Abstract
The paper is devoted to new applications of the ideas underlying Godunov’s method that was developed as early as in the 1950s for solving fluid dynamics problems. This paper deals with elastoplastic problems. Based on an elastic model and its modification obtained by introducing the Maxwell viscosity, a method for modeling plastic deformations is proposed.
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S. K. Godunov and I. M. Peshkov, “Symmetric Hyperbolic Equations in the Nonlinear Elasticity Theory,” Zh. Vychisl. Mat. Mat. Fiz. 48, 1034–1055 (2008) [Comput. Math. Math. Phys. 48, 975–995 (2008)].
I. M. Peshkov, “Numerical Simulation of Discontinuous Solutions in Nonlinear Elasticity Theory,” Zh. Prikl. Mekh. Tekh. Fiz. 5, 152–161 (2009) [J. Appl. Mech. Tech. Phys. 50, 858–865 (2009)].
S. K. Godunov, “A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations,” Mat. Sb. 47 (1959).
S. K. Godunov, “On the Concept of Discontinuous Solution,” Dokl. Akad. Nauk SSSR 134, 1279–1282 (1960).
S. K. Godunov, “On Nonunique Blurring of Discontinuities in Solutions of Quasi-Linear Systems”, Dokl. Akad. Nauk SSSR 136(2), 272–274 (1961) [Soviet Math. Dokl. 2, 43–44 (1961)].
S. K. Godunov, “An Interesting Class of Quasilinear Systems,” Dokl. Akad. Nauk SSSR 139(3), 520–523 (1961) [Soviet Math. 2, 947–949 (1961)].
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, (Nauka, Moscow, 1953; Pergamon Press, Oxford, 1987).
S. K. Godunov, Elements of Continuum Mechanics (Nauka, Moscow, 1978) [in Russian].
S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauchnaya Kniga, Novosibirsk, 1998; Kluwer, New York, 2003).
G. I. Gurevich, Deformability of Media and Propagation of Seismic Waves (Nauka, Moscow, 1974) [in Russian].
G. H. Miller and P. Colella, “A High-Order Eulerian Godunov Method for Elastic-Plastic Flow in Solids,” J. Comput. Phys. 167, 131–176 (2001).
V. D. Ivanov, V. I. Kondaurov, I. B. Petrov, and A. S. Kholodov, “Calculation of Dynamical Deformation and Destruction of Elastoplastic Body Using Mesh-Charachterictic Methods” Mat. Modelir. 2(11), 10–29 (1990).
P. T. Barton, D. Drikakis, and E. I. Romenski, “An Eulerian Finite-Volume Scheme for Large Elastoplastic Deformations in Solids,” Int. J. Numer. Meth. Engineering 81, 453–484 (2010).
S. K. Godunov and I. M. Peshkov, “Symmetrization of the Nonlinear System of Gas Dynamics Equations” Sib. Mat. Zh. 49, 1046–1052 (2008) [Sib. Math. J. 49, 829–834 (2008)].
T. Qin, “Symmetrizing Nonlinear Elastodynamic System,” J. Elasticity 50, 245–252 (1998).
D. H. Wagner, “Symmetric Hyperbolic Equations of Motion for a Hyperelastic Material,” J. Hyperbolic Diff. Equat. 6, 615–630 (2009).
A. G. Kulikovskii and E. I. Sveshnikova, Nonlinear Waves in Elastic Media (Moskovskii litsei, Moscow, 1998; CRC Press, Boca Raton, FL, 1995).
R. A. Horn and R. J. Johnson, Topics in Matrix Analysis (University Press, Cambridge, 1994).
O. Yu. Vorobiev, I. N. Lomov, A. V. Shutov, et al., “Application of Schemes on Moving Grids for Numerical Simulation of Hypervelocity Impact Problems,” Int. J. Impact Engineering 17, 891–902 (1995).
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Original Russian Text © S.K. Godunov, I.M. Peshkov, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 8, pp. 1481–1498.
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Godunov, S.K., Peshkov, I.M. Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium. Comput. Math. and Math. Phys. 50, 1409–1426 (2010). https://doi.org/10.1134/S0965542510080117
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DOI: https://doi.org/10.1134/S0965542510080117