## 1 Introduction
However, 3D models of the whole wheel also have been developed to further deepen the knowledge about the soil-compaction phenomenon [23] and to develop some indexes that, taking account of both the contact area and the volume of soil involved in the footprint, can be related to the compaction and traction behaviour of a considered tyre [24; 25]. Moreover, the technological evolution of the tyre structure brought about by the \"improved flexion\" and \"very high flexion\" tyres (namely, IF and VF tyres) and the concept of new tread profiles, different from classic ones, confirm the importance of continuing to improve the modelling of agricultural tyres to deepen the knowledge of their operative behaviour. An accurate study of the tyre vertical dynamics and, hence, of the vehicle dynamics, can require models of agricultural tyres that are more complex than a damper-spring system [26; 27], but which can also take into account the elastic, mechanical and geometric properties. This approach is present in the recent scientific literature concerning the automotive [28; 29] and, especially, in all the works using simulation software to study the driveability of the vehicle in order to develop Advanced Driver-Assistance Systems (ADAS). The main properties regarding on-road tyre performances (e.g., longitudinal, lateral forces or self-aligning moment) can be expressed by the so-called \"magic formula\", an empirical model proposed by [PERSON], commonly used to simulate steady-state tyre forces. The magic formula can be applied whatever the structure of the wheel: indeed, it has also been applied to a mechanical elastic wheel (ME-Wheel) for an off-road vehicle to set up an integratedyaw and rollover stability control [30]. However, it is also necessary to consider the geometry and mechanical properties of the tyres when developing models of agricultural vehicle aimed at studying both stability [31; 32], and traction and soil compacting [33; 34]. To this end, the approach based on a tyre equipped with strain sensors, also known as \"Intelligent Tyre\", seems particularly promising, thus allowing relevant vehicle dynamics information to be provided [35]. In order to simulate the behaviour of an agricultural tyre, the finite elements method (FEM) has been hence considered the most appropriate, e.g., for studying deformation of an off-road tyre based on experimental results of characterization of the material [36; 37]. The influence of tyre pressure on the deformations of an agricultural tractor wheel has been studied in [38] by means of a plane-symmetric FE model of the tyre in the frontal plane. A more detailed 3D FE model was used in [36]; this detailed multi-component model accurately reproduces the real structure of the tyre with its tread, inner fabric, carcass, sidewalls, circumferential and radial cords and bead cores. The modelling of the interaction of a tractor tyre with the soil surface also requires a high detail of this last, if the aim is to analyse the contact pressure distribution for several wheel loads and tyre inflation pressures, as in [39; 40]. In these cases, the soil structure has been modelled through a multilayered mesh with its attributes corresponding to the soil properties, and, the tyre-soil interface pressures having been validated by equipping a tyre thread with six miniature pressure transducers [39] or by using proper lab equipment [40]. The aim of this work is to study the elastic behaviour of an agricultural tractor tyre under a vertical load to identify which physical parameters (inflation pressure, mechanical characteristics of the tyre, geometry and dimensions) most influence the overall elastic coefficients of the tyre. In addition, the mathematical development of the global elastic force-vertical displacement response has been investigated to determine whether this development can be considered linear, linear in stretches or non-linear.