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Interaction of Plane Strain Waves in a Heteromodular Elastic Half-Space at the Stage of Forced Stopping of Its Boundary after Uniaxial Tension–Compression

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Abstract

The evolution of the wave pattern in a multimodulus elastic half-space with a boundary moving in nonstationary uniaxial piecewise linear “tension–compression–stop” mode is studied. The solution of the boundary value problem includes all cases of interaction between plane one-dimensional strain waves, including reflected weak-intensity fronts. A number of new features of one-dimensional elastic deformation dynamics in a multimodulus medium are revealed, some of which (e.g., the appearance of a reflected shock wave at a distance from the loaded boundary, cyclic transitions of a narrow moving zone from a compressed to rigid state and back, and a stepwise decrease in the tensile strain level in the near-boundary zone after the boundary is stopped) can be obtained with a given boundary loading only taking into account reflection effects.

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REFERENCES

  1. I. V. Baklashov and B. A. Kartozija, Mechanics of Rocks (Nedra, Moscow, 1975) [in Russian].

    Google Scholar 

  2. D. E. Bessonov, Yu. P. Zezin, and E. V. Lomakin, “Multimodulus behavior of the grained composites on the base of unsaturated polyethers,” Izv. Saratov.Gos. Univ. Nov. Ser. Ser. Mat. Mekh. Inf. 9 (4), 9–13 (2009) [in Russian]. https://doi.org/10.18500/1816-9791-2009-9-4-2-9-13

  3. E. V. Lomakin and B. N. Fedulov, “Nonlinear anisotropic elasticity for laminate composites,” Meccanica 50 (6), 1527–1535 (2015). https://doi.org/10.1007/s11012-015-0104-5

    Article  MathSciNet  Google Scholar 

  4. A. Makeev, Yihong He, P. Carpentier, and B. Shonkwiler, “A method for measurement of multiple constitutive properties for composite materials,” Composites. Part A: Appl. Sci. Manuf. 43 (12), 2199–2210 (2012). https://doi.org/10.1016/j.compositesa.2012.07.021

    Article  CAS  Google Scholar 

  5. A. M. Zhukov, “Moduli of elasticity of materials in extension and compression,” J. Appl. Mech. Tech. Phys. (26), 568–571 (1985). https://doi.org/10.1007/BF01101643

  6. J. L. Katz, P. Spencer, Yong Wang, A. Misra, O. Marangos, and L. Friis, “On the anisotropic elastic properties of woods,” J. Mater. Sci. (43), 139–145 (2008). https://doi.org/10.1007/s10853-007-2121-9

  7. Simin Li, Emrah Demirci, and V. V. Silberschmidt, “Variability and anisotropy of mechanical behavior of cortical bone in tension and compression,” J. Mech. Behav. Biomed. Mater. (21), 109–120 (2013). https://doi.org/10.1016/j.jmbbm.2013.02.021

  8. V. D. Kupradze, T. G. Gegelia, M. O. Bashelejshvili, and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity (North-Holland Publishing Co., New York, 1979).

    Google Scholar 

  9. E. V. Lomakin and Yu. N. Rabotnov, “A theory of elasticity for an isotropic body with different moduli in tension and compression,” Mech. Solids 13 (6), 25–30 (1978).

    MathSciNet  Google Scholar 

  10. S. A. Ambartsumyan, Theory of Elasticity for Materials with Variable Moduli (Nauka, Moscow, 1982) [in Russian].

    Google Scholar 

  11. I. Yu. Tsvelodub, “Multimodulus elasticity theory,” J. Appl. Mech. Tech. Phys. (49), 129–135 (2008). https://doi.org/10.1007/s10808-008-0019-1

  12. V. A. Lyakhovskii and V. P. Myasnikov, “On the behavior of elastic cracked solid,” Izv. Akad. Nauk SSSR. Fiz. Zemli (10), 71–75 (1984) [in Russian].

  13. V. P. Myasnikov and A. I. Oleinikov, Fundamentals of Mechanics of Heterogeneous-Resisting Media (Dal’nauka, Vladivostok, 2007) [in Russian].

    Google Scholar 

  14. O. V. Sadovskaya and V. M. Sadovskii, “The theory of finite strains of a granular material,” J. Appl. Math. Mech. (71), 93–110 (2007). https://doi.org/10.1016/j.jappmathmech.2007.03.005

  15. V. M. Sadovskii, O. V. Sadovskaya, and I. E. Petrakov, “On the theory of constitutive equations for composites with different resistance in compression and tension,” Compos. Struct. (268), 113921 (2021). https://doi.org/10.1016/j.compstruct.2021.113921

  16. V. E. Nazarov, “Elastic waves in media with bimodular nonlinearity taking into account the effects of reflection from shock fronts,” Tech. Phys. 67 (14), 2261–2269 (2022). https://doi.org/10.21883/TP.2022.14.55229.118-21

    Article  Google Scholar 

  17. S. N. Gavrilov and G. C. Herman, “Wave propagation in a semi-infinite heteromodular elastic bar subjected to a harmonic loading,” J. Sound Vib. 331 (40), 4464–4480 (2012). https://doi.org/10.1016/j.jsv.2012.05.022

    Article  ADS  Google Scholar 

  18. O. V. Dudko, A. A. Lapteva, and K. T. Semyonov, “About the distribution of plane one-dimensional waves and their interaction with a barrier in a media differently reacting to tension and compression,” Dal’nevost. Mat. Zh. 6 (1–2), 94–105 (2005) [in Russian].

    Google Scholar 

  19. O. V. Dudko, A. A. Lapteva, and V. E. Ragozina, “Nonstationary 1D dynamics problems for heteromodular elasticity with piecewise-linear approximation of boundary conditions,” Vestn. PNIPU. Mekh. (4), 37–47 (2019) [in Russian]. https://doi.org/10.15593/perm.mech/2019.4.04

  20. O. V. Dudko, A. A. Lapteva, and V. E. Ragozina, “Evolution of the wave pattern for piecewise linear uniaxial tension and compression of a heteromodular elastic bar,” J. Appl. Ind. Math. 16 (4), 645–658 (2022). https://doi.org/10.1134/S1990478922040068

    Article  MathSciNet  Google Scholar 

  21. Ali H. Nayfeh, Perturbation Methods (John Wiley & Sons, New York, 1973).

  22. M. Kuznetsova, M. Khludyakov, and V. Sadovskii, “Wave propagation in continuous bimodular media,” Mech. Adv. Mater. Struct. (2021). https://doi.org/10.1080/15376494.2021.1889725

  23. V. P. Maslov and P. P. Mosolov, “General theory of the solutions of the equations of motion of an elastic medium of different moduli,” J. Appl. Math. Mech. 49 (3), 322–336 (1985). https://doi.org/10.1016/0021-8928(85)90031-0

    Article  MathSciNet  Google Scholar 

  24. A. G. Kulikovskii and E. I. Sveshnikova, Nonlinear Waves in Elastic Media (CRC Press, New York, 1995).

    Google Scholar 

  25. Handbook of Physical Quantities, I. S. Grigoriev and E. Z. Meilikhov, Eds., (CRC Press, New York, 1997).

  26. O. V. Dudko and V. E. Ragozina, “On the motion of shock waves at constant speed in multimodulus elastic media,” Mech. Solids 53 (1), 111–119 (2018). https://doi.org/10.3103/S0025654418010132

    Article  ADS  Google Scholar 

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Institute of Automation and Control Processes, Far East Branch, Russian Academy of Sciences, Vladivostok, 690041, Russia

    O. V. Dudko, A. A. Lapteva & V. E. Ragozina

Authors
  1. O. V. Dudko
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  2. A. A. Lapteva
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  3. V. E. Ragozina
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Correspondence to O. V. Dudko, A. A. Lapteva or V. E. Ragozina.

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Translated by V. Potapchouck

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Dudko, O.V., Lapteva, A.A. & Ragozina, V.E. Interaction of Plane Strain Waves in a Heteromodular Elastic Half-Space at the Stage of Forced Stopping of Its Boundary after Uniaxial Tension–Compression. J. Appl. Ind. Math. 17, 710–723 (2023). https://doi.org/10.1134/S1990478923040038

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  • DOI: https://doi.org/10.1134/S1990478923040038

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