Elastic–Plastic Waves in a Loosened Material

Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 21)

Abstract

A priori estimates for solutions in characteristic cones, which provide assurance that a boundary-value problem with initial data and dissipative boundary conditions is well-posed in the framework of a model describing dynamic deformation of an elastic–plastic granular material, are obtained. Shock adiabatic curves for plane longitudinal compression waves, propagating in an unbounded body, are constructed for various combinations of mechanical parameters of a material. Computational algorithm for the analysis of propagation of shock waves of small amplitude in a granular material, based on the method of splitting with respect to physical processes and with respect to spatial variables, is proposed. The results of two-dimensional computations of interaction of signotons in an inhomogeneous loosened material accompanied by a transverse cumulative ejection as well as the results of modeling of the “dry boiling” process (formation of continuity jumps in a material under the action of periodic load and their collapse) are presented.

Keywords

Shock Wave Variational Inequality Granular Material Jacobi Equation Displacement Discontinuity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Annin, B.D., Sadovskii, V.M.: The numerical realization of variational inequality in the dynamics of elastoplastic bodies. Comput. Math. Math. Phys. 36(9), 1313–1324 (1996)Google Scholar
  2. 2.
    Burenin, A.A., Zinoviev, P.V.: On the problem of allocation of surfaces of discontinuities in numerical methods in the dynamics of deformable media. In: Klimov, D.M. (ed.) Problems of Mechanics: Collection of Articles (by the 90-th Anniversary of the Birth of A. Yu. Ishlinskii), pp. 146–155. Fizmatlit, Moscow (2003)Google Scholar
  3. 3.
    Burenin, A.A., Bykovtsev, G.I., Rychkov, V.A.: Surfaces of the velocity discontinuities in the dynamics of irreversibly compressible media. In: Probl. Mekh. Sploshnoi Sredy, pp. 116–127. IAPU DVO RAN, Vladivostok (1996)Google Scholar
  4. 4.
    Bykovtsev, G.I., Ivlev, D.D.: Teoriya Plastichnosti (Plasticity Theory). Dal’nauka, Vladivostok (1998)Google Scholar
  5. 5.
    Bykovtsev, G.I., Yarushina, V.M.: On the features of the model of unsteady creep based on the use of piecewise-linear potentials. In: Problems of Continuum Mechanics and Structural Elements: Proceedings (by the 60-th Anniversary of the Birth of G. I. Bykovtsev), pp. 9–26. Dal’nauka, Vladivostok (1998)Google Scholar
  6. 6.
    Clayton, R., Engquist, B.: Absorbing boundary conditions for acoustic and elastic wave equations. Bull. Seismol. Soc. Am. 67(6), 1529–1540 (1977)Google Scholar
  7. 7.
    Dudko, O.V., Lapteva, A.A., Semenov, K.T.: On the propagation of plane one-dimensional waves and their interaction with obstacles in a medium with different resistance to tension and compression. Dal’nevost. Mat. Zh. 6(1–2), 94–105 (2005)Google Scholar
  8. 8.
    Engquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Commun. Pure Appl. Math. 32, 313–357 (1979)CrossRefGoogle Scholar
  9. 9.
    Fomin, V.M., Vorozhtsov, E.V., Yanenko, N.N.: On the properties of curvilinear shock waves “smearing” in calculations by the particle-in-cell methods. Comput. Fluids 7(2), 109–121 (1979)CrossRefGoogle Scholar
  10. 10.
    Friedrichs, K.O.: Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Math. 7(2), 345–392 (1954)CrossRefGoogle Scholar
  11. 11.
    Fryazinov, I.V.: Economical symmetrization schemes for solving boundary value problems for a multi-dimensional equation of parabolic type. Zh. Vychisl. Mat. Mat. Fiz. 8(2), 436–443 (1968)Google Scholar
  12. 12.
    Godunov, S.K.: Uravneniya Matematicheskoi Fiziki (Equations of Mathematical Physics). Nauka, Moscow (1979)Google Scholar
  13. 13.
    Godunov, S.K., Zabrodin, A.V., Ivanov, M.Y., Kraiko, A.N., Prokopov, G.P.: Chislennoe Reshenie Mnogomernykh Zadach Gazovoi Dinamiki (Numerical Solving Many-Dimensional Problems of Gas Dynamics). Nauka, Moscow (1976)Google Scholar
  14. 14.
    Goldshtik, M.A.: Proczessy Perenosa v Zernistom Sloe (Transfer Processes in Granular Layer). Institut Teplofiziki SO RAN, Novosibirsk (1984)Google Scholar
  15. 15.
    Haar, A., von Kármán, T.: Zur Theorie der Spannungszustände in plastischen und sandartigen Medien. Nachrichten von der Königlichen Gesellschaft der Wissenschaften, pp. 204–218 (1909)Google Scholar
  16. 16.
    Higdon, R.L.: Radiation boundary conditions for elastic wave propagation. SIAM J. Numer. Anal. 27(4), 831–870 (1990)CrossRefGoogle Scholar
  17. 17.
    Il’gamov, M.A., Gil’manov, A.N.: Neotrazhayushhie Usloviya na Graniczakh Raschetnoi Oblasti (Nonreflecting Conditions on Boundaries of Computational Domain). Fizmatlit, Moscow (2003)Google Scholar
  18. 18.
    Ivanov, G.V., Volchkov, Y.M., Bogulskii, I.O., Anisimov, S.A., Kurguzov, V.D.: Chislennoe Reshenie Dinamicheskikh Zadach Uprugoplasticheskogo Deformirovaniya Tverdykh Tel (Numerical Solution of Dynamic Elastic–Plastic Problems of Deformable Solids). Sib. Univ. Izd., Novosibirsk (2002)Google Scholar
  19. 19.
    Kamenetskii, V.F., Semenov, A.Y.: Self-consistent allocation of discontinuities in the shock-capturing computations of gas-dynamic flows. Zh. Vychisl. Mat. Mat. Fiz. 34(10), 1489–1502 (1994)Google Scholar
  20. 20.
    Kolarov, D., Baltov, A., Bontcheva, N.: Mekhanika na Plastichnite Sredi (Mechanics of Plastic Media). Izd. Bulg. Akad. Nauk, Sofia (1975)Google Scholar
  21. 21.
    Kondaurov, V.I., Fortov, V.E.: Osnovy Termomekhaniki Kondensirovannoi Sredy (Fundamentals of the Thermomechanics of a Condensed Medium). Izd. MFTI, Moscow (2002)Google Scholar
  22. 22.
    Kukudzhanov, V.N.: Raznostnye Metody Resheniya Zadach Mekhaniki Deformiruemykh Tel (Finite Difference Methods for the Problems of Solid Mechanics). Izd. MFTI, Moscow (1992)Google Scholar
  23. 23.
    Kulikovskii, A.G., Pogorelov, N.V., Semenov, A.Y.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Monographs and Surveys in Pure and Applied Mathematics, vol. 118. Chapman& Hall, Boca Raton (2001)Google Scholar
  24. 24.
    Magomedov, K.M., Kholodov, A.S.: Setochno-Kharakteristicheskie Chislennye Metody (Grid-Characteristic Numerical Methods). Nauka, Moscow (1988)Google Scholar
  25. 25.
    Marchuk, G.I.: Methods of Numerical Mathematics. Springer, Berlin (1975)Google Scholar
  26. 26.
    Marchuk, G.I.: Metody Rasshhepleniya (Splitting Methods). Nauka, Moscow (1988)Google Scholar
  27. 27.
    Maslov, V.P., Mosolov, P.P.: 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)Google Scholar
  28. 28.
    Maslov, V.P., Myasnikov, V.P., Danilov, V.G.: Mathematical Modeling of the Chernobyl Reactor Accident. Springer, Berlin (1992)Google Scholar
  29. 29.
    Mosolov, P.P., Myasnikov, V.P.: Variaczionnye Metody v Teorii Techenii Zhestko-Vyazko-Plasticheskikh Sred (Variational Methods in the Theory of Flows of Rigid-Viscoplastic Media). Izd. Mosk. Univ., Moscow (1971)Google Scholar
  30. 30.
    Mosolov, P.P., Myasnikov, V.P.: Mekhanika Zhestkoplasticheskikh Sred (Mechanics of Rigid-Plastic Media). Nauka, Moscow (1981)Google Scholar
  31. 31.
    Nowacki, W.K.: Stress Waves in Non-Elastic Solids. Pergamon Press, Oxford (1977)Google Scholar
  32. 32.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985)Google Scholar
  33. 33.
    Richtmyer, R.: Principles of Advanced Mathematical Physics. Springer, New York (1978–1986)Google Scholar
  34. 34.
    Sadovskaya, O.V.: To the analysis of a velocities and stresses discontinuities in ideal granular elastic–plastic medium. Dal’nevost. Mat. Zh. 4(2), 242–251 (2003)Google Scholar
  35. 35.
    Sadovskaya, O.V.: Shock-capturing method as applied to the analysis of elastoplastic waves in a granular material. Comput. Math. Math. Phys. 44(10), 1818–1828 (2004)Google Scholar
  36. 36.
    Sadovskaya, O.V., Sadovskii, V.M.: Elastoplastic waves in granular materials. J. Appl. Mech. Tech. Phys. 44(5),741–747 (2003)Google Scholar
  37. 37.
    Sadovskii, V.M.: Razryvnye Resheniya v Zadachakh Dinamiki Uprugoplasticheskikh Sred (Discontinuous Solutions in Dynamic Elastic–Plastic Problems). Fizmatlit, Moscow (1997)Google Scholar
  38. 38.
    Sadovskii, V.M.: Problems of the dynamics of granular media. Mat. Modelirovanie 13(5), 62–74 (2001)Google Scholar
  39. 39.
    Sadovskii, V.M.: To the theory of elastic–plastic waves propagation in granular materials. Doklady Phys. 47(10), 747–749 (2002)Google Scholar
  40. 40.
    Samarskii, A.A.: Theory of Difference Schemes. Marcel Dekker, New York (2001)CrossRefGoogle Scholar
  41. 41.
    Sedov, L.I.: Mechanics of Continuous Media (in 2 vol.), Series in Theoretical and Applied Mechanics, vol. 4, 4th edn. World Scientific Publishing Company, Singapore (1997)Google Scholar
  42. 42.
    Shokin, Y.I., Yanenko, N.N.: Metod Differenczial’nogo Priblizheniya. Primenenie k Gazovoi Dinamike (Method of Differential Approximation. Application to Gas Dynamics). Nauka, SO RAN, Novosibirsk (1985)Google Scholar
  43. 43.
    Trusdell, C.: A First Course in Rational Continuum Mechanics. The John Hopkins University, Baltimore (1972)Google Scholar
  44. 44.
    Wilkins, M.L.: Calculation of elastic–plastic flow. In: Methods in Computational Physics. Fundamental Methods in Hydrodynamics, vol. 3, pp. 211–263. Academic Press, New York (1964)Google Scholar
  45. 45.
    Yakovlev, I.V., Kuzmin, G.E., Pai, V.V. (eds.): Volnoobrazovanie pri Kosykh Soudareniyakh: Sbornik Statei (Wave Formation in Oblique Impacts: Collection of Articles). Izd. IDMI SO RAN, Novosibirsk (2000)Google Scholar
  46. 46.
    Yanenko, N.N., Vorozhtsov, E.V., Fomin, V.M.: Differential analyzers of shock waves. Dokl. Akad. Nauk SSSR 227(1), 50–53 (1976)Google Scholar
  47. 47.
    Yang, W.H.: A useful theorem for constructing convex yield functions. Trans. ASME J. Appl. Mech. 47(2), 301–305 (1980)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.ICM SB RASKrasnoyarskRussia

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