Issue 33

M.A. Meggiolaro et alii, Frattura ed Integrità Strutturale, 33 (2015) 357-367; DOI: 10.3221/IGF-ESIS.33.39 363 The main difference among the several NLK hardening models proposed in the literature rests in the equation of the hardening surface translation direction i v   , which for most models can be condensed into the general equation [11]   Pr - cov * * [ (1 ) ( ) ] T i i i i i i i i i ager dynamic radial Ziegler re ery return v n r m n n                                 (7) where the scalar functions  i * and m i * are defined as | | * i i i i r            and | | , if 0 * 0, if 0 i m T T i i i i T i n n m n                                 (8) The calibration parameters for each surface i are the ratcheting exponent  i , the multiaxial ratcheting exponent m i , the ratcheting coefficient  i , and the multiaxial ratcheting coefficient  i , scalar values that are listed in Tab. 1 for several popular NLK hardening models. Note that several literature references represent the NLK hardening parameters  r i , p i , and  i as r (i) , c (i) , and  (i) , but this notation is not used in this work to avoid mistaking the (i) superscripts for exponents. Year Kinematic hardening model  i m i  i  i 1949 Prager [6] 0 0 0 1 1966 Armstrong-Frederick [4] 0 0 0   i  1 1 1967 Mróz [1] 0 0 1 1 1983 Chaboche [12] 1 0 1 1 1986 Burlet-Cailletaud [13] 0 0 0   i  1 0 1993 Ohno-Wang I [14-15] ∞ 1 1 1 1993 Ohno-Wang II [14-15] 0   i < ∞ 1 1 1 1995 Delobelle [16] 0 0 0   i  1 0   i  1 1996 Jiang-Sehitoglu [17-18] 0   i < ∞ 0 1 1 2004 Chen-Jiao [19] 0   i < ∞ 1 1 0   i  1 2005 Chen-Jiao-Kim [20] 0   i < ∞  ∞ < m i < ∞ 1 1 Table 1 : Calibration parameters for the general translation direction from Eq. (8). The translation direction i v   of each hardening surface i , shown in Eq. (7), can be separated into three components: (i) the Prager-Ziegler term, in the normal direction n   perpendicular to the yield surface at the current stress point s   ; (ii) the dynamic recovery term, in the opposite direction of the backstress component of the considered surface, which acts as a recall term that gradually erases plastic memory with an intensity proportional to the product of the ratcheting terms  i *  m i *   i   i ; and (iii) the radial return term, in the opposite direction of the normal vector n   , which mostly affects multiaxial ratcheting predictions. Fig. 4 shows the geometric interpretation of these three components. T WO -S URFACE K INEMATIC H ARDENING M ODEL he two-surface model proposed by Dafalias and Popov [21] and independently by Krieg [22] is an unconventional plasticity model based on the translation of only two moving surfaces: a single hardening surface ( i  2 ), usually called bounding or limit surface, and an inner yield surface ( i  1 ), shown in Fig. 5. The outer failure surface ( i  3 ) is also present, however it does not translate. T