## 2 Phenomenon Description  
Pyrotechnic pretensioning systems, with a cable, as shown in Figure 1, are found on PBP and PLP models. In terms of location, they also coincide with the seatbelt anchorage points in the area of the central tunnel of the vehicle body, respectively, on the B-pillar, in the lower area. Their components include the following:  
* The guide tube of the piston, which usually has a built-in pyrotechnic staple. It is the element that deforms in the process of blocking the anti-return system. * The piston, mounted at the end of the belt buckle cable, for the PBP model. It is pushed by the force of the gases generated by the pyrotechnic staple and reduces the slack in the belt webbing. It has a special shape, geometry similar to a truncated cone, with generators inclined at approximately 7 degrees. Its shape allows for the reverse movement of the belt cable to be blocked after the pyrotechnic staple has been triggered. * The balls of the anti-return system are mounted in a special plastic cage, similar to a bearing cage. They are placed around the conical area of the piston, on its generators. They make contact with both the piston and the tube. The process of blocking the pretensioning mechanism, type PBP, is carried out as follows:  
* In the event of a frontal impact, after activating the pyrotechnic staple and performing the tightening stroke of the seatbelt on the occupant's body, the passenger tends to move forward due to inertia. At this moment, the balls of the anti-return system come out of their resting position and tend to \"climb\" up the conical surface from a smaller diameter to a larger one, until the moment when the piston-balls-cylinder mechanism stops. * The intensity of the force with which the occupant's body acts on the belt straps will lead to the deformation of the piston and the cylinder, the balls leaving marks in them, as shown in Figure 2.  
Experimental tests were carried out on specially designed equipment, with various traction forces, using the pretensioning system from a seatbelt with a three-point attachment. The three-point seatbelt is, in accordance with the European Union's general approach to \"Vehicles\" in the Transatlantic Trade and Investment Partnership negotiations, equipped with a PBP (Pyrotechnical Buckle Pretensioner). This study considers, as a test case, the US and EU legislation on seatbelt anchorages. The presence of the occupant in the seat or on a special catapult-type stand is not necessary because only the deformations in the PBP piston-cylinder (tube) pretensioning system were analyzed. At the contact between the balls of the locking mechanism and the piston, respectively, the cylinder of the pretensioning tube, normal reactions will occur on the surface of the piston and cylinder due to the traction force in the belt. These will cause the human body to stop advancing after pretensioning by deforming the material of the piston and cylinder. The force distribution on the two loops of the strap and the effects on the human body are not the subject of this study; some of these aspects have been presented in various studies, such as [22; 26]. We identify the situation of spherical contact with an indentation in the plastic regime studied under various aspects in [27; 28; 29; 30; 31; 32]. These have been studied in numerous works, beginning with the early years of the 20 th century, in order to develop models for testing the hardness of materials [33; 34; 35], as well as to determine spherical indentation in the elastic-plastic regime [36; 37; 38; 39]. Throughout time, the study of indentation contact has attempted to develop models for testing material hardness. In indentation models, contact between a rigid sphere and a deformable flat is considered, as shown in Figure 3, where \"N\" is the normal force that acts on the indenter, \"D\" is the diameter of the indenter, \"\(d\)\" is the diameter of the indentation, and \"\(h\)\" is the depth of the indentation on a flat wall.