To better understand the geometry constraint effect, the predicted dissipated energy that can be determined by calculating the area in the hysteresis loop, Ah , are plotted in Fig.8a. The same principal was also utilized before in the miniature work of Chenet al. 58, and Peng et al .34. As can be seen from Fig.8a, an exponential increase is observed in dissipated energy against scaling factor values. Three exponential equations to determine the scaling factors are fitted and shown in Fig.8a. By using the fitted equations, the scaling factor can be possibly determined for a strain-range deformation required by the MTP high temperature LCF testing under small deformation framework, once the predicted SSFS responses are provided by the UVP model for FV566 at 600 °C. Furthermore, as the increase of scaling factor,β , apparent influence can be observed on the resultant dissipated energy, and this influence is found more pronounced for the finalized scaling factor value in MTP-3. Fig.8b provides the curve in terms of finalized scaling factors versus the ratios ofl0 /d0 . Due to the increase of l0 /d0 , the geometry constraint effect on the finalized identified scaling factors (see Table 2) is obvious, e.g., the geometry constraint effect becomes weaker asl0 /d0 increases.
Fig.8c shows the computed maximum stress evolution in every cycle until 58th. It is possible to observe that, when the same material properties are accommodated into the MTP FE models for tension and compression testing, the UMAT can gives the inconsistent softening responses due to the geometry constraint effect caused by the changes ofl0 /d0. A continuous predicted cyclic softening (a decrease in the stress amplitude) can be observed for all MTP specimens, which can be potentially regarded as an implication of accumulated damage and fatigue life2,17,59,60. The largest amount of softening with the highest decay-rate can be observed in MTP-1 with the determined,β = 4.0, due to the strongest geometry constraint effect. The intermediate case can be found in MTP-2, while the amount of softening and decay-rate decelerate in MTP-3. By increasingl0 /d0 to 2.0, the geometry constraint effect on the decay-rate of softening in MTP-3 has a relatively small impact comparing with MTP-1 and MTP-2.