Issue 49

H. Berrekia et alii, Frattura ed Integrità Strutturale, 49 (2019) 643-654; DOI: 10.3221/IGF-ESIS.49.58 650 3 2 D ij p ij eq        Si: 0 f f    1 1 1 ij e kk ij ij D D                e p ij ij          2 0 2 1 F f H p p ES D      (6) The normality rule provides the evolution laws of: With: 1 3 2 3 p ij ij            (8) The damage evolution law (7.b) can be written: D S    If 0 p p  (9) When replacing Y by its value of (4) we obtain:       2 2 2 2 1 3 1 2 3 2 1 eq H eq D S D                             (10) In summary, the constitutive laws written for Newton's numerical method [16]: With: 2 2 eq v D R S     If: 0 p p  (11) where : S: is a constant characterizing the damage, depends on the material and the temperature. H: Heavyside function. 0 p : is the accumulated plastic strain, when the damage is zero (the threshold of damage) with:   0 0 H p p   If: eq s      0 0 H p p   If: 0 p p  p F E      (7.a)   0 . F D P H p p S          (7.b) Where : λ: is the plastic multiplier, determined by the consistency condition 0 f f    P: is the accumulated plastic strain ( 0 p p  if D = 0)