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Formability of Metallic Materials pp 119–172Cite as

Anisotropy of Sheet Metal

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Part of the book series: Engineering Materials ((ENG.MAT.))

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

$$r = \frac{{{\varepsilon _2}}}{{{\varepsilon _3}}} $$
(4.1)

where ε 2; ε 3 are the strains in the width and thickness directions, respectively. Eq. 4.1 can be written in the form

$$r = \frac{{In\frac{b}{{{b_0}}}}}{{In\frac{t}{{{t_0}}}}} $$
(4.2)

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

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  1. D. Banabic
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Editors and Affiliations

  1. Dept. of Mechanical Engineering, Technical University Cluj-Napoca, Bd. Muncii 103-105, Ro-3400, Cluj-Napoca, Romania

    Dorel Banabic (Professor) (Professor)

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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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  • DOI: https://doi.org/10.1007/978-3-662-04013-3_4

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