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Thermodynamic Consistency and Mathematical Well-Posedness in the Theory of Elastoplastic, Granular, and Porous Materials

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Abstract

Mathematical models of the dynamics of elastoplastic, granular, and porous media are reduced to variational inequalities for hyperbolic differential operators that are thermodynamically consistent in the sense of Godunov. On this basis, the concept of weak solutions with dissipative shock waves is introduced and a priori estimates of smooth solutions in characteristic conoids of operators are constructed, which suggest the well-posedness of the formulation of the Cauchy problem and boundary value problems with dissipative boundary conditions. Additionally, efficient shock-capturing methods adapted to solution discontinuities are designed.

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REFERENCES

  1. S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, A. N. Kraiko, and G. P. Prokopov, Numerical Solution of Multidimensional Problems in Gas Dynamics (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  2. S. K. Godunov, Equations of Mathematical Physics (Nauka, Moscow, 1971) [in Russian].

    Google Scholar 

  3. S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauchnaya Kniga, Novosibirsk, 1998; Kluwer, New York, 2003).

  4. S. K. Godunov and I. M. Peshkov, “Symmetric hyperbolic equations in the nonlinear elasticity theory,” Comput. Math. Math. Phys. 48 (6), 975–995 (2008).

    Article  MathSciNet  Google Scholar 

  5. K. O. Friedrichs, “Symmetric hyperbolic linear differential equations,” Commun. Pure Appl. Math. 7 (2), 345–392 (1954).

    Article  MathSciNet  Google Scholar 

  6. L. M. Martyushev and V. D. Seleznev, The Maximum Entropy Production Principle in Physics and Related Areas (Ural. Gos. Tekh. Univ., Yekaterinburg, 2006) [in Russian].

    Google Scholar 

  7. V. M. Sadovskii, Discontinuous Solutions in the Dynamics of Elastoplastic Media (Nauka, Moscow, 1997) [in Russian].

    Google Scholar 

  8. V. M. Sadovskii, M. A. Guzev, O. V. Sadovskaya, and Ch. Qi, “Modeling of plastic deformation based on the theory of an orthotropic Cosserat continuum,” Fiz. Mezomekh. 22 (2), 59–66 (2019).

    Google Scholar 

  9. O. V. Sadovskaya and V. M. Sadovskii, Mathematical Modeling in Mechanics of Granular Materials (Fizmatlit, Moscow, 2008; Springer-Verlag, Berlin, 2012).

  10. V. M. Sadovskii and O. V. Sadovskaya, “Analyzing the deformation of a porous medium with account for the collapse of pores,” J. Appl. Mech. Tech. Phys. 57 (5), 808–818 (2016).

    Article  MathSciNet  Google Scholar 

  11. V. I. Smirnov, A Course of Higher Mathematics (Addison-Wesley, Reading, Mass., 1964; Fizmatlit, Moscow, 1981), Vol. 4, Part 2.

  12. L. V. Ovsyannikov, Lectures on Fundamentals of Gas Dynamics (Inst. Komp’yut. Issled., Moscow, 2003) [in Russian].

    Google Scholar 

  13. V. M. Sadovskii, “Hyperbolic variational inequalities in problems of the dynamics of elastoplastic bodies,” J. Appl. Math. Mech. 55 (6), 927–935 (1991).

    Article  MathSciNet  Google Scholar 

  14. V. N. Kukudzhanov, “Study of equations of elastoplastic dynamics under finite strains,” Nonlinear Deformation Waves (Tallinn, 1977), Vol. 2, pp. 102–105 [in Russian].

    Google Scholar 

  15. G. I. Bykovtsev and L. D. Kretova, “Shock wave propagation in elastic-plastic media,” J. Appl. Math. Mech. 36 (1), 94–103 (1972).

    Article  Google Scholar 

  16. V. M. Sadovskii, “Elastoplastic waves of strong discontinuity in linearly hardening media,” Mech. Solids 32 (6), 88–94 (1997).

    Google Scholar 

  17. A. G. Kulikovskii and A. P. Chugainova, “Study of discontinuities in solutions of the Prandtl–Reuss elastoplasticity equations,” Comput. Math. Math. Phys. 56 (4), 637–649 (2016).

    Article  MathSciNet  Google Scholar 

  18. V. P. Maslov, V. P. Myasnikov, and V. G. Danilov, Mathematical Modeling of the Emergency Reactor at the Chernobyl Nuclear Power Plant (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

  19. V. M. Sadovskii, “On the theory of the propagation of elastoplastic waves through loose materials,” Dokl. Phys. 47 (10), 747–749 (2002).

    Article  MathSciNet  Google Scholar 

  20. M. L. Wilkins, “Calculation of elastic-plastic flow,” in Methods in Computational Physics: Fundamental Methods in Hydrodynamics (Academic, New York, 1964), Vol. 3, pp. 211–263.

    Google Scholar 

  21. B. D. Annin and V. M. Sadovskii, “The numerical realization of a variational inequality in the dynamics of elastoplastic bodies,” Comput. Math. Math. Phys. 36 (9), 1313–1324 (1996).

    MathSciNet  MATH  Google Scholar 

  22. B. N. Pshenichnyi and Yu. M. Danilin, Numerical Methods in Extremal Problems (Nauka, Moscow, 1975; Mir, Moscow, 1978).

  23. A. V. Zhiltsov and R. V. Namm, “The Lagrange multiplier method in a finite-dimensional convex programming problem”, Dal’nevost. Mat. Zh. 15 (1), 53–60 (2015)

    MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Russian Foundation for Basic Research, the Government of the Krasnoyarsk Territory, and the Krasnoyarsk Regional Fund of Science, research project no. 18-41-242001-r_mk.

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Authors and Affiliations

  1. Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, 660036, Krasnoyarsk, Russia

    V. M. Sadovskii

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  1. V. M. Sadovskii
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Correspondence to V. M. Sadovskii.

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Dedicated to Academician S.K. Godunov on the occasion of his 90th birthday

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Sadovskii, V.M. Thermodynamic Consistency and Mathematical Well-Posedness in the Theory of Elastoplastic, Granular, and Porous Materials. Comput. Math. and Math. Phys. 60, 723–736 (2020). https://doi.org/10.1134/S0965542520040156

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