Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Kovalev V. A. Covariant field equations and d-tensors of hyperbolic thermoelastic continuum with fine microstructure. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 2, pp. 60-68. DOI: 10.18500/1816-9791-2013-13-2-1-60-68, EDN: SJJAYJ

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
27.02.2013
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Russian
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539.374
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SJJAYJ

Covariant field equations and d-tensors of hyperbolic thermoelastic continuum with fine microstructure

Autors: 
Kovalev Vladimir Aleksandrovich, Moscow City Government University of Management Moscow, Russia
Abstract: 

A non-linear mathematical model of hyperbolic thermoelastic continuum with fine microstructure is proposed. The model is described in terms of 4-covariant field theoretical formalism. Fine microstructure is represented by d-tensors, playing role of extra field variables. A Lagrangian density for hyperbolic thermoelastic continuum with fine microstructure is given and the corresponding least action principle is formulated. 4-covariant field equations of hyperbolic thermoelasticity are obtained. Constitutive equations of microstructural hyperbolic thermoelasticity are discussed. Virtual microstructural inertia is added to the considered action density. It is also concerned to the thermal inertia. Variational symmetries of the thermoelastic action are used to formulate covariant conservation laws in a plane space-time. 

References: 
  1. Kovalev V. A., Radaev Yu. N. Elementy teorii polia: variatsionnye simmetrii i geometricheskie invarianty [Elements of the field theory: variational symmetries and geometric invariants]. Moscow, Fizmatlit, 2009, 156 p. (in Russian).
  2. Kovalev V. A., Radaev Yu. N. Volnovye zadachi teorii polia i termomekhanika [Wave problems of the field theory and thermomechanics]. Saratov, Saratov Univ. Press, 2010, 328 p. (in Russian).
  3. Ovsiannikov L. V. Gruppovoi analiz differentsial’nykh uravnenii [Group analysis of differential equations]. Moscow, Nauka, 1978, 400 p. (in Russian).
  4. Toupin R. A. Theories of Elasticity with Couple-stress. Arch. Ration. Mech. Anal., 1964, vol. 17, no. 5, pp. 85– 112.
  5. Cosserat E., Cosserat F. Th´eorie des corps d´eformables. Paris, Librairie Scientifique A. Hermann et Fils, 1909, 226 p. (in French).
  6. Kovalev V. A., Radaev Yu. N. Derivation of energy- momentum tensors in the theories of hyperbolic micropolar thermoelasticity. Mech. Sol. 2011, vol. 46, no. 5, pp. 705—720.
  7. Kovalev V. A., Radaev Yu. N. Teoretiko-polevye formulirovki i modeli nelineinoi giperbolicheskoi mikropoliarnoi termouprugosti [Covariant field formulations and models of non-linear hyperbolic micropolar thermoelasticity]. XXXVI Dal’nevostochnaia matematicheskaia shkola- seminar im. akad. E. V. Zolotova. Vladivostok, 2012, pp. 137–142 (in Russian).
  8. Kovalev V. A., Radaev Yu. N. On precisely conserved quantities of coupled micropolar thermoelastic field. Izv. Sarat. Univ. N. S. Ser. Math. Mech. Inform., 2012, vol. 12, iss. 4, pp. 71–79 (in Russian).
  9. Kovalev V. A., Radaev Yu. N. Kovariantnaia forma uravnenii sovmestnosti na poverkhnostiakh sil’nogo razryva v mikropoliarnom termouprugom kontinuume: giperbolicheskaia teoriia [Covariant forms of jump equations on shock surfaces in micropolar thermoelastic continuum: a hyperbolic theory]. Trudy XVI Mezhduna- rodnoi konferentsii «Sovremennye problemy mekhaniki sploshnoi sredy». vol. II. Rostov on Don, 2012, pp. 99– 103 (in Russian).
Received: 
01.09.2012
Accepted: 
17.01.2013
Published: 
27.02.2013
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