## 2 Materials and Methods  
### Specim Parameters and Experimental Data Collection  
The study involved three different materials: aluminum 2024-T3, control mix concrete, and steel fiber-reinforced concrete (SFRC). Figure 1 displays the geometry of the various specimens manufactured, while the experimental data were obtained from prior experiments. To conduct the experiments, the specimen was affixed onto the shaker and heatedto multiple temperatures. The shaker applied mechanical loads, while impact tests were performed to determine the fundamental frequency of the specimen. Measurements were taken using a laser vibrometer. The vibration test was then conducted at the fundamental frequency. If crack propagation occurred, the beam tip's displacement amplitude was reduced, and the shaker was halted to record the new frequency. Further impact tests were executed to determine the new fundamental frequency, which was then set on the shaker. This process was reiterated until the specimen failed catastrophically due to crack propagation. In addition, Fused Deposition Modelling (FDM) 3D printed ABS was also used as representative material and tested in a similar manner.  
ABS, concrete, and aluminum are widely used materials in industries, such as automotive, construction, and aerospace. Understanding crack propagation in these materials is of practical significance for ensuring the reliability and durability of structures made from them [16; 17; 18; 19; 20; 21; 22; 23]. By evaluating the performance of machine learning models on these materials, the study can provide insights and guidance for real-world applications, aiding in the design and maintenance of structures involving ABS, concrete, and aluminum. While ABS, concrete, and aluminum differ in nature, the comparative analysis allows researchers to gain a comprehensive understanding of crack propagation across materials with diverse properties. This broader evaluation provides insights into the strengths and weaknesses of the machine learning models, facilitating the development of more robust and accurate prediction methods applicable to a range of materials. It is important to acknowledge the differences between these materials and interpret the results accordingly, considering the unique characteristics of each material type [1; 2].  
The collected data (as shown in Table 1) were plotted using scatter, pair, heatmap, and grid plots to understand the relationships between the features and the predicted crack depth. The data showed non-linearity and overlapping nature, and the problem was a regression problem with small and moderate datasets.  
An empirical model had been developed earlier to relate crack depth/location and structural dynamic response. However, this model contained a large number of coefficients and was of a high order, making it challenging to interpret. To overcome this limitation, a more concise and suitable model was necessary that could accurately predict crack propagation and provide physical insight into the coefficients' meaning. Accordingly, selected models were proposed that met these requirements.  
\begin{table}
\begin{tabular}{c c c} \hline
**Material** & **Temperature \({}^{\circ}\)C** & **Crack Location mm** & **Structural Response** \\ \hline Composite & 20 & \\ Control Mix & 40 & 5 \\ SFRC & 60 & \\ \hline Polymer & 20 & \\  & 50 & 5 \\ ABS & 60 & 15 \\  & 70 & 25 \\ \hline Metal & & \\  & 20–25 & 4.5 \\ aluminum 2024-T3 & 50–100 & 5 \\  & 150–200 & 10 \\ \hline \end{tabular}
\end{table}
Table 1: The experimental conditions and parameters.