On the use of the Stieltjes integral for calculating of mechanical work in relation to adhesive contact

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Аннотация

A procedure for regularization of the Stieltjes integral in the case of a common breaking point for integrand functions is described. Using this procedure, it is possible to determine the Stieltjes integral, which represents mechanical work in accordance with the energy conservation law. The physical validity of the obtained results is confirmed by a number of examples. In particular, the regularization procedure makes it possible to calculate the energy dissipated during an abrupt change in the state of the elastic suspension.

Авторлар туралы

I. Soldatenkov

Ishlinsky Institute for Problems in Mechanics RAS

Хат алмасуға жауапты Автор.
Email: iasoldat@hotmail.com
Moscow, Russia

Әдебиет тізімі

  1. Bland D.R. The Theory of Linear Viscoelasticity. Oxford, New York: Pergamon Press, 1960.
  2. Christensen R.M. Theory of Viscoelasticity. An introduction. New York: Academic Press, 1971.
  3. Rabotnov Yu.N. Elements of Hereditary Solid Mechanics. M: Mir, 1980.
  4. Kantorovich L.V. Application of the Stieltjes integral to the calculation of beams lying on elastic supports // Proc. Leningrad Institute of Industrial Construction Engineers. 1934. № 1. P. 17–34.
  5. Fikhtengol’ts G.M. Course of Differential and Integral Calculus. V. 3. M: FIZMATLIT, 2003 (in Russian).
  6. de La Vallée Poussin C.J. Leçons de Mécanique Analytique. T. 1. Louvain: Librairie universitare, 1924.
  7. Hahn H.G. Elastizitätstheorie. Grundlagen der linearen Theorie und Anwendungen auf eindimensionale, ebene und räumliche Probleme. Stuttgart: Teubner, 1985.
  8. Israelachvili J.N. Intermolecular and Surface Forces. 3-rd ed. London: Academic, 2011.
  9. Overbeek J.T.G., Sparnaay M.J. Classical coagulation. London-van der Waals attraction between macroscopic objects // Discuss. Faraday Soc. 1954. V. 18. P. 12–24. https://doi.org/10.1039/DF9541800012
  10. Muller V.M., Yushchenko V.S., Derjaguin B.V. On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane // J. Coll. Interface Sci. 1980. V. 77. № 1. P. 91–101.
  11. Attard P., Parker J.L. Deformation and adhesion of elastic bodies in contact // Phys. Rev. A. 1992. V. 46. № 12. P. 7959–7971. https://doi.org/10.1103/PhysRevA.46.7959
  12. Greenwood J.A. Adhesion of elastic spheres // Proc. R. Soc. Lond. A. 1997. V. 453. № 1961. P. 1277–1297. https://doi.org/10.1098/rspa.1997.0070
  13. Soldatenkov I.A. Contact with intermolecular interaction for a viscoelastic layer (self-consistent approach): Energy dissipation under indentation and friction force // Mechanics of Solids. 2022. V. 57. № 7. P. 1701–1716. https://doi.org/10.3103/S0025654422070160
  14. Soldatenkov I.A. Contact with intermolecular interaction forces for a viscoelastic layer (self-consistent approach): the energy balance for the system of indenter–layer–substrate // PMM. 2024. V. 88. № 3. P. 456–482 (in Russian). https://doi.org/10.31857/S0032823524030093
  15. Medvedev F.A. Development of the Integral Concept. M: Nauka, 1974 (in Russian).
  16. Glivenko V.I. The Stieltjes Integral. Leningrad: ONTI, 1936 (in Russian).
  17. Smirnov V.I. A Course of Higher Mathematics. V. 5. Oxford: Pergamon Press, 1964.
  18. Rodionov V.I. Application of Algebraic Systems in the Theory of Differential Equations. Izhevsk: Udmurt State University, 2021 (in Russian).
  19. Derr V.Ya. On the extension of a Rieman–Stieltjes integral // Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki. 2019. V. 29. № 2. P. 135–152 (in Russian). https://doi.org/10.20537/vm190201
  20. Lukashenko T.P., Skvortsov V.A., Solodov A.P. Generalized Integrals. M: LIBROKOM, 2011 [in Russian].
  21. Hanung U.M., Tvrdý M. On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil // Mathematica Bohemica. 2019. V. 144. № 4. P. 357–372. https://doi.org/10.21136/MB.2019.0015-19
  22. Derr V.Ya. A generalization of Riemann–Stieltjes integral // Functional Differential Equations. 2002. V. 9. № 3–4. P. 325–341. https://campuscore.ariel.ac.il/wp/fde/wp-content/uploads/sites/97/2020/02/2002-3-4.pdf
  23. Kolmogorov A.N., Fomin S.V. Elements of the Theory of Functions and Functional Analysis. V. 1: Metric and Normed Spaces. Rochester: Graylock Press, 1952. V. 2: Measure. The Lebesgue Integral. Hilbert Space. Albany: Graylock Press, 1961.
  24. Tsalyuk V.Z. Multivalued Stieltjes integral for discontinuous functions of bounded variation // Functional Differential Equations. 2002. V. 9. № 3–4. P. 551–576. https://campuscore.ariel.ac.il/wp/fde/wp-content/uploads/sites/97/2020/02/2002-3-4.pdf
  25. Fikhtengol’ts G.M. Course of Differential and Integral Calculus. V. 1. M.: FIZMATLIT, 2003 [in Russian].
  26. Zorich V.A. Mathematical Analysis I. Berlin: Springer, 2004.
  27. Soldatenkov I.A. The contact problem with the bulk application of intermolecular interaction forces (a refined formulation) // J. Appl. Math. Mech. 2013. V. 77. № 6. P. 629–641. https://doi.org/10.1016/j.jappmathmech.2014.03.007
  28. Johnson K.L., Greenwood J.A. An adhesion map for the contact of elastic spheres // Journal of Colloid and Interface Science. 1997. V. 192. № 2. P. 326–333. https://doi.org/10.1006/jcis.1997.4984
  29. Soldatenkov I.A. Contact with intermolecular interactions for a viscoelastic layer (self-consistent approach): Feature analysis of the indenter approach/retract process // Mechanics of Solids. 2021. V. 56. № 7. P. 1259–1276. https://doi.org/10.3103/S0025654421070232
  30. Malkin A.Ya., Isayev A.I. Rheology: Concepts, Methods and Applications. Toronto: ChemTec Publishing. 2012.
  31. Vladimirov V.S. Equations of Mathematical Physics. New York: Marcel Dekker, 1971.

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