predicting the press performance of deep-drawing sheets. Transaction ASM 42:1196–1232

    Google Scholar 

  2. Wech PI, Radtke L, Bunge HJ (1983) Comparison of plastic anisotropy parameters. Sheet Metal Industries 60:594–597

    Google Scholar 

  3. Siegert K, TALAT Lecture 3705 (http://www.eaa.net/eaa/education/TALAT/index.htm)

  4. Choi Y, Walter ME, Lee JK, Han CS (2006) Observations of anisotropy evolution and identification of plastic spin parameters by uniaxial tensile tests. International Journal of Solids and Structures 1:303–325

    Google Scholar 

  5. Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, Choi SH, Pourboghrat F, Chu E, Lege DJ (2003) Plane stress yield function for aluminium alloy sheets – Part 1: Theory. International Journal of Plasticity 19:297–319

    Google Scholar 

  6. Banabic D, Wagner S (2002) Anisotropic behaviour of aluminium alloy sheets. Aluminium 78:926–930

    Google Scholar 

  7. Pöhlandt K, Banabic D, Lange K (2002) Equi-biaxial anisotropy coefficient used to describe the plastic behavior of sheet metal. Proceedings of the ESAFORM Conference, Krakow, 723–727

    Google Scholar 

  8. Findley WN, Michno MJ (1976) A historical perspective of yield surface investigations for Metals. International Journal of Non-Linear Mechanics 11:59–80

    CrossRef  MATH  Google Scholar 

  9. Zyczkowski M (1981) Combined loadings in the theory of plasticity. Polish Scientific Publishers, Warsaw

    MATH  Google Scholar 

  10. Barlat F, Lege DJ, Brem JC (1991) A six-component yield function for anisotropic materials. International Journal of Plasticity 7:693–712

    CrossRef  Google Scholar 

  11. Tresca H (1864) On the yield of solids at high pressures. Comptes Rendus Academie des Sciences 59:754 (in French)

    Google Scholar 

  12. Huber MT (1904) Przyczynek do podstaw wytorymalosci. Czasopismo Techniczne 22: 34–81

    Google Scholar 

  13. Mises R (1913) Mechanics of solids in plastic state. Göttinger Nachrichten Mathematical Physics 4:582–592 (in German)

    Google Scholar 

  14. Hencky H (1924) On the theory of plastic deformations. Zeitschrift für Angewandte Mathematik und Mechanik 4:323–334 (in German)

    CrossRef  Google Scholar 

  15. Timoshenko SP (1953) History of strenght of materials. McGraw-Hill, New York, NY

    Google Scholar 

  16. Drucker DC (1949) Relations of experiments to mathematical theories of plasticity. Journal of Applied Mechanics 16:349–357

    MathSciNet  MATH  Google Scholar 

  17. Norton FH (1929) The creep of steel at high temperatures. McGraw-Hill, New York, NY

    Google Scholar 

  18. Bailey RW (1929) Creep of steel under simple and compound stresses and the use of high initial temperature in steam power plants. Transmission in Tokyo Section Meeting World Power Conference, Konai-kai Publishing, Tokyo

    Google Scholar 

  19. Hershey AV (1954) The plasticity of an isotropic aggregate of anisotropic face centred cubic crystals. Journal of Applied Mechanics 21:241–249

    MATH  Google Scholar 

  20. Hosford WF (1972) A generalised isotropic yield criterion. Journal of Applied Mechanics 39:607–609

    CrossRef  Google Scholar 

  21. Karafillis AP, Boyce MC (1993) A general anisotropic yield criterion using bounds and a transformation weighting tensor. Journal of the Mechanics and Physics of Solids 41:1859–1886

    CrossRef  MATH  Google Scholar 

  22. Yu MH (2002) Advances in strength theories for materials under complex stress state in the 20th century. Applied Mechanics Reviews 55:198–218

    CrossRef  Google Scholar 

  23. von Mises RV (1928) Mechanics of plastic deformation of crystals. Zeitschrift für Angewandte Mathematik und Mechanik 8:161–185 (in German)

    CrossRef  MATH  Google Scholar 

  24. Olszak W, Urbanowski W (1956) The orthotropy and the non-homogeneity in the theory of plasticity. Polska Archiwum Mechaniki Stosowanej 8:85–110

    MathSciNet  MATH  Google Scholar 

  25. Hill R (1948) A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society London A 193:281–297

    Google Scholar 

  26. Woodthrope J, Pearce R (1970) The anomalous behaviour of aluminium sheet under balanced biaxial tension. International Journal of Mechanical Sciences 12:341–347

    CrossRef  Google Scholar 

  27. Pearce R (1968) Some aspects of anisotropic plasticity in sheet metals. International Journal of Mechanical Sciences 10:995–1001

    CrossRef  Google Scholar 

  28. Banabic D, Müller W, Pöhlandt K (1998) Determination of yield loci from cross tensile tests assuming various kinds of yield criteria. Sheet Metal Forming Beyond 2000, Brussels, 343–349

    Google Scholar 

  29. Hill R (1979) Theoretical plasticity of textured aggregates. Mathematical Proceedings of the Cambridge Philosophical Society 85:179–191

    Google Scholar 

  30. Lian J, Zhou D, Baudelet B (1989) Application of Hill’s new theory to sheet metal forming – I. Hill’s 1979 criterion and its application to predicting sheet forming limits. International Journal of Mechanical Sciences 31:237–244

    CrossRef  MATH  Google Scholar 

  31. Hosford WF (1979) On yield loci of anisotropic cubic metals. In: Proceedings of the 7th North American Metalworking Conference (NMRC), SME, Dearborn, MI, 191–197

    Google Scholar 

  32. Chu E (1995) Generalization of Hill’s 1979 anisotropic yield criteria. Journal of Materials Processing Technology 50:207–215

    CrossRef  Google Scholar 

  33. Zhou W (1990) A new non-quadratic orthotropic yield criterion. International Journal of Mechanical Sciences 32:513–520

    CrossRef  Google Scholar 

  34. Müller W (1996) Characterization of sheet metal under multiaxial load. Berichte aus dem Institut für Umformtechnik, Universität Stuttgart, Nr. 123, Springer, Berlin (in German)

    Google Scholar 

  35. Bassani JL (1977) Yield characterisation of metals with transversally isotropic plastic properties. International Journal of Mechanical Sciences 19:651–654

    CrossRef  Google Scholar 

  36. Bishop JFW, Hill R (1951) A theory of the plastic distortion of polycrystalline agregates under combined stress. Philosophical Magazine 42:414–427

    MathSciNet  MATH  Google Scholar 

  37. Zhou W (1994) A new orthotropic yield function describable anomalous behaviour of materials. Transactions of Nonferrous Metals Society of China 4:431–449

    Google Scholar 

  38. Montheillet F, Jonas JJ, Benferrah M (1991) Development of anisotropy during the cold rolling of aluminium sheet. International Journal of Mechanical Sciences 33:197–209

    CrossRef  Google Scholar 

  39. Hill R (1990) Constitutive modelling of orthotropic plasticity in sheet metals. Journal of the Mechanics and Physics of Solids 38:405–417

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. Lin SB, Ding JL (1996) A modified form of Hill’s orientation-dependent yield criterion for orthotropic sheet metals. Journal of the Mechanics and Physics of Solids 44:1739–1764

    CrossRef  Google Scholar 

  41. Chung K, Shah K (1992) Finite element simulation of sheet metal forming for planar anisotropic metals. International Journal of Plasticity 8:453–476

    CrossRef  Google Scholar 

  42. Leacock AG (2006) A mathematical description of orthotropy in sheet metals. Journal of the Mechanics and Physics of Solids 54:425–444

    CrossRef  MATH  Google Scholar 

  43. Hill R (1993) A user-friendly theory of orthotropic plasticity in sheet metals. International Journal of Mechanical Sciences 15:19–25

    CrossRef  Google Scholar 

  44. Stout MG, Hecker SS (1983) Role of geometry in plastic instability and fracture of tubes sheet. Mechanics of Materials 2:23–31

    CrossRef  Google Scholar 

  45. Logan R, Hosford WF (1980) Upper-bound anisotropic yield locus calculations assuming (111) – Pencil glide. International Journal of Mechanical Sciences 22:419–430

    CrossRef  Google Scholar 

  46. Hosford WF (1985) Comments on anisotropic yield criteria, International Journal of Mechanical Sciences 27:423–427

    CrossRef  Google Scholar 

  47. Hosford WF (1993) The mechanics of crystals and textured polycrystals. Oxford University Press, Oxford

    Google Scholar 

  48. Hosford WF (1996) On the crystallographic basis of yield criteria. Texture and Microstructures 26–27:479–493

    CrossRef  Google Scholar 

  49. Barlat F, Richmond O (1987) Prediction of tricomponent plane stress yield surfaces and associated flow and failure behaviour of strongly textured FCC polycrystalline sheets. Materials Science and Engineering 91:15–29

    CrossRef  Google Scholar 

  50. Barlat F, Lian J (1989) Plastic behaviour and stretchability of sheet metals (Part I): A yield function for orthotropic sheet under plane stress conditions. International Journal of Plasticity 5:51–56

    CrossRef  Google Scholar 

  51. Banabic D, Balan T, Comsa DS (2000) A new yield criterion for orthotropic sheet metals under plane-stress conditions. Proceedings of the 7th Conference ‘TPR2000’, Cluj Napoca, Romania, 217–224

    Google Scholar 

  52. Banabic D, Kuwabara T, Balan T, Comsa DS, Julean D (2003) Non-quadratic yield criterion for orthotropic sheet metals under plane-stress conditions. International Journal of Mechanical Sciences 45:797–811

    CrossRef  MATH  Google Scholar 

  53. Banabic D, Aretz H, Comsa DS, Paraianu L (2005) An improved analytical description of orthotropy in metallic sheets. International Journal of Plasticity 21:493–512

    CrossRef  MATH  Google Scholar 

  54. Choi SH et al. (1999) Prediction of yield surfaces of textured sheet metals. Metallurgical and Materials Transactions 30A:377–379

    CrossRef  Google Scholar 

  55. Lian J, Chen J (1991) Isotropic polycrystal yield surfaces of BCC and FCC metals: Crystallographic and continuum mechanics approaches. Acta Metallurgica 39:2285–2294

    CrossRef  Google Scholar 

  56. Barlat F et al. (1997) Yielding description for solution strengthened aluminium alloys. International Journal of Plasticity 13:185–401

    CrossRef  Google Scholar 

  57. Hayashida Y et al. (1995) FEM analysis of punch stretching and cup drawing tests for aluminium alloys using a planar anisotropic yield function. In: Shen SF, Dawson PR (eds) Simulation of materials processing-theory, methods and applications. AA Balkema, Rotterdam, 712–722

    Google Scholar 

  58. Barlat F et al. (1997) Yield function development for aluminium alloy sheets. Journal of the Mechanics and Physics of Solids 45:1727–1763

    CrossRef  Google Scholar 

  59. Yoon JW et al. (1999) A general elasto-plastic finite element formulation based on incremental deformation theory for planar anisotropy and its application to sheet metal forming. International Journal of Plasticity 15:35–67

    CrossRef  MATH  Google Scholar 

  60. Bron F, Besson J (2004) A yield function for anisotropic materials. Application to aluminium alloys. International Journal of Plasticity 20:937–963

    CrossRef  MATH  Google Scholar 

  61. Wang D-A, Pan J (2006) A non-quadratic yield function for polimeric foams. International Journal of Plasticity 22:434–458

    CrossRef  MathSciNet  MATH  Google Scholar 

  62. Budiansky B (1984) Anisotropic plasticity of plane-isotropic sheets. In: Dvorak GJ, Shield RT (eds) Mechanics of material behavior. Elsevier, Amsterdam, 15–29

    Google Scholar 

  63. Tourki Z et al. (1994) Orthotropic plasticity in metal sheets. Journal of Materials Processing Technology 45:453–458

    CrossRef  Google Scholar 

  64. Gotoh M (1977) A theory of plastic anisotropy based on a yield function of fourth order. International Journal of Mechanical Sciences 19:505–520

    CrossRef  MATH  Google Scholar 

  65. Hill R (1950) The mathematical theory of plasticity. Oxford University Press, Oxford

    MATH  Google Scholar 

  66. Barlat F et al. (2000) Constitutive modeling for aluminium sheet forming simulations. In: Khan AS, Zhang H, Yuan Y (eds) Plastic and viscoplastic response of materials and metal forming. Proceedings of the 8th International Symposium in Plasticity and its Current Applications, Whistley, Canada, Neat Press, Fulton, MD, 591–593

    Google Scholar 

  67. Aretz H, Barlat F (2004) General orthotropic yield function based on linear stress deviator transformations. In: Ghosh S, Castro GC, Lee JK (eds) Materials processing and design: Modelling, simulation and applications. Proceedings of the NUMIFORM 2004 Conference, Columbus, OH, 147–151

    Google Scholar 

  68. Barlat F et al. (2005) Linear transformation-based anisotropic yield functions. International Journal of Plasticity 21:1009–1039

    CrossRef  MATH  Google Scholar 

  69. Yoon JW et al. (2006) Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function. International Journal of Plasticity 22:174–193

    CrossRef  MATH  Google Scholar 

  70. Cosovici GA (2006) Implementation of the new yield criteria in the FE programs for sheet metal forming simulation. PhD Thesis, Cluj Napoca, Romania (in Romanian)

    Google Scholar 

  71. Paraianu L (2006) Modelling of the FLC using the large deformation theory. PhD Thesis, Cluj Napoca, Romania (in Romanian)

    Google Scholar 

  72. Press WH et al. (1992) Numerical recipes in C. The art of scientific computing. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  73. Aretz H (2005) A non-quadratic plane stress yield function for orthotropic sheet metals. Journal of Materials Processing Technology 168:1–9

    CrossRef  Google Scholar 

  74. Barlat F, Yoon JW, Cazacu O (2006) On linear transformations of stress tensors for the description of plastic anisotropy. International Journal of Plasticity 22:876–896

    Google Scholar 

  75. Cazacu O, Barlat F (2001) Generalization of Drucker’s yield criterion in orthotropy. Mathematics and Mechanics of Solids 6:613–630

    CrossRef  MATH  Google Scholar 

  76. Cazacu O, Barlat F (2003) Application of representation theory to describe yielding of anisotropic aluminium alloys. International Journal of Engineering Science 41:1367–1385

    CrossRef  Google Scholar 

  77. Cazacu O, Barlat F (2004) A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals. International Journal of Plasticity 20:2027–2045

    CrossRef  MATH  Google Scholar 

  78. Liu C, Huang Y, Stout MG (1997) On the asymmetric yield surface of plastically orthotropic materials: A phenomenological study. Acta Materialia 45:2397–2406

    CrossRef  Google Scholar 

  79. Cazacu O, Plunkett B, Barlat F (2006) Orthotropic yield criterion for hexagonal close packed metals. International Journal of Plasticity 22:1171–1194

    CrossRef  MATH  Google Scholar 

  80. Vegter H, Drent P, Huetink J (1995) A planar isotropic yield criterion based on material testing at multi-axial stress state. In: Shen SF, Dawson PR (eds) Simulation of materials processing-theory, methods and applications. AA Balkema, Rotterdam, 345–350

    Google Scholar 

  81. Vegter D, van den Boogaard AH (2006) A plane stress yield function for anisotropic sheet material by interpolation of biaxial stress states. International Journal of Plasticity 22: 557–580

    CrossRef  MATH  Google Scholar 

  82. Mollica F, Srinivasa AR (2002) A general framework for generating convex yield surfaces for anisotropic metals. Acta Mechanica 154:61–84

    CrossRef  MATH  Google Scholar 

  83. Hu W (2003) Characterized behaviors and corresponding yield criterion of anisotropic sheet metals. Materials Science and Engineering A345:139–144

    CrossRef  Google Scholar 

  84. Hu W (2005) An orthotropic yield criterion in a 3-D general stress state. International Journal of Plasticity 21:1771–1796

    CrossRef  MATH  Google Scholar 

  85. Hu W (2007) A novel quadratic yield model to describe the feature of multi-yield-surface of rolled sheet metals. International Journal of Plasticity 23:2004–2028

    CrossRef  MATH  Google Scholar 

  86. Comsa DS (2006) Numerical simulation of the sheet metal forming prodesses using a new yield criterion. PhD Thesis, Tehnical University of Cluj Napoca (in Romanian)

    Google Scholar 

  87. Comsa DS, Banabic D (2007) Numerical simulation of sheet metal forming processes using a new yield criterion. Key Engineering Materials 344:833–840

    CrossRef  Google Scholar 

  88. Soare S (2007) On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet metal forming. PhD Thesis, University of Florida

    Google Scholar 

  89. Soare S, Yoon JW, Cazacu O (2007) On using homogeneous polynomials to design anisotropic yield functions with tension/compression symmetry/asymmetry. In: Cesar de Sa JMA, Santos AD (eds) Materials processing and design: Modeling, simulation and applications. Proceedings of the NUMIFORM 2007 Conference, Porto, 607–612

    Google Scholar 

  90. Oller S, Car E, Lubliner J (2003) Definition of a general implicit orthotropic yield criterion. Computer Methods in Applied Mechanics and Engineering 192:895–912

    CrossRef  MATH  Google Scholar 

  91. Comsa DS, Banabic D (2008) Plane-stress yield criterion for highly-anisotropic sheet metals. In: Hora P (ed) Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, NUMISHEET 2008, Interlaken, Switzerland, 43–48

    Google Scholar 

  92. Banabic D, Cazacu O, Barlat F, Comsa DS, Wagner S, Siegert K (2002) Recent anisotropic yield criteria for sheet metals. Proceedings of the Romanian Academy 3:91–99

    Google Scholar 

  93. Barlat F, Banabic D, Cazacu O (2002) Anisotropy in sheet metals. In: Yang D-Y, Oh SI, Huh H, Kim YH (eds) Design inovation through virtual manufacturing. Proceedings of the NUMISHEET 2002 Conference, Jeju Island, Korea, 515–524

    Google Scholar 

  94. Paraianu L, Banabic D (2006) Predictive accuracy of different yield criteria. Proceedings of the SISOM Conference, Bucharest, 465–574

    Google Scholar 

  95. Chaparro BM, Alves JL, Menezes LF, Fernandes JV (2007) Optimization of the phenomenological constitutive models parameters using genetic algorithms. In: Banabic D (ed) Advanced methods in material forming, Springer, Heidelberg, 35–54

    CrossRef  Google Scholar 

  96. Barlat F, Cazacu O, Zyczkowski M, Banabic D, Yoon J-W (2004) Yield surface plasticity and anisotropy. In: Raabe D, Chen L-Q, Barlat F, Roters F (eds) Continuum scale simulation of engineering materials fundamentals-microstructures-process applications. Wiley-VCH, Weinheim, 145–185

    Google Scholar 

  97. Banabic D, Tekkaya EA (2006) Forming Simulation. In: Hirsch J (ed) Virtual fabrication of aluminum alloys: Microstructural modeling in industrial aluminum production. Wiley-VCH, Weinheim, 275–302

    Google Scholar 

  98. Yoon JW, Barlat F (2006) Modeling and simulation of the forming of aluminium sheet alloys. In: Semiatin SL (ed) ASM handbook, Vol 14B, Metalworking: Sheet forming. ASM International, Materials Park, OH, 792–826

    Google Scholar 

  99. Barlat F (2007) Constitutive modeling for metals. In: Banabic D (ed) Advanced methods in material forming, Springer, Heidelberg, 1–18

    CrossRef  Google Scholar 

  100. Banabic D, Barlat F, Cazacu O, Kuwabara T (2007) Anisotropy and formability. In: Chinesta F, Cueto E (eds) Advances in material forming-ESAFORM 10 years on. Springer, Heidelberg, 143–173

    CrossRef  Google Scholar 

  101. Barlat F (2007) Constitutive descriptions for metal forming simulations, In: Cesar de Sa JMA, Santos AD (eds) Materials processing and design: Modeling, simulation and applications. Proceedings of the NUMIFORM 2007 Conference, Porto, 3–22

    Google Scholar 

  102. ABAQUS (2004), Analysis user’s manual, Version 6.4. Hibbit, Karlsson & Sorensen Inc., Providence, RI

    Google Scholar 

  103. Krasovskyy A (2005) Verbesserte Vorhersage der Rückfederung bei der Blechumformung durch weiterentwickelte Werkstoffmodelle. PhD Thesis, University of Karlsruhe (in German)

    Google Scholar 

  104. Yoshida F, Urabe M, Toropov VV (1998) Identification of material parameters in constitutive model for sheet metals from cyclic bending test. International Journal of Mechanical Sciences 40:237–249

    CrossRef  Google Scholar 

  105. Yoshida F, Uemori T, Fujiwara K (2002) Elastic-plastic behavior of steel sheets under in-plane cyclic tension-compression at large strain. International Journal of Plasticity 18: 633–659

    CrossRef  MATH  Google Scholar 

  106. Cleveland RM, Ghosh AK (2002) Inelastic effects on springback in metals. International Journal of Plasticity 18:769–785

    CrossRef  MATH  Google Scholar 

  107. Peeters B, Kalidindi SR, Teodosiu C, Van Houtte P, Aernoudt E (2002) A theoretical investigation of the influence of dislocation sheets on evolution of yield surfaces in single-phase B.C.C. polycrystals. Journal of the Mechanics and Physics of Solids 50:783–807

    CrossRef  MATH  Google Scholar 

  108. Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, Cambridge

    CrossRef  MATH  Google Scholar 

  109. Armstrong PJ, Frederick CO (1996) A mathematical representation of the multiaxial Bauschinger effect. GEGB Report RD/B/N731. Berkeley Nuclear Laboratories

    Google Scholar 

  110. Chaboche JL, Rousselier G (1983) On the plastic and viscoplastic constitutive equations – Part 1: Rules developed with internal variable concept. Journal of Pressure Vessel Technology 105:153–158

    CrossRef  Google Scholar 

  111. Yoshida F, Uemori T (2002) A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation. International Journal of Plasticity 18:661–686

    CrossRef  MATH  Google Scholar 

  112. Kubli W, Krasovskyy A, Sester M (2008) Advanced modeling of reverse loading effects for sheet metal forming processes. In: Hora P (ed) Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, NUMISHEET 2008, Interlaken, Switzerland, 479–484

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technical University of Cluj-Napoca, Research Centre on Sheet Metal Forming – CERTETA, 27 Memorandumului, 400114, Cluj Napoca, Romania

    Prof. Dr. Ing. Dorel Banabic

Authors
  1. Prof. Dr. Ing. Dorel Banabic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dorel Banabic .

Rights and permissions

Reprints and Permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Banabic, D. (2010). Plastic Behaviour of Sheet Metal. In: Sheet Metal Forming Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88113-1_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-88113-1_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88112-4

  • Online ISBN: 978-3-540-88113-1

  • eBook Packages: EngineeringEngineering (R0)