## 3 Results and Discussion
### Ultrasonic Nonlinearity Tests and Characteristics  
For a better understanding of the relationship between microstructure and nonlinear ultrasonic parameter (\(\beta\)), the magnitudes of 5 and 10 MHz were measured to give an estimated \(\hat{a}^{\prime}\) value with Equation (9). Figures 15 and 16 show the changes in the \(\hat{a}^{\prime}\) values for cast stainless steel specimens with aging time and deformation. It is observed from Figure 15 that the variations in \(\hat{a}^{\prime}\) for HF and LF specimens with aging time are in a similar trend. The value was noted to increase in the early stage of the aging process. After that, a decrease with aging time was observed. For the specimens machined, the \(\hat{a}^{\prime}\) increased both HF and LF samples which were in reasonable agreement with the present investigation [24]. This suggests that the variations in the microstructural change would affect the nonlinear ultrasonic parameter. In order to elucidate the relationship between microstructures and nonlinear ultrasonic parameter, cast stainless steels with two different ferrite contents were used. From Figure 15, it can be seen that the HF-0h and LF-0h specimens show variations in ultrasonic nonlinearity. This implies that the variation of phase content will induce the change of ultrasonic nonlinearity. According to [PERSON] et al. [25], a stress-dependence model has been proposed to depict the change in the ultrasonic nonlinearity parameter with the applied stress  
\[\Delta\beta\propto|\sigma| \tag{12}\]  
where \(\Delta\beta\) is the change of nonlinearity parameter, and \(\sigma\) is the applied stress. The stress generated by the change of lattice structure can be calculated by the following expression  
\[\sigma\approx 2\mu\varepsilon^{\prime}f_{p} \tag{13}\]  
where \(f_{p}\) is the volume fraction of the dispersed phase, \(\mu\) is the shear modulus of the matrix, and \(\varepsilon^{\prime}\) is the coherency strain. For assuming a two-phase material, the coherency strain resulting from a phase embedded into the other phase matrix is described as  
\[\varepsilon^{\prime}=\frac{3K\delta}{3K+2E(1+\
u)} \tag{14}\]  
where \(K\) is the bulk modulus of the embedded phase, \(\delta\) is the embedding parameter, \(E\) is the [PERSON]'s modulus of the matrix, and \(\
u\) is the Poisson's ratio of the matrix. The embedding parameter \(\delta\) can be used to represent the relationship between embedded phase and the matrix as  
\[\delta=\frac{2(a_{e}-a_{m})}{(a_{e}+a_{m})} \tag{15}\]  
where \(a_{e}\) is the lattice parameter of the embedded phase and \(a_{m}\) is the lattice parameter of the matrix. The nonlinear ultrasonic parameter evaluated by the precipitate-matrix model is a function of the volume fraction of precipitates and the misfit parameter. As an example, there is a stainless steel containing austenite and ferrite phases. The lattice parameter of austenite is 0.36 nm, and that for ferrite is 0.28 nm. When the content of ferrite increases, a variation of \(\delta\) is obtained for the steel with the ferrite grains dispersed in the austenite matrix. This indicates that a phase embedded into the matrix would change the nonlinear ultrasonic parameter. From Figure 15, the ferrite content increases with the nonlinearity decrease. In the similar way, the nonlinearity would change by either the phase transformation of spinodal decomposition in the a-ferrite islands due to the aging or the dislocation in the austenite matrix due to the deformation. From Figure 15, the aging behaviors could be separated into two stages. The nonlinear parameter curve shows different trends in stages I and II, corresponding to the different states of microstructural change. From [PERSON] et al. [26], the study shows that the nonlinearity parameter decreases due to the phase precipitation. It means that some precipitations would decrease the nonlinearity parameter. Similarly, the increase of ferrite phases in the austenite matrix would decrease the nonlinearity.