Introduction

It is well known that there are three main contact models for classical contact problems. In the first it is assumed that the contact surfaces are smooth, as a result of which the shear contact stresses are equal to zero. The second model assumes that the shear contact stresses are related to the normal contact pressure by Coulomb's law. Finally, the third method takes into account the presence of complete adhesion between the contact surfaces [1-4]. It is also known that the second model is used only in the case when, in addition to the normal, compressive load, the stamp is also subject to a shear load. Since it is not correct to consider the model without taking into account shear loadUsing the third model, oscillating singularities arise near the ends of the stamps, as a result of which, near the endpoints of the contact zone, the modulus of shear stresses becomes greater than the contact pressure, thereby violating the adhesion condition [3].

To eliminate this defect, in 1945 Galin L.A. proposed a new contact model, which assumes that in the middle part of the contact zone there is complete adhesion, and at the ends of the stamp there is sliding, and the shear stresses are related to the contact pressure by Coulomb's law [5]. The Galin’s contact model is actually a combination of the second and third contact models and more accurately describes the contact process. Based on this model, efficient solutions to many contact problems for classical foundations were obtained [6-10]. However, the use of this contact model, especially in cases of non-classical foundations, often leads to mathematical and computational complexities.

Based on the above, we have proposed a new contact model taking into account static friction. It is assumed that the shear stresses acting in the contact zone are related to the normal contact pressure under the stamp according to the formula of dry friction in which the friction coefficient depends on the coordinates of the points of the contacting surfaces and is directly proportional to them. It should be noted that the proposed model, like the Galin contact model, is close to the model of complete adhesion in the central part of the contact zone and to the model of Coulomb friction at the end points of the stamp. A characteristic feature of this model is that solutions of contact problems for classical foundations within the framework of this model are reduced to solving singular integral equations with variable coefficients. However, for a number of classical bases, simple exact solutions can be obtained.

In [11], the indicated contact model is justified and an exact solution for the problem of pressing a stamp with a flat base into an elastic half-plane is constructed. In [12], the proposed model is generalized for the case where, in addition to the normal compressive load, a shear load also acts on the stamp, and an exact solution to the problem in quadratures is obtained. This model is also generalized for a rigid interphase inclusion located in a piecewise homogeneous plane [13].

Based on the proposed model, closed and efficient solutions were also obtained for a number of plane and axisymmetric contact problems for classical foundations such as a half-plane [14], a half-space [15], a homogeneous and composite plane with a crack [16-18], and a homogeneous space with a disc-shaped crack [19]. Namely, the papers [20-22] discuss the exact solutions for plane and axisymmetric contact problems of pressing a stamp of arbitrary shape into an elastic half-plane and half-spaces with a variable contact zone. In most of the cited works, numerical calculations were carried out and the obtained results were compared with the results obtained for similar problems within the framework of the Galin’s model. Comparisons showed that the obtained results are very close. Note that the proposed model is not difficult to apply to any non-classical foundations, since the contact conditions are the same throughout the entire contact zone.

References

1. Galin LA (1980) Contact Problems in the Theory of Elasticity and Viscoelasticity. Nauka: 304.
2. Shtaerman Iya (1949) Contact Problem of Elasticity Theory. Gostekhteorizdat: 270.
3. Muskhelishvili NI (1966) Some Problems of Mathematical Theory of Elasticity. Nauka: 708.
4. Hakobyan VN (2022) Stress Concentrators in Continuous Deformable Bodies. Advanced Structured Materials 181: 397p.
5. Galin LA (1945) Indentation of a Punch in the Presence of Friction and Adhesion. Prikl Mat Mekh 9(5): 413-424.
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10. Korotov SV, Kononov DP, Pakulina EV (2021) Stress State in the Presence of Slip and Adhesion. Proceedings of Petersburg Transport University, saint Petersburg, Petersburg State Transport University 18: 2.
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12. Hakobyan VN, Amirjanyan HA, Grigoryan AM (2024) On the Contact of an Absolutely Rigid Stamp with a Half-Plane Taking Into Account Static Friction. J Phys Conf Ser 2817: 012004.
13. Hakobyan VN, Amirjanyan AA, Khachikyan AS (2025) On a Contact Problem of a Thin Absolutely Rigid Interphase Inclusion with a Piecewise Homogeneous Plane in the Presence of Static Friction. Proc NAS RA Mechanics 78(3-4): 17-29 p.
14. Hakobyan VN, Hakobyan LV, Dashtoyan LL (2024) On Two Contact Problems for a Half-Plane with Static Friction J Phys Conf Ser 2817: 012004.
15. Hakobyan VN, Dashtoyan LL, Amirjanyan HA (2024) On an Indentation of a Circular Cylindrical Punch into an Elastic Half-Space Under Friction. ZAMM 104(5).
16. Hakobyan VN, Amirjanyan AA, Hakobyan LV (2024) On a Contact Problem for a Homogeneous Plane with a Finite Crack under Friction. Jour Mechanics of Solid 59: 2711-722 pp.
17. Hakobyan V, Dashtoyan L, Amirjanyan H (2024) Contact Problem for a Piecewise Homogeneous Plane with a Finite Crack Under the Static Friction. State of the Art and Future Trends in Materials Modelling 2. Advanced Structured Materials 200: 243-253.
18. Dashtoyan LL, Hakobyan LV (2025) Contact problem for piecewise homogeneous plane with interphase slit under static friction. Proc of IX international conference Topical Problems of Continuum Mechanics Tsakhkadzor: 24-27 Pp.
19. Hakobyan VN, Sahakyan AV, Dashtoyan LL, Amirjanyan HA (2024 )Axisymmetric Contact Problem for a Homogeneous Space with a Circular Disk-Shaped Crack Under Static Friction. Jour of Elasticity 156: 899-916.
20. Hakobyan VN, Dashtoyan LL, Amirjanyan HA, Hakobyan LV (2025) On the Plane Contact Problem of a Punch with a Previously Unknown Contact Area in the Presence of Static Friction In Current Developments in Solid Mechanics and Their Applications. Advanced Structured Materials vol 223: 255-266p.
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22. Hakobyan VN, Dashtoyan LL, Amirjanyan HA, Hakobyan LV (2025) On an Axisymmetric Contact Problem for a Half-Space with a Variable Contact Region in the Presence of Static Friction. Acta Mech.