## 2 Calculations  
In this section, the significance of the twinning boundaries in LUS measurements is examined by considering the effect of twinning density in austenitic stainless and HSLA steels. Several approaches have been proposed over the years to model the relationship of the twinning boundaries with the parent grains in face-centered cubic materials [7; 15; 16; 17]. [PERSON] et al. [15] determined twinning density \(p\) as the number of twin intersections over a unit of length. Their experimental work indicates a simple relation between the twinning density, \(p\), and grain size \(D\):  
\[p=\frac{B}{D}\log\frac{D}{D_{0}} \tag{1}\]  
in which \(D\) is the grain size determined as the mean linear intercept, \(D_{0}\) is a grain size below which twins can no longer be found, and \(B\) is an empirical constant dependent on the material, affected strongly by the stacking fault energy of the alloying composition. Linear intercept values measured from either optical metallographs or orientation maps lend themselves to the calculation of twinning density \(p\), provided that values both including and excluding twins have been measured. The twinning density is then given via  
\[p=\frac{1}{D}-\frac{1}{D_{n}} \tag{2}\]  
where \(D_{n}\) is the mean linear intercept value calculated when the intersections occurring at twin boundaries are disregarded. Combining Equations (1) and (2), the relationship of \(D\) with \(D_{n}\) is given via  
\[D_{n}=\left(\frac{1}{D}-\frac{B}{D}\log\frac{D}{D_{0}}\right)^{-1} \tag{3}\]  
Using Equation (3), \(D_{n}\) values were calculated for a range of grain sizes using the hypothetical values \(B\) = 0.2 and \(B\) = 0.1 and \(D_{0}\) = 0.1 \(\upmu\)m. Figure 1 shows the calculated \(D_{n}\) values with respect to \(D\). Both relationships are very close to linear, with the non-twinned grain size being approximately 2.4 times that of the twinned grain size for \(B\) = 0.2 and 1.5 times for \(B\) = 0.1. The apparently nearly linear relationship over a wide range of grain sizes indicates that LUS can be calibrated to make reasonably accurate predictions of grain sizes \(D_{n}\) disregarding twins, using data from optical micrographs insensitive to twinning boundaries. However, the figure also clearly shows that this assumption holds only for materials with a similar relationship between twinning density \(p\) and grain size \(D\). From an industrial perspective, it would be desirable to spend as little effort as possible on calibration in order to reduce downtime and to quickly implement the method for new products. In the case of LUS measurements in low-alloy steels, the question is how much difference in the twinning behavior can be expected between various low-alloy compositions. The alloying composition does have a significant effect on the twinning density by affecting the energy stored in the twinning interfaces. The interfacial energy is commonly known as the stacking fault energy (SFE). [PERSON] and [PERSON] [18] suggested the following equation for calculating the SFE:  
\[\Gamma=2\rho\big{(}\Delta G^{\gamma\to\varepsilon}+E_{m}^{str}\big{)}+2 \sigma^{\gamma/\varepsilon}\]  
where \(\rho\) is the molar surface density, \(\Delta G^{\gamma\to\varepsilon}\) is the free energy change of the austenite to epsilon martensite transformation, \(E_{m}^{str}\) is a strain energy term, and \(\sigma^{\gamma/\varepsilon}\) is the interfacial energy between the austenite and epsilon martensite phases. Figure 2 shows the term \(\Delta G^{\gamma\to\varepsilon}\) calculated with Thermo-Calc R2022b [19] with the TCFE11 database for the temperature range 800-1300 \({}^{\circ}\)C for three low-alloy high-strength steels, a tool steel, and a high-Si quenching and partitioning composition. Despite the relative differences, each alloy had less than 5 wt% total alloying elements. A comparative value is shown for austenitic 304 stainless steel.