Issue 39

O. Daghfas et alii, Frattura ed Integrità Strutturale, 39 (2017) 263-273; DOI: 10.3221/IGF-ESIS.39.24 267 The behavior model is defined by: Yield function In particular, we will assume that the elastic range evolves homothetically, the yield criterion is then written as follows:       p p c s f , 0        q q (1) c  : Equivalent stress is given by the Barlat criterion 91[12]:     m c 1 m m m 1 2 2 3 1 3 = q - q + q - q + q - q  q (2) where k q 1,2,3  are the eigenvalues of a modified stress deviator tensor q defined as follows: D :  q Α  (3) D  is the deviator of the Cauchy stress tensor (incompressible plasticity). The fourth order tensor Α carries the anisotropy by 6 coefficients c1, c2, c3, c4, c5, c6.   p s   : Isotropic hardening function; where p  is the equivalent plastic strain. Hardening law Using as a hardening function respectively a Hollomon and Voce laws [17]: Hollomon law     n p p s K     (4) K and n: the Hollomon parameters to be identified Voce law       p p s s 1 exp        σ ε (5) This law introduces a hardening saturation s  ,  and  describe the non-linear part of the curve during the onset of plasticity where 0<  <1 and  <0) Evolution law The direction of the plastic strain rate p   is perpendicular to the yield surface and is given by: p D f     ε σ   (6) With  plastic multiplier that can be determined from the consistency condition f 0   Lankford coefficient In the characterization of thin sheets, the plastic anisotropy with different directions is frequently measured by the Lankford coefficient r  that is given by the following expression: