PSI - Issue 28

Jelena Srnec Novak et al. / Procedia Structural Integrity 28 (2020) 53–60 Author name / Structural Integrity Procedia 00 (2019) 000–000

55 3

3. Theoretical background of cyclic plasticity The yield criterion determines the stress level at which yielding occurs. In this work is considered the von Mises yield criterion defined in terms of the yield function f , Dunne and Petrinic (2005):

2 3

: 0       R σ σ

0

(1)

f



where σ՛ is the deviatoric stress tensor, R is the drag stress and σ 0 is the initial yield stress (bold symbol indicate tensor). The variable R describes the expansion of the radius of the yield surface in the stress space. As can be noticed from Eq. (1), the evolution of the yield surface is described by scalars R and σ 0 . Variable R describes isotropic hardening or softening, while σ 0 represents the size of the elastic area. The initial size of the yield surface is determined by using the criterion R =0. 3.1. Voce nonlinear isotropic model Materials subjected to cyclic loading may exhibit cyclic hardening or softening behavior that can be captured numerically with an isotropic model. The isotropic model assumes that the center of the yield surface remains always at the origin and the surface enlarges or reduces homothetically in size as plastic strain ε pl develops. In other words, according to Lemaitre and Chaboche (1990), evolution of the loading surface is governed only by one scalar variable, in this case, the accumulated plastic strain ε pl,acc . The evolution equation of the commonly adopted nonlinear isotropic model (Voce) has the form:     pl,acc 1- exp- b ε R R   (2) The governing equation is described by two parameters: b is the speed of stabilization, R ∞ is the saturation stress that can be positive or negative representing either cyclic hardening ( R ∞ > 0) or softening ( R ∞ < 0), respectively. Evolution of the R is possible to express as the relative change of the maximum stress stress σ max,i in the N th cycle with respect to the maximum stress in the first ( σ max,1 ) and in the stabilized ( σ max,s ) cycle:   pl,acc max,1 max, max,1 max, 1 exp      b R R s i        (3)

Fig. 1. Sensitivity of Eq. (3) for different values of the speed of stabilization.