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Anisotropy of Sheet Metal

  • D. Banabic
Chapter
Part of the Engineering Materials book series (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.

Keywords

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

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 = 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

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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© Springer-Verlag Berlin Heidelberg 2000

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  • D. Banabic

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