# Measurement of Strain and Strain Rate during the Impact of Tennis Ball Cores
## 2 Materials and Methods
### Data Processing  
Images were imported into GOM Correlate Professional 2017 where a surface component was created. Maximal surface coverage was achieved with a facet size in the region of 15 pixels and point difference in the region of 10 pixels, whereby a facet is a square section of the image and the point difference is the distance, in pixels, between the centre points of adjacent facets. The surface parameters were coherent with GOM recommendations whereby the point distance should be approximately two thirds the facet size. An image approximately five frames prior to impact was selected as the reference frame, against which all other strain calculations were compared. It was therefore assumed the ball core was under no strain at this point in time. Assuming zero strain was deemed valid due to the non-contact nature of the pneumatic cannon, subjecting the ball to minimal deformation on firing. Furthermore, on visual inspection and through use of GOM, no significant change in ball shape or surface strain was apparent prior to impact. Major and minor nominal strain, major and minor nominal strain rate and velocity were calculated in GOM and exported as CSV files. The frame of first and last contact with the impact plate were noted and used to define the frames of interest when analysing strain and strain rate.  
Each CSV file contained the three-dimensional coordinates of each surface point alongside the strain, strain rate and velocity values for an individual frame. The number of surface points exported per file varied depending on how much of the surface defined in the reference frame was visible in that particular frame. Results were further processed in Matlab R2015b (Mathworks, Natick, MA, USA) to assess the frequency distribution of strain and strain rate during impact.  
Major and minor strains were calculated by GOM using the principal axis transformation of the strain tensor. The principal axis transformation provided eigenvalues and eigenvectors for the new orthonormal basis of the strain tensor. The larger eigenvalue defined the major strain and the smaller eigenvalue defines the minor strain, with the direction of the strains defined by the corresponding eigenvectors (Figure 4). Major and minor strain rate were defined as the change in corresponding strain divided by the change in time.  
To determine the maximum absolute strain for each element of the surface, the major and minor strains of the element were compared to determine the greatest absolute strain value. The value of greatest magnitude was selected for strain and the corresponding strain rate value was also selected based on the magnitude of the major and minor strain. The mean and standard deviation of strain and strain rate were calculated across all elements for all frames during impact, across all repeats. The frequency distribution of strain and strain rate was also assessed. The mean velocity of all elements of the surface prior to impact was used to define the pre-impact ball velocity.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Impact Angle (\({}^{\circ}\))** & **Reasoning** \\ \hline
18.9 & Mean surface impact angle \\
27.6 & Mean surface impact angle plus one standard deviation \\
40 & Intermediate case between surface impact angles results and normal impacts \\
90 & Simplified impact case, comparable to previous FE simulations \\ \hline
**Ball Speed (m\(\cdot\)s\({}^{-1}\))** & **-** \\ \hline
13.7 & Mean pre-surface impact ball speed minus one standard deviation \\
22.1 & Mean pre-surface impact ball speed \\
30.5 & Mean pre-surface impact ball speed plus one standard deviation \\
43.1 & Mean second serve ball speed (Men’s) \\
52.5 & Mean 1 st serve ball speed (Men’s) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Impact condition selection.