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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, Issue 1(34), Pages 66–85 (Mi vsgtu1310)  

This article is cited in 7 scientific papers (total in 7 papers)

Mechanics of Solids

On Nonlinear Strain Vectors and Tensors in Continuum Theories of Mechanics

V. A. Kovaleva, Yu. N. Radayevb

a Moscow City Government University of Management Moscow, Moscow, 107045, Russian Federation
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526, Russian Federation

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. For micropolar type-II thermoelastic Lagrangians following the usual procedure independent rotationally invariant functional arguments are obtained. Objective forms of the Lagrangians satisfying the frame indifference principle are given. Those are derived by using extra strain vectors and tensors.

Keywords: thermoelasticity, microstructure, field, extra field, action, covariance, conservation law, $d$-tensor, $4$-current, energy–momentum tensor, kinematic constraint, Lagrange multiplier, rotation, frame indifference principle, extrastrain tensor

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00139
This work is partially supported by RFBR, project no. 13–01–00139 “Hyperbolic Thermal Waves in Solid Bodies with Microstructure”.


DOI: https://doi.org/10.14498/vsgtu1310

Full text: PDF file (683 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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Bibliographic databases:

UDC: 539.3
MSC: 74A60, 74F05
Original article submitted 19/I/2014
revision submitted – 21/II/2014

Citation: V. A. Kovalev, Yu. N. Radayev, “On Nonlinear Strain Vectors and Tensors in Continuum Theories of Mechanics”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1(34) (2014), 66–85

Citation in format AMSBIB
\Bibitem{KovRad14}
\by V.~A.~Kovalev, Yu.~N.~Radayev
\paper On Nonlinear Strain Vectors and Tensors in Continuum Theories of Mechanics
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2014
\vol 1(34)
\pages 66--85
\mathnet{http://mi.mathnet.ru/vsgtu1310}
\crossref{https://doi.org/10.14498/vsgtu1310}
\zmath{https://zbmath.org/?q=an:06968826}
\elib{http://elibrary.ru/item.asp?id=22813961}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. A. Kovalev, Yu. N. Radaev, “Ob'ektivnye rotatsionno-invariantnye formy termouprugikh lagranzhianov”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:2 (2015), 325–340  mathnet  crossref  zmath  elib
    2. V. A. Kovalev, E. V. Murashkin, Yu. N. Radaev, “Ekstradeformatsii mikrostrukturnogo kontinuuma s odnim svobodnym direktorom”, Vestnik Chuvashskogo gosudarstvennogo pedagogicheskogo universiteta im. I. Ya. Yakovleva. Seriya: Mekhanika predelnogo sostoyaniya, 2015, no. 3 (25), 61–65  elib
    3. Yu. N. Radaev, V. A. Kovalev, “Giperbolicheskaya termomekhanika kontinuuma”, KhI Vserossiiskii s'ezd po fundamentalnym problemam teoreticheskoi i prikladnoi mekhaniki, Sbornik dokladov, Kazan, 2015, 3177–3178  elib
    4. V. A. Kovalev, E. V. Murashkin, Yu. N. Radaev, “Konechnye deformatsii kontinuuma s odnim svobodnym mikrostrukturnym direktorom”, Mekhanika predelnogo sostoyaniya i smezhnye voprosy, Materialy Vserossiiskoi nauchnoi shkoly-konferentsii, posvyaschennoi 85-letiyu professora D. D. Ivleva, 2015, 26–30  elib
    5. V. A. Kovalev, Yu. N. Radaev, “Neprivodimye sistemy ob'ektivnykh tenzorov deformatsii mikropolyarnogo kontinuuma”, Mekhanika deformiruemogo tverdogo tela, sbornik trudov IX Vserossiiskoi konferentsii, 2016, 55–59  elib
    6. “Mikropolyarnye kontinuumy s dopolnitelnymi mikrostrukturnymi svyazyami by V. A. Kovalev, Yu. N. Radaev”, Mekhanika deformiruemogo tverdogo tela, sbornik trudov IX Vserossiiskoi konferentsii, 2016, 187–192  elib
    7. E. V. Murashkin, Yu. N. Radaev, “Kontinuumy s inertsiei mikrostruktury”, Trudy desyatoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem: v 3-kh tomakh, Matematicheskoe modelirovanie i kraevye zadachi, 2016, 152–155  elib
  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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