## 2 Materials and Methods
### Experimental Design  
The primary goal of this study was to estimate the effects that different types of alloy rolling processes and post-rolling treatments have on the composition of the Cu-Al-Mn SMA sheets, and by extension, on the phase transformation temperatures and overall transformation capacity. In addition to the consideration of the primary elements--copper, aluminum, and manganese, special consideration is given also to the influence of the presence of impurities in the form of phosphorus within the casting. The two rolling approaches used were described in detail in the previous section with the main difference being the use of only hot rolling or a combination of hot and cold rolling. Prior to application, the Cu-Al-Mn SMA sheet samples would be treated by aging to ensure stable phase transformation temperatures [18]. Beyond this post-rolling heat treatment, two additional thermomechanical treatments are considered: abnormal grain growth treatment and tensile cycling treatment (i.e., SMA training). When dealing with as-cast Cu-Al-Mn with a high material thickness, it is possible to obtain a high degree of grain growth through annealing in the single \(\beta\)-phase region [23]. However, in thinner specimens, such as the sheet in this study, it has been observed that once the grain size reaches about two to three times the sheet thickness, there is a stagnation in further grain growth, which has been attributed to the reduction in the dimensionality of the grain structure of the alloy from a three-dimensional to a two-dimensional structure, at which point the tension balance between the free surface and the grain boundaries causes thermal grooving of the boundaries that in turn pins the grain boundary migration [23]. To obtain high relative grain size in the Cu-Al-Mn SMA thin sheets, the authors subjected sheet samples to an abnormal grain growth (AGG) heat treatment [24,25]. The growth in this case occurs by having a few grains engulf their neighboring grains and increase to considerable dimensions. This can be realized by annealing the Cu-Al-Mn thin sheets to a temperature within the dual phase region \(\alpha\) (fcc) + \(\beta\) (bcc), see Figure 1, in this case, up to temperatures up to \(\sim\)700 \({}^{\circ}\)C, then following this up by heating the alloy to the single \(\beta\)-phase region, in this case around 900 \({}^{\circ}\)C [24]. These steps of the heat treatment process can be seen in Figure 2. The two annealing treatments were denoted by AGG (abnormal grain growth) and BET (betatization) with each lasting for 10 min and being performed at 550 and 900 \({}^{\circ}\)C, respectively. After the betatization, the samples were quenched in cold water. As with all samples, an aging heat treatment at 200 \({}^{\circ}\)C for 10 min was performed to stabilize the phase transformation temperatures of the SMA [26]. For each sample, the effect of the heat treatment in achieving grain growth was estimated using a Nikon ECLIPSE LV150 NL optical microscope. From the captured images, the relative grain size was calculated as r/t, where r and t are defined as the mean grain size and the SMA sheet thickness, respectively. The final thermomechanical treatment performed during the study was tensile cycling for SMA training [2]. A pseudoelastic loading cycle was employed, whereby the sample was kept at a constant temperature above the austenitic finish temperature A\({}_{\text{f}}\), and then a tensile load was applied to induce phase transformation from the austenitic to the de-twinned martensitic phase. The schematic representation of the loading cycle is given in Figure 3, which is based on the so-called Brinson model used to describe the one-dimensional constitutive behavior of shape memory alloys, with the arrows indicating the changing state of the alloy in the \(\sigma\)-T diagram during loading [27,28].