# Stick-Slip Phenomena and Acoustic Emission in the Hertzian Linear Contact
## 1 Introduction  
The stick-slip phenomenon is characterised as a jerky motion at low and very low driving speeds in a frictional couple. During sliding, this phenomenon occurs if two types of conditions are met--necessary and sufficient. The necessary conditions are given by a decrease in the friction kinetic coefficient with an increase in the driving speed and the dependence of the friction static coefficient on the sticking time. At the same time, the value of the static friction coefficient between two contact surfaces must be greater than the value of the kinetic friction coefficient. A quantitative relationship provides sufficient conditions among the driving speed (order of magnitude from microns/s to mm/s), the moving body mass, and the finite stiffness system in the sliding direction. [1; 2; 3]. The theory established by [PERSON] for the normal non-conforming contact with friction between two bodies with elliptical profiles represents a landmark in linear elasticity [4]. Although the theory of [PERSON] refers only to elastic contact under normal force or normal force with adhesion, the knowledge of the pressure distribution, the deformations at the connection between two bodies, and the mechanical properties of the materials allow for an extension to a compound analysis with tangential and friction forces generated by sliding or rolling. [5]. Contact friction between solid bodies under sliding conditions gives rise to various waveforms and oscillations within the contact, resulting in sound radiation, including acoustic emission [6]. Acoustic emission refers to transitory mechanical waves in the high-frequency range that are produced by elastic stress energy released in a solid material under mechanical stress. The propagation of elastic waves and oscillations due to the appearance of microcracks, deformations, and fibre breakage in solid bodies are manifested by acoustic emission [7] and can be detected by AE-devoted transducers placed on or near the monitored interface [8]. Initially, in the tribology field, AE has been proposed to monitor the operating condition of bearings [9], cutting tools [10], surface finishing processes [11], sanding [12], and rotary machines [13]. The AE could assess the behaviour of machining tools during operation [14; 15]. The AE measurement method is appropriate for discovering initial structural changes in materials. For example, for complex systems, such as the piston-ring-cylinder system of diesel engines, the sources of acoustic emissions are signalled when the engine speed, load, and lubrication conditions change [16]. The active monitoring and diagnosis of various machine components, such as bearings, gears, pumps, and motors, are assessed by AE evaluation over time [17; 18]. In addition, the generation of AE at different pressures and sliding speeds has been evaluated by basic methods for rough/finish turning [19; 20], detection of the breakdown of a machine tool device [21; 22], or in the case of disc brake friction couple components [23]. Regarding the acoustic emission related to the stick-slip phenomena, experimental research has shown the occurrence of AE during the sliding and rubbing of flat surfaces of various materials, such as rock [24], granite [25], and composite materials [26]. Studies on the stick-slip phenomenon in O-ring sealing samples [27] and the contact between a mild steel sample and hardened steel clamps [28] have shown the importance of AE utility in tribology. The origin of acoustic emission is the internal modifications in the materials due to an external stimulus. These modifications are usually composed of discrete events so that the emission of elastic waves is in the form of bursts or pulses of distinctive characteristics. Therefore, the analysis of signals captured in the AE test is usually performed considering this discrete nature, using a rather complex range of parameters related to the wave pulses, such as the number of counts, the rise and disappearance time, or frequency content-related variables [29; 30; 31; 32; 33].