PSI - Issue 61

Necdet Ali Özdür et al. / Procedia Structural Integrity 61 (2024) 277–284

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N.A. O¨ zdu¨r et al. / Structural Integrity Procedia 00 (2024) 000–000

as introduced by Kalidindi (1998). A Voce-type hardening law governs the evolution of energy storage mechanisms in slip systems. In contrast, a linear hardening law is assumed for tensile twin volume fractions. Rate dependency for plastic slip is introduced through a power-law type dissipation potential, while the twins are assumed to be rate independent. For more model details and model parameters, readers can refer to the original paper of Chang and Kochmann (2015). Variational constitutive updates are calculated implicitly by minimizing the incremental form of the internal energy density. This is iteratively carried out in two nested minimizations in a predictor-corrector fashion. At each time step, first the strain is held fixed, and the incremental change in internal variables (slip and twin volume fraction increments) are predicted by minimizing the internal energy density using a projected gradient method developed by Bertsekas (1982). Then a correction step in the form of a Newton-Raphson iteration is used to update the total strain for the current values of the internal variables, similar to Chang and Kochmann (2015). Fig. 3(a) shows the combined stress and change in temperature ( θ , with respect to the initial temperature) as function of strain. Up until 0.2% strain, the temperature of the specimen sharply increased by 120 mK due to thermoelastic e ff ects. After the yield-point, the twin plateau characteristic to the sharply textured Magnesium (see Lou et al. (2007)) was observed during when the temperature dropped down to about 60 mK above its initial temperature and stabilized. At about 2.5% strain, the material hardens further as the macroscopic shear bands probably encompassed the entire gage section. After 16 seconds, the specimen Mg is loaded by nearly 3.2% strain (Fig. 2(b)). Regarding the temperature loss estimates using Eq. 8, h 1 and h 2 are di ffi cult to characterize individually, despite the fact that the dimensions of the specimen are known. Therefore, τ eq is estimated directly through an exponential fit to the temperature decay following the end of the experiment, which is found to be about 0.95 seconds. This is presented in Fig. 3(b). Fig. 4 shows a decomposition of each energy component given in Eq. 7 as calculated directly from experimental measurements. By rearranging these energy terms, the fraction of plastic dissipation to the applied plastic work, β , can be calculated (the ratio between black and blue curves in Fig. 4). This is separately plotted for the entire duration of the experiment in Fig. 5(a). The value of β is commonly chosen to be about 0.9 for most engineering metals. Here, 3. Results and Discussion

Fig. 4. The energetic contributions obtained from the experimental results. See Eq. 7 for their definitions.