PSI - Issue 57

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

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and p is the accumulated plastic strain: ̇ = | ̇ |

(13) Figure 5 illustrates the translation of the elastic domain | − | and the evolution of the backstress  . The yielding condition (given by + 0 ) cannot be violated at any time [8].

Fig. 5. The combined kinematic and isotropic behavior of Chaboche’s model.

It is worth noting that the parameters described so far, i.e., Young’s modulus E, yield stress 0 , the isotropic hardening parameters ∞ ,  , the kinematic hardening parameters (H i , b i ) and the viscosity parameters Z and n are all temperature dependent in the context of thermomechanical fatigue analysis. 1.3. Calibration of Chaboche’s Model The integration of equation 10 results in equation 14, which is a simple analytical solution for Chaboche’s backstress under uniaxial loading. = ∑ +( 0 − ) =1 (− ( − 0 )) (14) 0 and 0 are the initial values of backstress and plastic strains, while =±1 provides the direction of the plastic flow ( tension or compression ). Chaboche [9] presents an expression that relates the stress and plastic strain amplitudes of a stabilized cycle, which is obtained from experiments where mean stresses are zero ( i.e., symmetrical hysteresis loops ): Δ 2 =∑ ℎ ( Δ 2 )+ =1 0 + (15) Various methods and strategies have been proposed for the fitting of Equation 15 to the data ( plastic strain and stress amplitudes ). Some examples can be found in [25], [6], [26] and [5]. The Python library SciPy brings a convenient nonlinear least square implementation that can also be used to calibrate Chaboche. Consider that each blue dot in figure 6 represents a hysteresis loop of different magnitude ( , ). Python “least squares” function in this case results in the following parameters: H 1 = 15648.8 , b 1 = 1025.4 , H 2 = 1137.9 , b 2 = 26.2 , H 3 = 3432.9 , b 3 = 175.5 , which fits well the data as shown in figure 6.