Mathematical modeling of dynamic deformation of elasto-viscoplastic shells of finite lenght by a ray method

N. D. Verveiko ; M. V. Egorov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 22 (2018) no. 2, pp. 325-343.

Résumé

The paper presents a mathematical modeling of dynamic stress-strain state of the rotation of the shell elasto-viscoplastic material. We solve the modified system of S. P. Timoschenko's partial differential equations by constructing a system of equations on moving surfaces of the gap with the initial conditions in the form of a shock at the end, written in the form of a power series in time, whose coefficients have initial conditions for the differential equations. The solution is presented in the form of the Taylor's row series up to the fourth order in the shell coordinate. To simulate the waves reflected from the boundaries, the conditions at the boundary of two types (rigidly restrained and stress-free), independent of time, are introduced. A set of programs written in Fortran 90 on the Code::Blocks platform is developed. Two programs for the simulation of dynamic deformation of shell in elastic and elasto-viscoplastic state are implemented. We use the difference representation of the derivatives, the calculation of the integrals by the trapezoid method with a given step of partitioning the segment. The result of the programs is the grid functions of the coefficients of the Taylor rows, which are used to construct the displacement graphs as functions of time and the longitudinal coordinate of the shell.

Détail

Keywords: dynamic deformation, rotating shell, ray method, reflected wave, modeling, elasticity, viscosity, plasticity.

Bibliographie

[1] Timoshenko S., Woinowsky–Krieger S., Theory of Plates and Shells, McGraw-Hill Book Comp., New York, 1959, 580+xiv pp.

[2] Sagomonian A. Ya., Volny napriazheniia v sploshnykh sredakh [Stress Waves in Continuous Media], Moscow State Univ., Moscow, 1985, 416 pp. (In Russian)

[3] Ivlev D. D., Teoriia ideal'noi plastichnosti [Theory of Ideal Plasticity], Nauka, Moscow, 1966, 332 pp. (In Russian) | MR

[4] Egorov M. V., “Dynamic deformation of an axisymmetric shell of rotation from an elasto-viscoplastic material near shock waves”, Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I. Ia. Iakovleva. Ser. Mekhanika predel'nogo sostoianiia, 2016, no. 2, 132–143 (In Russian)

[5] Verveiko N. D., Luchevaia teoriia uprugoviazkoplasticheskikh voln i voln gidroudara [Ray Theory of Elasto-Viscoplastic Waves and Hydrodynamic Waves], Voronezh State Univ., Voronezh, 1997, 204 pp. (In Russian)

[6] Marchuk G. I., Methods of Numerical Mathematics, Application of Mathematics, 2, New York, Springer-Verlag, 1982, xiii+510 pp. | DOI | MR | Zbl

[7] Sedov L. I., Similarity and Dimensional Methods in Mechanics, Academic Press, New York, London, 1959, 363 pp. | DOI | MR | MR | Zbl

[8] Samarskii A. A., Gulin A. V., Chislennye metody [Numerical Methods], Nauka, Moscow, 1989, 429 pp. (In Russian)

[9] Burenin A. A., Sevastyanov G. M., Shtuka V. I., “On the localization of discontinuities in calculations of incompressible elastic media”, Computational Continuum Mechanics, 9:4 (2016), 400–411 (In Russian) | DOI

[10] Gerasimenko Ye. A., Zavertan A. V., Ragozina V. Ye., “The use of near-front ray expansions in deformation dynamics”, J. App. Math. Mech., 73:2 (2009), 204–208 | DOI | Zbl

[11] Shtuka V. I., “The ray metod application for stain-stress state of elastic cylindrical layer with preliminary deformations definition”, Uchenye zapiski KnAGTU, 2017, no. 2, 40–44 (In Russian)

[12] Babicheva L. A., Bykovtsev G. I., Verveiko N. D., “Ray method of solving dynamic problems in elastic-viscoplastic media”, J. Appl. Math. Mech., 37:1 (1973), 132–141 | DOI | Zbl

[13] Egorov M. V., A program for numerical calculation of dynamic stress-strain state of the shell of revolution made of elasto-viscoplastic material when strikes the butt end: Certificate of state registration of the computer program, Registered no. 2017660609. 22.09.2017 (In Russian)