Plane unsteady contact problem for a rigid stamp and an elastic half-space with a cavity


DOI: 10.34759/trd-2020-113-02

Аuthors

Arutyunyan A. M.1*, Kuznetsova E. L.1**, Fedotenkov G. V.2***

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. ,

*e-mail: 89057254188@mail.ru
**e-mail: vida_ku@mail.ru
***e-mail: greghome@mail.ru

Abstract

Plane non-stationary contact problems are being considered for absolutely rigid bodies with rectangular sections in plan (stamps) and an elastic half-space with recessed cavities of arbitrary geometry.

The problems setting includes the equations of plane motion of an elastic medium (Lamé’s equations), Hooke’s law, Cauchy relations, zero initial conditions, boundary conditions of the free edge on the boundary of the internal cavity in a half-space, conditions of contact between the boundary of the half-space and the stamp. To close the problem, the equations of translational motion of the centers of mass of the stamp are added. Cases of free slip and rigid adhesion are considered as contact conditions. We assume that outside the contact zone, the surface of the half-space is free of stresses. All equations and relations of the mathematical formulation of the problem are written in a Cartesian rectangular coordinate system.

At the initial time instant, a vertical load is applied to the stamp with a predetermined law of changing in time, which resultant is passing through the center of mass of this body.

The dynamic work reciprocity theorem is used to solve the problem. Application of the reciprocity theorem of works leads to the two-dimensional boundary integral equations, which kernels of are the transient functions. These functions represent solutions for a non-stationary problem for an elastic plane under the impact of concentrated mass forces. Integral Laplace transforms in time and Fourier transforms in spatial coordinates are used to build a solution to this problem.

The direct boundary element method with discretization in time is used to solve the two-dimensional boundary resulting integral equations. The linear interpolations in time herewith are used for displacements, and piecewise constant approximations for stresses. Special quadrature formulas based on the canonical regularization method are employed for singular integrals calculation.

As the result, a statement is presented and a method for solving new plane non-stationary contact problems for absolutely rigid stamps and an elastic half-space containing a recessed cavity with a smooth boundary of arbitrary geometry is developed. A resolving boundary integral equation is built and its discrete analogue is proposed.

Keywords:

unssteady contact problems, elastic half-space, cavity, boundary integral quations, Green's functions, generalized functions, rigid stamp, dynamic reciprocity theorem, integral transformations

References

  1. Tarlakovskiy D.V., Fedotenkov G.V. Analytic investigation of features of stresses in plane nonstationary contact problems with moving boundaries, Journal of Mathematical Sciences, 2009, vol. 162, no. 2, pp. 246 – 253. DOI: 10.1007/s10958-009-9635-4.

  2. Mikhailova E.Yu., Fedotenkov G.V. Nonstationary Axisymmetric Problem of the Impact of a Spherical Shell on an Elastic Half-Space (Initial Stage of Interaction), Mechanics of Solids, 2011, vol. 46, no. 2, pp. 239 – 247. DOI: 10.3103/S0025654411020129.

  3. Tarlakovskii and G.V. Fedotenkov Two-Dimensional Nonstationary Contact of Elastic Cylindrical or Spherical Shells, Journal of Machinery Manufacture and Reliability, 2014, vol. 43, no. 2, pp. 145 – 152. DOI: 10.3103/S1052618814010178.

  4. Gregory V. Fedotenkov, Elena Yu. Mikhailova, Elena L. Kuznetsova, Lev N. Rabinskiy Modeling the unsteady contact of spherical shell made with applying the additive technologies with the perfectly rigid stamp, International Journal of Pure and Applied Mathematics, 2016, vol. 111, no. 2, pp. 331 - 342. DOI: 10.12732/ijpam.v111i2.16.

  5. Fedotenkov G.V., Suvorov Ye.M., Tarlakovskii D.V. The plane problem of the impact of a rigid body on a half-space modelled by a Cosserat medium, Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 5, pp. 511 - 518. DOI: https://doi.org/10.1016/j.jappmathmech.2012.11.015

  6. Rabinskiy L.N., Tushavina O.V., Fedotenkov G.V. Plain non-stationary problem of the effect of a surface load on an elastic-porous half-space, Asia Life Sciences, 2019, vol. 28, no. 1, pp. 149 - 162.

  7. Mikhailova E.Yu., Tarlakovskii D.V., Fedotenkov G.V. Transient contact problem for spherical shell and elastic half-space, Shell Structures: Theory and Applications, 2017, vol. 4, pp. 301 - 304. DOI: https://doi.org/10.1201/9781315166605-67

  8. Mikhailova E.Yu., Tarlakovskii D.V., Fedotenkov G.V. The impact of liquid filled concentric spherical shells with a rigid wall, Shell Structures: Theory and Applications, 2017, vol. 4, pp 305 - 308. DOI: https://doi.org/10.1201/9781315166605-68

  9. Fedotenkov G.V., Kalinchuk V.V., Mitin A.Y. Three-Dimensional Non-stationary Motion of Timoshenko-Type Circular Cylindrical Shell, Lobachevskii Journal of Mathematics, 2019, vol. 40, no. 3, pp. 311 – 320. DOI: https://doi.org/10.1134/S1995080219030107

  10. Mitin A.Yu., Tarlakovskii D.V., Fedotenkov G.V. Trudy MAI, 2019, no. 107. URL: http://trudymai.ru/eng/published.php?ID=107884

  11. Lur'e S.A., Solyaev Yu.O., Nguen K., Medvedskii A.L., Rabinskii L.N. Trudy MAI, 2013, no 71. URL: http://trudymai.ru/eng/published.php?ID=47084

  12. Lyapin A.A., Seleznev M.G., Seleznev N.M. Ekologicheskii vestnik nauchnykh tsentrov ChES, 2008, no. 4, pp. 70 - 75.

  13. Aleksandrov V.M., Mark A.V. Quasistatic periodic contact problem for a viscoelastic layer, a cylinder, and a space with a cylindrical cavity, Journal of Applied Mechanics and Technical Physics, 2009, vol. 50, no. 5, pp. 866 - 871. DOI: 10.1007/s10808-009-0117-8

  14. Bozhkova L.V., Noritsina G.I., Ryabov V.G. Izvestiya moskovskogo gosudarstvennogo tekhnicheskogo universiteta MAMI, 2015, vol. 4, no. 4 (26), pp. 9 - 13.

  15. Pozharskii D.A., Pozharskaya E.D. Contact problems for an elastic inhomogeneous body with a cylindrical cavity, PNRPU Mechanics Bulletin, 2018, no. 4, pp. 200 - 208. DOI: 10.15593/perm.mech/2018.4.18

  16. Evdokimova O.V., Babeshko O.M., Babeshko V.A. Ekologicheskii vestnik nauchnykh tsentrov chernomorskogo ekonomicheskogo sotrudnichestva, 2017, vol. 14, no. 4-1, pp. 30 - 39.

  17. Rozhkova E.V., Abdukadyrov F.E., Ruzieva N.B. Prilozhenie matematiki v ekonomicheskikh i tekhnicheskikh issledovaniyakh, 2019, vol. 1, no. 9, pp. 125 - 129.

  18. Kalentev E.A. Stress-strain state of an elastic half-space with a cavity of arbitrary shape, International Journal of Mechanical and Materials Engineering, 2018, vol. 13, no. 8, DOI: https://doi.org/10.1186/s40712-018-0094-x

  19. Pushchin R.V., Pykhalov A.A. Trudy MAI, 2020, no. 110. URL: http://trudymai.ru/eng/published.php?ID=112862. DOI: 10.34759/trd-2020-110-11.

  20. Turanov R.A., Pykhalov A.A. Trudy MAI, 2019, no. 104. URL: http://trudymai.ru/eng/published.php?ID=102119

  21. Alielahi H., Kamalian M., Adampira M. A BEM investigation on the influence of underground cavities on the seismic response of canyons, Acta Geotechnica, 2016, vol. 11, pp. 391 – 413. DOI: https://doi.org/10.1007/s11440-015-0387-7

  22. Alielahi H., Adampira M. Seismic Effects of Two-Dimensional Subsurface Cavity on the Ground Motion by BEM: Amplification Patterns and Engineering Applications, International Journal of Civil Engineering, 2016, vol. 14, pp. 233 – 251. DOI: https://doi.org/10.1007/s40999-016-0020-7

  23. Igumnov L.A., Litvinchuk S.Yu., Petrov A.N. A numerical study of wave propagation on poroelastic half-space with cavities by use the BEM and Runge-Kutta method, Materials Physics and Mechanics, 2016, vol. 28, no. 1-2, pp. 96 - 100.

  24. Igumnov L.A., Markov I.P. Problemy prochnosti i plastichnosti, 2017, vol. 79, no. 3, pp. 348 - 356. DOI: https://doi.org/10.32326/1814-9146-2017-79-3-348-356

  25. Zhao J., Vollebregt E.A., Oosterlee, C.W. Extending the BEM for Elastic Contact Problems Beyond the Half-Space Approach, Mathematical Modelling and Analysis, 2016, vol. 21, no. 1, pp. 119 - 141. DOI: https://doi.org/10.3846/13926292.2016.1138418

  26. Schanz M., Rüberg T., Kielhorn L. Time-domain BEM: Numerical Aspects of Collocation and Galerkin Formulations, Recent Advances in Boundary Element Methods: A Volume to Honor Professor Dimitri Beskos, 2009, vol. 1, pp. 415 - 432. DOI:10.1007/978-1-4020-9710-2_27

  27. Carrer J.A.M., Pereira W., Mansur W.J. Two-dimensional elastodynamics by the time-domain boundary element method: Lagrange interpolation strategy in time integration, Engineering Analysis With Boundary Elements, 2012, vol. 36, pp. 1164 - 1172. DOI:10.1016/J.ENGANABOUND.2012.01.004

  28. Weidong Lei, Duofa Ji, Guopeng Zhu. Time-domain boundary element method with von Mises model for solving 2-D elastoplastic dynamic problems, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2019, vol. 41, pp. 1 - 13. DOI:10.1007/s40430-019-1770-3

  29. Delfim Soares. Dynamic analysis of elastoplastic models considering combined formulations of the time-domain boundary element method, Engineering Analysis With Boundary Elements, 2015, vol. 55, pp. 28 - 39. DOI:10.1016/J.ENGANABOUND.2014.11.014

  30. Schanz M., Antes H. Application of ‘Operational Quadrature Methods’ in Time Domain Boundary Element Methods, Meccanica, 1997, vol. 32, pp. 179 – 186. DOI: https://doi.org/10.1023/A:1004258205435

  31. Kielhorn L., Schanz M. Convolution Quadrature Method based symmetric Galerkin Boundary Element Method for 3-d elastodynamics, International Journal for Numerical Methods in Engineering, 2008, vol. 76, no. 11, pp. 1724 – 1746. DOI: http://dx.doi.org/10.1002/nme.2381

  32. Schanz M., Ye W., Xiao J. Comparison of the convolution quadrature method and enhanced inverse FFT with application in elastodynamic boundary element method, Computational Mechanics, 2016, vol. 57, pp. 523 – 536. DOI: https://doi.org/10.1007/s00466-015-1237-z

  33. Gorshkov A.G., Rabinskii L.N., Tarlakovskii D.V. Osnovy tenzornogo analiza i mekhanika sploshnoi sredy (Fundamentals of tensor analysis and continuum mechanics), Moscow, Nauka, 2000, 214 p.

  34. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnykh sredakh (Waves in continuum media: textbook for universities), Moscow, Fizmatlit, 2004, 472 c.

  35. Tarlakovskii D.V., Fedotenkov G.V. Obshchie sootnosheniya i variatsionnye printsipy teorii uprugosti (General relations and variation principles of the theory of elasticity), Moscow, Izd-vo MAI-PRINT, 2009, 144 p.


Download

mai.ru — informational site MAI

Copyright © 2000-2024 by MAI

 Вход