## 2 Materials and Methods
### Rubber Hysteresis  
Hysteresis is the ability of a material to expand and contract in the same way, that is, to follow the same path for expansion and contraction on a force-displacement graph. The most common examples of this behavior involve rubber and similar polymers. For instance, it's become warm when driven on a road. The heat is not generated by friction but is due to hysteresis in the rubber as it flexes (becomes momentarily flat on the road). The drag due to this conversion of kinetic energy to heat is described as rolling resistance [24].  
Figure 1: Research methodology and procedure.  
Rubber hysteresis is a very significant property of rubber that influences friction and absorption in shoe soles. There is static (measured and explained in this paper) and dynamic hysteresis (Yearsley hysteresis). Static hysteresis in this experiment is measured using an internal procedure by the Faculty of Mechanical Engineering in Nis. This procedure has been performed for 20 years and there is a lot of testing with different types of rubber (different hardnesses, mixtures) [25]. Rubber static hysteresis measured using this method yields very similar results to testing using the Yearsley method as there is no significant result dissipation. Measuring the static hysteresis of rubber is a simpler method than others (Yearsley), does not require special equipment, and is performed on a uniaxial testing machine. In these experimental tests, three different compounds of rubber with a hardness of 65 ShA (Method A) and one with 60 ShA were used. The testing samples for rubber hysteresis determination were of cylindrical shape with \(\Phi\)35.7 mm diameter and 18 mm height. The internal labels were OB202, OB290, OB2280, and OB221, for example. Rubber samples were produced by a local shoe sole manufacturer.  
The rubber hysteresis sample testing, shown in Figure 2, was performed at the Faculty of Mechanical Engineering in Nis on a uniaxial testing machine (Shimadzu AGS-X) with a maximum load of 10 KN (Figure 2a). Before testing, the samples rested at the testing temperature for a minimum of 24 h. Prior to the start of the testing process, the dimensions of all samples were verified. The samples were inserted between two sheets of sandpaper, as shown in the schematics given in Figure 2b, to increase the friction coefficient between the machine pressing plates and the rubber sample and to ensure that there was no sliding between the compression plates and the rubber sample. As the pressure was applied axially, the rubber cylindrical sample deformed in the direction of the applied force, causing the sides to deform outward, taking on a barrel-like shape, as shown in Figure 2b. As the samples were new, for this rubber hysteresis testing it was necessary to perform sample conditioning. The conditioning process was performed to bring the polymer chains into a stable state, i.e., conditioning breaks the weak bonds between polymer chains formed during the vulcanization of rubber. This process included five cycles of applying a vertical pressure force with a speed of 0.25 mm/s until half of the height deformation (9 mm) was reached, then keeping the machine stroke for 20 s, and after that releasing the pressure on the sample with the same speed in the opposite direction. This operation was performed for all samples. There were three testing samples for every rubber mixture. After the conditioning process, the samples were allowed to rest for 30 min. The rubber hysteresis testing process consisted of applying a vertical pressure force with a speed of 5 mm/min while reaching the half height (9 mm) stroke and then releasing the force with the same speed. During testing, the data on force and deformation were acquired to obtain the mechanical response of the samples.  
Figure 2: Rubber hysteresis sample testing: (a) sample placed between sandpaper sheets on a uniaxial testing machine, (b) schematics of the testing arrangement.