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Sheet Metal Forming Processes pp 27–140Cite as

Plastic Behaviour of Sheet Metal

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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 \({\boldsymbol{\dot A}} = {{d{\textbf{A}}} \mathord{\left/ {\vphantom {{d{\textbf{A}}} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}\).

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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

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

    Prof. Dr. Ing. Dorel Banabic

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  1. Prof. Dr. Ing. Dorel Banabic
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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

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

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