# Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites
## 6 Mechanics of Shape Memory Alloy Reinforced Self-Healing Metal Matrix Composites  
To date, no analytical models have been developed for designing and synthesizing the microstructure of SHMMCs with long, short, or nanosized NiTi and other SMA fibers, where the crack can be closed or its propagation stopped, or where the crack can be filled with a healing material. It is expected that an accurate analytical model will be able to successfully predict the optimum volume fraction of SMA fibers needed to yield the matrix and prevent damage to the SMA fibers during healing. Analytical models need to be developed to calculate the pre-strain of SMA fibers and predict the optimum orientation and distribution of SMA fibers in the metal matrix for the most effective healing of long SMA-reinforced SHMMCs. The importance of bonding and load transfer between SMA fibers and the matrix, effect of pre-strain in SMA fibers for load transfer between SMA fibers and the matrix, the rate of crack propagation, rate of crack closure, and shape recovery of SMA-reinforced SHMMCs need to be quantified. [PERSON] [39] developed an analytical model to determine the minimum volume fraction of SMA wires required to induce a clamping force at the crack interface during a healing cycle. The healing of the SMA-reinforced SHMMCs samples is greatly affected by the volume fraction of SMA reinforcement, compressive yield strength of the matrix, and parent phase flow stress of the SMA reinforcement. [PERSON]'s model presented in this section is based on the SMA wire's transformation temperatures and the matrix's compressive yield strength at elevated temperatures. The Johnson-Cook relationship is a mathematical equation that characterizes the response of materials when subjected to compressive loads at high temperatures. It mathematically represents the best fit curve through the temperature-dependent matrix compressive flow stress, which is presented in Equation (3) [119]. \[\sigma_{flow}\!=\!\left[\text{A}+\text{B}\varepsilon^{n}\right]\left[1+\text{ Cln}\;\hat{\varepsilon}^{*}\right]\left[1-\left(\frac{T-T_{room}}{T_{melt}-T_{ room}}\right)^{m}\right] \tag{3}\]  
where \(\sigma_{flow}\) = flow stress of matrix alloy, \(\varepsilon\) = equivalent plastic strain, \(\hat{\varepsilon}^{*}\) = normalized plastic strain rate, \(T_{room}\) = room temperature, \(T_{melt}\) = melting temperature of matrix alloy, and A, B, C, \(n\), and \(m\) are empirical constants that depend on the material being. The constants A, B, C, \(n\), and \(m\) are typically determined through experimental testing and calibration for each specific material of interest to accurately represent its behavior under such conditions. The flow stress of an alloy depends on work-hardening, strain rate, and thermal effects, which are presented in the above equation by the three groups of bracketed terms. However, if the focus is solely on the temperature effects on stress, Equation (3) can be simplified to the following expression:  
\[\sigma_{flow}\!=\text{A}\ \left[1-\left(\frac{T-T_{room}}{T_{melt}-T_{room}} \right)^{m}\right] \tag{4}\]  
Flow stress will be zero at the melting temperature (\(T_{melt}\)). Since the matrix alloy contains the eutectic phase, the alloy's eutectic temperature (\(T_{cut}\)) is used as a substitute for \(T_{melt}\) because of the alloy softening that takes place during healing in the partially liquefied alloy. Therefore, the Johnson-Cook relationship can be modified as Equation (5), considering the alloy's melting temperature as the eutectic healing temperature. \[\sigma_{flow}\!=\text{A}\ \left[1-\left(\frac{T-T_{room}}{T_{cut}-T_{room}} \right)^{m}\right] \tag{5}\]  
The matrix is clamped back together by the applied force of SMA wires at elevated temperatures at the equilibrium condition. The applied forces (\(F\)) between the matrix and SMA wires are equal at this condition.