# Recovery Behavior of Fe-Based Shape Memory Alloys under Different Restraints
## 4 Result and Discussion
### Fe-SMA Behavior under Rigid Restraint
#### 4.3.2 Effect of Heating Temperature  
Figure 8 shows the comparison of temperature-recovery stress curves by heating temperature under rigid restraint. The maximum recovery stress (\(\sigma_{m}\)) and final recovery stress (\(\sigma_{f}\)) in the FSMA-A specimen increased by 64 MPa as the heating temperature rose every 40 \({}^{\circ}\)C. As presented in Table 2, the maximum recovery stress (\(\sigma_{m}\)) and final recovery stress (\(\sigma_{f}\)) in the FSMA-A specimen was the highest and short-term relaxation nearly developed. The maximum recovery stress (\(\sigma_{m}\)) and final recovery stress (\(\sigma_{f}\)) in the FSMA-B specimen grew approximately 35 MPa and 24 MPa as the heating temperature rose every 40 \({}^{\circ}\)C. However, the maximum recovery stress (\(\sigma_{m}\)) and final recovery stress (\(\sigma_{f}\)) over 200 \({}^{\circ}\)C increased less than 3% even though the heating temperature rose by 40 \({}^{\circ}\)C. The short term relaxation in the FSMA-B specimen was 5 MPa at a heating temperature of 160 \({}^{\circ}\)C. As the heating temperature reached 200 \({}^{\circ}\)C or 240 \({}^{\circ}\)C, the short-term relaxation developed to roughly 28 MPa. In contrast with the FSMA-B specimen, the FSMA-C specimen had additional heat treatment. The maximum recovery stress (\(\sigma_{m}\)) and final recovery stress (\(\sigma_{f}\)) were in a range of 120 \({}^{\circ}\)C to 240 \({}^{\circ}\)C that linearly increased by 32.33% and 45.88% as the heating temperature rose every 40 \({}^{\circ}\)C. The short term relaxation in the FSMA-C specimen was 0.72 MPa at a heating temperature of 120 \({}^{\circ}\)C. However, the short term relaxation in the range of 160 \({}^{\circ}\)C to 200 \({}^{\circ}\)C was nearly 16 MPa and developed to 46 MPa at 240 \({}^{\circ}\)C.  
In the same pre-strain and heating temperature level, the recovery stress of the FSMA-A was greater than one of the FSMA-B and FSMA-C. Therefore, it is considered that FSMA-A is more suitable for application as a prestressing tendon that requires high recovery stress. Also, [PERSON] et al. [22] claimed that temperatures higher than 160 \({}^{\circ}\)C delay the formation of ettringite at the interface between the cement and fine aggregate or destroys the formation of ettringite. Therefore, when Fe-SMA is used as the prestressing tendon, the upper limit of the heating temperature is considered to be 160 \({}^{\circ}\)C.  
Figure 8: _Cont._The effective prestressing force (\(P_{i}\)) for Fe-SMA tendon is calculated from Equations (1) to (8). As shown in Figure 9, when the recovery stress is activated by heating Fe-SMA tendon, the compressive force that generates elastic shortening acts on the concrete. At this time, the concrete compressive strain at centroid of Fe-SMA tendon is same as the amount of change in the Fe-SMA strain \(\Delta\varepsilon_{sma}\), and it can be expressed as Equation (1). Subsequently, Equation (1) is expressed as Equation (3) by applying [PERSON]'s law and the ratio of elastic modulus in Equation (4). \[\varepsilon_{e}=\Delta\varepsilon_{sma} \tag{1}\]  
\[\frac{\sigma_{cs}}{E_{c}}=\frac{\Delta\sigma_{sma}}{E_{sma}} \tag{2}\]  
\[\Delta\sigma_{sma}=n\sigma_{cs} \tag{3}\]  
\[n=\frac{E_{sma}}{E_{c}} \tag{4}\]  
where \(\varepsilon_{e}\) is concrete compressive strain at centroid of Fe-SMA, \(\Delta\varepsilon_{sma}\) is strain reduction of Fe-SMA due to elastic shortening, \(\sigma_{cs}\) is concrete compressive stress at centroid of Fe-SMA, MPa; \(E_{c}\) is elastic  
Figure 8: Comparison of temperature-recovery stress curves by heating temperature under rigid restraint. modulus of concrete, MPa; \(\Delta\sigma_{sma}\) is stress reduction of Fe-SMA due to elastic shortening, MPa; \(E_{sma}\) is elastic modulus of Fe-SMA, MPa; and \(n\) is elastic modulus ratio. At this time, concrete compressive stress at centroid of Fe-SMA can be summarized in Equation (5). The effective pre-stressing force by Fe-SMA tendon can expressed as Equation. (6).