Nonlocal solutions of the theory of elasticity problems for an infinite space loaded with concentrated forces

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Resumo

Two classical problems of the theory of elasticity are considered in the paper. The first is the Kelvin problem for an infinite space loaded with a concentrated force. The classical solution is singular and specifies an infinitely high displacement of the point of the force application which has no physical meaning. To obtain a physically consistent solution, the nonlocal theory of elasticity is used, which, in contrast to the classical theory, is based on the equations derived for an element of continuum that has small but finite dimensions, and allows one to obtain regular solutions for traditional singular problems. The equations of the nonlocal theory include an additional experimental constant, which has the dimension of length and cannot be determined for a space problem. Consequently, the second problem for an infinite plane loaded with two concentrated forces lying on the same straight line and acting in the opposite directions is considered. The classical solution of this problem is also singular and specifies an infinitely high elongation of the distance between the forces, irrespective of their magnitude. The solution of this problem is also obtained within the framework of the nonlocal theory of elasticity, which specifies a regular dependence of this distance on the forces magnitude. This solution also includes an additional constant which is determined experimentally for a plane problem.

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Sobre autores

V. Vasiliev

Central Research Institute of Special Engineering

Autor responsável pela correspondência
Email: vvvas@dol.ru
Rússia, Khotkovo

S. Lurie

Institute of Applied Mechanics RAS

Email: salurie@mail.ru
Rússia, Moscow

V. Salov

Central Research Institute of Special Engineering

Email: snegiricentral@yandex.ru
Rússia, Khotkovo

Bibliografia

  1. Vasil’ev V.V., Lurie S.A. Generalized Theory of Elasticity // Mech. Solids. 2015. V. 50. № 4. Р. 379–388. https://doi.org/10.3103/S0025654415040032
  2. Vasiliev V.V., Lurie S.A. Differential equations and the problem of singularity of solutions in applied mechanics and mathematic // Journal of Applied Mechanics and Technical Physics. 2023. V. 64. № 1. Р. 98–109. https://doi.org/10.1134/S002189442301011X
  3. Vasiliev V.V., Lurie S.A. To the problem of discontinuous solutions in applied mathematics // Mathematics. 2023. V. 11. P. 3362. https://doi.org/10.3390/math.11153362
  4. Nowacki W. Theory of Elasticity. M.: Nauka, 1975, 836 p.
  5. Polyanin A.D. Handbook of linear equations of mathematical physics. M.: Fizmatlit, 2001. 576 p.
  6. Handbook on special functions with formulas, graphs and mathematical tables. M.: Nauka, 1979. 832 p.

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2. Fig. 1. Unbounded plate loaded by two forces

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3. Fig. 2. Dependence of relative displacement on x/l

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4. Fig. 3. Disc stretched by concentrated forces

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5. Fig. 4. Dependences of displacement on distance l, corresponding to formula (3.12) (───) and experiment (●).

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Declaração de direitos autorais © Russian Academy of Sciences, 2024