# Recovery Behavior of Fe-Based Shape Memory Alloys under Different Restraints
## 4 Result and Discussion
### Fe-SMA Behavior under Partial Restraint
Figure 11: Comparison of FSMA-C’s temperature-recovery stress curves under partial restraint; (a) Heating, (b) Cooling  
Figure 10: Comparison of FSMA-A’s temperature-recovery stress curves under partial restraint; (a) Heating; (b) Cooling  
Figure 12 presents the recovery stress-recovery strain relation when the FSMA-A and FSMA-C that have 0.04 pre-strain were heated to 200 \({}^{\circ}\)C. At the same contact temperature, the recovery strain in the FSMA-C specimen was greater than one in the FSMA-A specimen. Therefore, the FSMA-C type of Fe-SMA is practical to be used in pipe joining that requires a high recovery strain for work usability. The recovery strains in the FSMA-A specimen and FSMA-C specimen at a 200 \({}^{\circ}\)C contact temperature were developed over 50% when compared to those under the free restraint. Also, the recovery stresses in the FSMA-A and FSMA-C specimens at a 200 \({}^{\circ}\)C contact temperature developed over 50% compared to ones under initial restraint. [PERSON] et al. [PERSON] et al. (2019) presented that the recovery stress of the Fe-SMA is hardly developed during cooling. Therefore, it is thought that the recovery stress and strain during cooling are dependent on the thermal contraction and that the management of the contact temperature in the heating phase encourages enhancement of the recovery stress and strain. Equations (9) to (16) are equations for calculating the effective confining pressure acting by a pipe coupler made of FSMA-C. As shown in Figure 13, if the internal diameter of the non-expanded Fe-SMA coupler is \(D_{0}\) and internal diameter of the expanded Fe-SMA coupler is \(D_{1}\), the internal circumference of expanded Fe-SMA coupler can be expressed as Equation (11). \[D_{1}=(1+\varepsilon_{res})D_{0} \tag{9}\]  
\[l_{1}=(1+\varepsilon_{res})l_{0} \tag{10}\]  
\[l_{1}=\pi D_{1} \tag{11}\]  
where, \(D_{0}\) is internal diameter of non-expanded Fe-SMA coupler, mm; \(\varepsilon_{res}\) is residual strain of Fe-SMA, \(D_{1}\) is internal diameter of expanded Fe-SMA coupler, mm; \(l_{0}\) is internal circumference of non-expanded Fe-SMA coupler, mm; and \(l_{1}\) is internal circumference of expanded Fe-SMA coupler, mm. As shown in Figure 14, the initial gap between the internal diameter of the Fe-SMA coupler and the external diameter of the connected pipe can be taken as Equation (12). For example, if initial gap is 0, the internal circumference of Fe-SMA coupler is same as the external circumference of pipe. Also, the strain of internal circumference of Fe-SMA coupler is expressed by contact strain (\(\varepsilon_{cl}\)) like Equation (13). Additionally, the confining strain of Fe-SMA coupler is same as Equation (14). Substituting Equations (10), (12), and (13) into Equation (14) yields Equation (15). \[x=\frac{D_{1}-D_{2}}{2} \tag{12}\]  
\[l_{2}=(1+\varepsilon_{cl})l_{0} \tag{13}\]  
Figure 12: Recovery strain- stress relationship of FSMA-A and FSMA-C.  
\[\varepsilon_{rec}=\varepsilon_{res}-\varepsilon_{ct} \tag{14}\]  
Figure 11:0  
where, \(x\) is initial gap of Fe-SMA coupler and pipe, \(D_{2}\) is external diameter of pipe, mm; \(l_{2}\) is external circumference of pipe, mm; \(\varepsilon_{ct}\) is contact strain, and \(\varepsilon_{rec}\) is recovery strain of Fe-SMA. Thereafter, in order to calculate effective confining stress, the recovery stress- recovery strain relationship of FSMA-C on Figure 15 was assumed as bilinear like the dotted line. The dotted lines can be expressed as Equations (16) to (17). Figure 11:1  
Figure 11:2  
where, \(\sigma_{e,\ rec}\) is effective recovery stress of Fe-SMA coupler. It was confirmed through the proposed formula and experimental results that the FSMA-C pipe coupler has the following characteristics. Namely, when FSMA-C is used for the pipe coupler, the internal diameter of coupler should not be larger than 102% of the external diameter of connected pipe.