×

Surface of discontinuity in anisotropic reduced Cosserat continuum: uniqueness theorem for dynamic problems with discontinuities. (English. Russian original) Zbl 1461.74002

Mech. Solids 55, No. 7, 1051-1056 (2020); translation from Prikl. Mat. Mekh. 84, No. 1, 77-84 (2020).
Summary: An isolated surface that moves relative to the micropolar media and across which the first derivatives of variables are discontinuous is considered. The reduced Cosserat continuum is an elastic medium where all translations and rotations are independent. Moreover, the force stress tensor is asymmetric and the couple stress tensor is equal to zero. Continuity conditions were established and it is shown that the first derivative of the rotation vector cannot have discontinuities. It is demonstrated that the solution in this case is unique.

MSC:

74A35 Polar materials
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kulesh, M. A.; Matveenko, V. P.; Shardakov, I. N., Constructing an analytical solution for Lamb waves using the Cosserat continuum approach, J. Appl. Mech. Techn. Phys., 48, 119-126 (2007) · Zbl 1150.74464
[2] Varygina, M. P.; Sadovskaya, O. V.; Sadovskii, V. M., Resonant properties of moment Cosserat continuum, J. Appl. Mech. Techn. Phys., 51, 405-414 (2010) · Zbl 1272.74019
[3] Suvorov, E. M.; Tarlakovskii, D. V.; Fedotenkov, G. V., The plane problem of the impact of a rigid body on a half-space modeled by a Cosserat medium, J. Appl. Math. Mech., 76, 511-518 (2012) · Zbl 1270.74153
[4] Zdanchuk, E. V.; Kuroedov, V. V.; Lalin, V. V., Variational formulation of dynamic problems for a nonlinear Cosserat medium, J. Appl. Math. Mech., 81, 66-70 (2017) · Zbl 1440.74074
[5] Schwartz, L. M.; Johnson, D. L.; Feng, S., Vibrational modes in granular materials, Phys. Rev. Lett., 52, 831-834 (1984)
[6] E. F. Grekova and G. C. Herman, “Wave propagation in rock modeled as reduced Cosserat continuum with weak anisotropy,” in Proc. 67th Europ. Assoc. Geosci. Engin., EAGE Conf. and Exhibition, Incorporating SPE Europe 2005 (Feria de Madrid, June 13-16, 2005), pp. 2643-2646.
[7] Harris, D., Double-slip and spin: a generalization of the plastic potential model in the mechanics of granular materials, Int. J. Eng. Sci., 47, 208-1215 (2009) · Zbl 1213.74064
[8] Kulesh, M. A.; Grekova, E. F.; Shardakov, I. N., The problem of surface wave propagation in a reduced Cosserat medium, Acoust. Phys., 55, 218-227 (2009)
[9] Grekova, E. F.; Kulesh, M. A.; Herman, G. C., Waves in linear elastic media with microrotations. Part 2: Isotropic reduced Cosserat model, Bull. Seismol. Soc. Am., 99, 1423-1428 (2009)
[10] Grekova, E. F., Linear reduced Cosserat medium with spherical tensor of inertia, where rotations are not observed in experiment, Mech. Solids, 47, 538-544 (2012)
[11] Eremeev, V. A., Conditions of acceleration waves’ propagation in thermoelastic micropolar media, Vestn. Yuzhn. Nauchn. Tsentra Ross. Akad. Nauk, 3, 10-13 (2007)
[12] Altenbach, H.; Eremeyev, V. A.; Lebedev, L. P.; Rendon, L. A., Acceleration waves and ellipticity in thermoelastic micropolar media, Arch. Appl Mech., 80, 217-227 (2010) · Zbl 1271.74251
[13] Lalin, V. V.; Zdanchuk, E. V., Conditions on the surface of discontinuity for the reduced Cosserat continuum, Mater. Phys. Mech., 31, 28-31 (2017)
[14] Lalin, V. V.; Zdanchuk, E. V., The initial boundary-value problem for a mathematical model for granular medium, Appl. Mech. Mater., 725-726, 863-868 (2015)
[15] Lurie, A. I., Nonlinear Theory of Elasticity (2012) · Zbl 1247.74009
[16] Petrashen’, G. I., Propagation of Waves in Anisotropic Elastic Media (1980), Leningrad: Nauka, Leningrad
[17] Poruchikov, V. B., Methods of the Classical Theory of Elastodynamics (1993), Berlin: Springer, Berlin
[18] Casey, J., On the derivation of jump conditions in continuum mechanics, Int. J. Struct. Changes Solids, 3, 61-84 (2011)
[19] Slattery, J. C., Momentum, Energy, and Mass Transfer in Continua (1972), Tokyo: McGraw-Hill Kogakusha Ltd., Tokyo
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.