Issue 38

A. Znaidi et alii, Frattura ed Integrità Strutturale, 38 (2016) 135-140; DOI: 10.3221/IGF-ESIS.38.18

σ

σ

D

D

c  , ) (  



( )  

f (

)

(4)

s

D σ is the deviator of the Cauchy stress tensor (incompressible plasticity). Using the special setup of the space deviators, the general form of the equivalent orthotropic plan stress, is thus:

σ

σ

D

D

c 







f / ( , 2 )  

(

)

1 2 3 (X , X , X )

(5)

c

σ

σ

σ

D

D

D

With 1 2 3 X cos ; X sin cos2 ; X      Any type of criterion (4) can be written in the form:



sin sin2 



σ

D

f ( , 2 )  

/ ( )  



(6)

s

Where  is the angle that defines the test and  the off-axis angle [12-14].

Expansions Equibiaxes (E.E)

Test

Simple Traction (S.T) Large Traction (L.T)









Table 1 : Values of θ relative to the various tests

I DENTIFICATION PROCEDURES

n this section we focus on the phenomenology of plastic behavior; especially modeling plasticity and hardening based on experimental data represented as families of hardening curves, and Lankford coefficient data. In order to simplify our identification process, the following assumptions are adopted: Identification through “small perturbations” process, the tests used are treated as homogeneous tests, we neglect the elastic deformation; the behavior is considered rigid plastic incompressible, the plasticity surface evolves homothetically (isotropic hardening)and all tests are performed in the plane of the sheet resulting in a plane stress condition. The identified model is defined by an equivalent stress σ c ( A: σ D ) when σ c is an isotropic function; it is assumed that the shape is defined by coefficient of the form m, A is the 4 th order orthotropic tensor defined by anisotropy coefficients f, g, h, n’ and the hardening curve  s (  ). Knowing that hardening’s curve ( )   is determined from experimental tests as r (  ). We can also begin our identification procedures using the Lankford coefficient; as determined from the tensile test by:

.

.

.

.



2 3 / -1/ (1 / )       1 2

r

(7)

We can notice that the Lankford coefficient is independent of  . This coefficient is equal to one in the case of isotropy, and remains constant in the case of transverse isotropy. However, in the case of orthotropy, it varies depending on the off-axis angle  . This coefficient completely characterizes the anisotropy of the sheet when loaded in its plane.

R ESULTS AND DISCUSSIONS

T

his identification strategy requires:  Experimental database. 

Criterion for anisotropic plasticity.

137