PSI - Issue 42

A. Kostina et al. / Procedia Structural Integrity 42 (2022) 425–432 A. Kostina / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 4 presents comparison of the numerical and experimental values of energy dissipation per loading cycle against crack length for the considered titanium alloys subjected to cyclic loading with two constant maximum forces. Results of the simulation are in agreement with the experimental data obtained by contact heat flux sensor. However, it can be seen that for crack lengths less than 11 mm the numerical results give less values than experimental data. This mismatch can be explained by the stationary crack approach. This method takes into account plastic dissipation induced by the reversed plastic zone and neglects effects of crack extension and its contribution to the total plastic dissipation. Therefore, the less the crack length is, the more this effect can play a significant role. a b

Fig. 4. Energy dissipation vs crack length for: (a) Ti-5Al-2V; (b) Grade-2 (markers are experimental data; solid lines are results of numerical simulation). 6. Conclusion This work is devoted to the numerical simulation of energy dissipation during cyclic loading. For this purpose, mathematical model and numerical algorithm were developed. The mathematical model is based on the phenomenological form of free energy which is in isothermal case a function of elastic strain, structural parameter and hardening variables. It is assumed that strain induced by structural parameter is small and its contribution to the total strain can be neglected. This assumption implies that most part of the plastic work converts into the heat. In this case structural parameter loses meaning of the additional strain and becomes internal variable responsible for the structure evolution. Constitutive relations for structural parameter and hardening variables are obtained within the framework of quasi-standard thermodynamic approach with two dissipation potentials. Equation for structural parameter is obtained using rate-dependent approach. The value of plastic strain is calculated according to rate-independent approach and associated flow rule with yield surface which can expand and translate during deformation process. Calculation of structural parameter enables estimation of stored energy value while plastic strain is used to define plastic work. Therefore, dissipated energy is computed as the difference between plastic work and stored energy values. Numerical implementation of the model is based on the stationary crack approach. According to the method, energy balance is calculated for several discrete crack lengths. It should be noted that energy balance is evaluated within the sequential coupling framework. This means that initially, a stress-strain state of the sample is defined. For each crack length several loading cycles are considered and value of the plastic work per stabilized loading cycle is defined. Then, values of the stress tensor components are transferred to compute components of structural strain tensor and stored energy value per stabilized cycle. This procedure is carried out for each of the previously chosen crack length. Thus, the dependence of energy balance per loading cycle on crack length is reconstructed. Application of the proposed model and numerical algorithm is demonstrated by simulation of energy dissipation during fatigue crack propagation for two titanium alloys. The obtained results are in agreement with the experimental data. However, for crack lengths less than 11 mm the model underestimate dissipated energy. Possible explanation of the disagreement is related to the stationary crack approach which takes into account reversed plastic zone ahead of the crack but neglects effects related to the crack extension. For materials in which these effects are important the obtained results can be viewed as a first approximation.