# Effect of friction in the interaction of an anisotropic strip with a rigid base

## Abstract

Relevance. Different models of contact between bodies are used in determining the stressed and deformed state in the strip lying on the base. It is necessary to evaluate the qualitative and quantitative nature of the change in stress in the strip depending on the coupling of the strip and base. The aim of the work - to analyze the effect of the coefficient of friction on the value of stresses in an anisotropic band when interacting with a rigid base. Methods. The solution is based on the equations of the plane problem of the theory of elasticity of an anisotropic body under the conditions that the band is closely adjacent to the base and the tangent force on the contact plane is proportional to the normal pressure. Displacements and stresses at any point of the strip are written in the form of the method of initial functions through the functions of displacements and forces on the lower plane, which depend on the nature of the load applied on the upper plane and the conditions of contact between the strip and the base. After the transformations, the calculation formulas for displacements and stresses are expressed using the Fourier integral transform through the normal surface load in the form of improper integrals. Results. Formulas for determining displacements and stresses are obtained for the variant of loading a strip with a concentrated force. These formulas are used to construct influence functions for the problem of equilibrium of an anisotropic strip lying on a rigid base, taking into account friction. Graphs of the effect of the coefficient of friction and the direction of the anisotropy axes of the material on the stress state of the strip are presented. The results of stress calculation are compared using anisotropic and isotropic models.

## Keywords

### Sergey G. Kudryavtsev

Volga State University of Technology

Author for correspondence.
Email: KudryavcevSG@volgatech.net

Candidate of Technical Sciences, Associate Professor of the Department of Resistance of Materials and Applied Mechanics

3 Lenina Sq, Yoshkar-Ola, 424000, Mari El Republic, Russian Federation

### Julia M. Buldakova

Volga State University of Technology

senior lecturer of the Department of Resistance of Materials and Applied Mechanics

3 Lenina Sq, Yoshkar-Ola, 424000, Mari El Republic, Russian Federation

## References

1. Shehter O.Y. Raschet beskonechnoi fundamentalnoi pliti, lejaschei na uprugom osnovanii konechnoi i beskonechnoi moschnosti i nagrujennoi sosredotochennoi siloi [Calculation of an infinite fundamental plate lying on an elastic base of finite and infinite power and loaded with a concentrated force]. Sbornik trudov nauchno-issledovatelskogo sektora tresta glubinnih rabot. 1939:133–139. (In Russ.)
2. Rappoport R.M. Zadacha Bussineska dlya sloistogo uprugogo poluprostranstva [The Boussinesq problem for a layered elastic half-space]. Trudi Leningradskogo politehnicheskogo instituta. 1948;(5):3–18. (In Russ.)
3. Vlasov V.Z., Leontev N.N. Balki, pliti i obolochki na uprugom osnovanii [Beams, plates and shells on elastic base]. Moscow, Gos. izd. fiz.-mat. lit-ry Publ.; 1960. (In Russ.)
4. Uflyand Ya.S. Integralnie preobrazovaniya v zadachah teorii uprugosti [Integrated transformations in tasks of the theory of elasticity]. Moscow, Leningrad, Izd-vo AN SSSR Publ.; 1963. (In Russ.)
5. Harr M.E. Osnovy teoreticheskoi mekhaniki gruntov [Foundations of theoretical soil mechanics]. Moscow, Stroiizdat Publ.; 1971. (In Russ.)
6. Smirnov A.V., Malyshev A.A., Agalakov Yu.A. Mehanika ustoichivosti i razrushenii dorojnih konstrukcii [Mechanics of stability and destruction of road structures]. Omsk, SibADI Publ.; 1997. (In Russ.)
7. Potelezhko V.P. Zadacha Flamana dlya dvuhsloinoi poluploskosti [The Flaman problem for a two-layer half-plane]. Mechanics and Physics of Processes on the Surface and in the Contact of Solids, Parts of Technological and Power Equipment. 2005;(1):29–33. (In Russ.)
8. Torskaya E.V., Lushnikov N.A., Lushnikov P.A. Analysis of stress-strain state of multi-layer pavements. Journal of Friction and Wear. 2008;29(2):204–210. (In Russ.)
9. Tarntira K., Senjuntichai T., Keawsawasvong S. Multilayered Elastic Medium under Axisymmetric Loading and Surface Energy. Advanced Materials and Engineering Materials VIII. 2019;(814):320–326.
10. Lehnickii S.G. K zadache ob uprugom ravnovesii anizotropnoi polosi [On the problem of elastic equilibrium of an anisotropic band]. Prikladnaya mehanika i matematika. 1963;(1):142–149. (In Russ).
11. Pan E. Static response of transversely isotropic and layered half-space to general surface loads. Phys. Earth Planet Inter. 1989;(54):353–363.
12. Kudryavcev S.G., Buldakova J.M. Solution of the plane problem of elasticity theory for an orthotropic stripe. The scientific and practical conference: Innovations in the educational process. 2010;(8):118–123. (In Russ.)
13. Krupoderov A.V. Fundamental solutions for transversely isotropic multilayered. News of the Tula State University. Sciences of Earth. 2011;(1):137–146. (In Russ.)
14. Kudryavcev S.G., Buldakova J.M. Interaction of anisotropic band and rigid base. Structural Mechanics of Engineering Constructions and Buildings. 2012;(4):29–35. (In Russ.)
15. Fabrikant V.I. Tangential contact problems for several transversely isotropic elastic layers bonded to an elastic foundation. Journal of Engineering Mathematics. 2013; 81(1):93–126.
16. Liu J., Zhang P., Lin G., Li C., Lu S. Elastostatic solutions of a multilayered transversely isotropic piezoelectric system under axisymmetric loading. Acta Mechanica. 2017;228(1):107–128.
17. Kulagina M.F., Ivanova V.I. Pervaya osnovnaya zadacha teorii uprugosti dlya oblasti sostoyaschei iz polosi i poluploskosti [The first major problem of elasticity theory for a region consisting of band and half-plane]. Journal of Samara State Technical University. Ser. Physical and Mathematical Sciences. 2003;(19):89–96. (In Russ.)
18. Lehnickii S.G. Teoriya uprugosti anizotropnogo tela [Theory of elasticity of an anisotropic body]. Moscow, Nauka Publ.; 1977. (In Russ.)
19. Nowacki W. Teoriya uprugosti [Theory of elasticity]. Moscow, Mir Publ.; 1975. (In Russ.)

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