PSI - Issue 57

Giovanni M. Teixeira et al. / Procedia Structural Integrity 57 (2024) 670–691 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

672

3

described by equation 4 ( illustrated in figure 1 ), where the evolution of plastic strains and back stresses are linearly related by the kinematic hardening coefficient H.

2 d Hd 3

(4)

= 

pl

Fig. 1. Example of cyclic stress strain curve predicted by Prager’s model. Young’s modulus E= 119080, yielding limit Sy=100, kinematic hardening coefficient H=10000.

Prager’s model is not adequate to predict ratcheting strains in the presence of mean stresses [5] and therefore should not be used in such complex situations as TMF. Frederick and Armstrong (1966) added a dynamic recovery term for the evolution of kinemati c hardening (equation 5), addressing some of the shortcomings of Prager’s model. = 2 3 − ̇ (5) The second term in the equation above ( − ̇ ) is responsible for the saturation of the kinematic hardening, where the parameter “b” controls the rate the hardening coefficient decreases as the plastic strains accumulate (parameter ̇ ). As figure 2 shows, unlike Prager, Frederick- Armstrong’s model can successfully represent the nonlinear part of the hysteresis curve and therefore can accurately simulate the ratcheting in metal alloys. For the combined ( isotropic and kinematic ) hardening model, Voce’s law [ 6 ] is used to account for the evolution of the yield surface size in accordance with equation 6: = ( ∞ − ) , (6)