# Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
## 6 Mechanics of Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
Equal forces can be represented by normalized stresses (\(\sigma\)) over the area (\(A\)) for the matrix and SMA wires:  
\[F_{SMA}=F_{matrix}\ \rightarrow\ \sigma_{SMA}A_{SMA}=\sigma_{matrix}A_{matrix} \tag{6}\]where \(F_{SMA}\) = applied force by the SMA wire, \(F_{matrix}\) = applied force by the matrix, \(\sigma_{SMA}\) = recovery stress SMA wire, \(\sigma_{matrix}\) = stress in the matrix during healing, \(A_{SMA}\) = cross-sectional area of SMA wire, and \(A_{matrix}\) = cross-sectional area of the matrix. Dividing Equation (6) by the total cross-sectional area, the following relationship can be obtained:  
\[\sigma_{SMA}V_{f}=\sigma_{matrix}(1-V_{f}) \tag{7}\]  
where \(V_{f}\) = volume fraction of SMA wires. During healing of the composite, the stress in the SMA wires is determined by the reversion stress (\(\sigma_{R}\)) of the SMA wires, and the stress in the matrix is governed by the matrix compressive yield stress (\(\sigma_{MCYS}\)) at elevated temperatures under compression. The SMA wires must yield the matrix to prevent plastic deformation during healing. Equation (7) can be modified as the following expression:  
\[\sigma_{R}=\sigma_{MCYS}\Bigg{(}\frac{V_{f}}{1-V_{f}}\Bigg{)} \tag{8}\]  
Equation (8) demonstrates how the volume fraction of the SMA wires in the SHMMC indirectly relates \(\sigma_{R}\) to the \(\sigma_{MCYS}\) of the matrix alloy. Since the flow stress of a matrix alloy is equal to its compressive yield strength (i.e., \(\sigma_{flow}\) = \(\sigma_{MCYS}\)), Equation (5) can be written as:  
\[\sigma_{MCYS}\mathrm{=A}\left[1-\left(\frac{T-T_{room}}{T_{cut}-T_{room}} \right)^{m}\right] \tag{9}\]  
The analytical model for the optimal volume fraction of SMA wires in the SHMMC to induce a clamping force at the crack interface during a healing cycle is derived by substituting the value of \(\sigma_{MCYS}\) from Equation (9) (the modified Johnson-Cook model) into Equation (8):  
\[\sigma_{R}\mathrm{=A}\left[1-\left(\frac{T-T_{room}}{T_{melt}-T_{room}} \right)^{m}\right]\Bigg{(}\frac{V_{f}}{1-V_{f}}\Bigg{)} \tag{10}\]  
[PERSON] [39] has found that for a Sn-based proof-of-concept composite, a 1% volume fraction of Ti-49.4 at.%Ni SMA wire is needed for healing the composite. However, a 30% volume fraction of Ti-49.4 at.%Ni SMA wire is needed for a Mg-based proof-of-concept composite. The reason for the higher volume fractions of SMA wires needed in healing the Mg-based composite than the Sn-based composite is that matrix strength increases for a Mg-based composite, which resists the recovery force of SMA wires. [PERSON] [33] has verified the above model for Al-Cu-Si and Al-Si alloys. It is found that for an Al-4.1 Cu-2 Si (at%) composite, the minimum NiTi SMA wire volume percentage required is determined to be 55%. This implies that the SHMMC would need to contain more SMA wires than matrix alloy to achieve effective healing. Similar data is obtained for the SMA-wire-reinforced Al-3 at%Si composite. For this composite, the volume percentage of SMA wires needed for the healing is calculated to be 25%. However, [PERSON] has reported that healing occurred in Al-Si composites with less than 2.5 vol% SMA wires, demonstrating that the Manuel model is inadequate in determining minimum SMA volume fractions in Al-based composites. The large difference between the calculated value and experimental value of the minimum volume fractions of SMA wires required to induce a clamping force at the crack interface during a healing cycle was attributed to the assumptions made in the model, such as equal strain and no plasticity in the SHMMC [33]. In addition, increased toughness of synthesized SHMMC can also be a reason to decrease the required minimum volume fraction of SMA wires for effective healing. Therefore, a more accurate analytical model needs to be developed, which will be able to address the mentioned issues and minimize the difference between the calculated value and experimental value of the required volume fractions of SMA wires needed for the most effective healing of SHMMs.