Geodesic
Reduction of the Cosserat-type nonlinear equations to the system of Godunov's form
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 1, pp. 55-64.

Voir la notice de l'article provenant de la source Math-Net.Ru

The complete system of equations for the dynamics of a Cosserat-type continuum with finite strains and particle rotations in Lagrangian variables is reduced to a compatible system of conservation laws in the Godunov sense. This system allows one to analyse generalized solutions with surfaces of strong discontinuity of stresses and velocities and admits integral estimates. They guarantee the uniqueness and continuity of solutions of the Cauchy problem and boundary-value problems with dissipative boundary conditions in relation to initial data.
Keywords: elasticity, Cosserat continuum, couple stress tensor, curvature tensor. 
@article{JSFU_2024_17_1_a6,
     author = {Vladimir M. Sadovskii},
     title = {Reduction of the {Cosserat-type} nonlinear equations to the system of  {Godunov's} form},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {55--64},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2024_17_1_a6/}
}
TY  - JOUR
AU  - Vladimir M. Sadovskii
TI  - Reduction of the Cosserat-type nonlinear equations to the system of  Godunov's form
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2024
SP  - 55
EP  - 64
VL  - 17
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2024_17_1_a6/
LA  - en
ID  - JSFU_2024_17_1_a6
ER  - 
%0 Journal Article
%A Vladimir M. Sadovskii
%T Reduction of the Cosserat-type nonlinear equations to the system of  Godunov's form
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2024
%P 55-64
%V 17
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2024_17_1_a6/
%G en
%F JSFU_2024_17_1_a6
Vladimir M. Sadovskii. Reduction of the Cosserat-type nonlinear equations to the system of  Godunov's form. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 1, pp. 55-64. http://geodesic.mathdoc.fr/item/JSFU_2024_17_1_a6/

[1] S.K.Godunov, Equations of Mathematical Physics, Nauka, M., 1979 (in Russian) | MR

[2] S.K.Godunov, E.I.Romenskii, Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic /~Plenum Publishers, New York – Boston – Dordrecht – London – Moscow, 2003 | DOI | MR | Zbl

[3] S.K.Godunov, I.M.Peshkov, “Symmetric hyperbolic equations in the nonlinear elasticity theory”, Comput. Math. Math. Phys., 48:6 (2008), 975–995 | DOI | MR | Zbl

[4] E.Cosserat, F.Cosserat, “Théorie des Corps Déformables”, Chwolson's Traité Physique, Librairie Scientifique A.Hermann et Fils, Paris, 1909, 953–1173

[5] E.L.Aero, E.V.Kuvshinskii, “Basic equations of the theory of elasticity of media with rotational interaction of particles”, Fizika Tverd. Tela [Phys. Solids], 2:7 (1960), 1399–1409 | MR

[6] V.A.Pal'mov, “Fundamental equations of the theory of asymmetric elasticity”, J. Appl. Math. Mech., 28:3 (1964), 496–505 | DOI | MR

[7] W.Pietraszkiewicz, V.A.Eremeyev, “On natural strain measures of the non-linear micropolar continuum”, Int. J. Solids Struct., 46:3–4 (2009), 774–787 | DOI | MR | Zbl

[8] V.M.Sadovskii, O.V.Sadovskaya, “Modeling of elastic waves in a blocky medium based on equations of the Cosserat continuum”, Wave Motion, 52 (2015), 138–150 | DOI | MR | Zbl

[9] V.M.Sadovskii, O.V.Sadovskaya, “On the acoustic approximation of thermomechanical description of a liquid crystal”, Phys. Mesomech., 16:4 (2013), 312–318 | DOI

[10] V.M.Sadovskii, O.V.Sadovskaya, I.V.Smolekho, “Modeling of the dynamics of a liquid crystal under the action of weak perturbations”, J. Appl. Mech. Tech. Phys., 62:1 (2021), 170–182 | DOI | MR

[11] V.M.Sadovskii, “On thermodynamically consistent form of nonlinear equations of the Cosserat theory”, Engng. Trans., 65:1 (2017), 201–208

[12] V.I.Kondaurov, V.E.Fortov, Fundamentals of Thermomechanics of Condensed Matter, MIPT, M., 2002 (in Russian)

[13] O.Sadovskaya, V.Sadovskii, Mathematical Modeling in Mechanics of Granular Materials, Advanced Structured Materials, 21, ed. H. Altenbach, Springer, Heidelberg–New York–Dordrecht–London, 2012 | DOI | MR | Zbl

[14] A.G.Kulikovskii, N.V.Pogorelov, A.Yu.Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Monographs and Surveys in Pure and Applied Mathematics, 118, Chapman Hall/CRC, Boca Raton–London–New York–Wasington, 2001 | DOI | MR | Zbl

[15] A.D.Resnyansky, E.I.Romensky, N.K.Bourne, “Constitutive modeling of fracture waves”, J. Appl. Phys., 93:3 (2003), 1537–1545 | DOI

[16] A.A.Belozerov, E.I.Romenski, N.A.Lebedeva, “Numerical modeling of gas – liquid compressible pipe flow based on the theory of thermodynamically compatible systems”, J. Math. Sci., 228:4 (2018), 357–371 | DOI | MR