On precisely conserved quantities of coupled micropolar thermoelastic field

V. A. Kovalev ; Yu. N. Radayev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 12 (2012) no. 4, pp. 71-79

Résumé

The paper is devoted to the $4$-covariant formulation in fourdimensional space-time of dynamics of non-linear hyperbolic micropolar thermoelastic continuum. Theory ofmicropolar continuum are due to E. Cosserat and F. Cosserat and their study of 1909. The complement microdeformations and microrotations of an element are described by a non-rigid trihedron (the case of deformable micropolar directors). Hyperbolic micropolar type-II thermoelastic continuum is considered as a physical field theory with the action density taking account of wave nature of heat transport (the second sound phenomenon in solids) according to the Green type-II model. The principle of the least action for a micropolar thermoelastic field is formulated. The canonical Euler–Lagrange field equations are derived from the principle of least action. These equations include a hyperbolic heat transport equation. Currents corresponding $4$-translations and $4$-rotations of the four-dimensional space-time are obtained. The $4$-covariant representations are rewritten in threedimensional forms as usual for continuum mechanics. The currents are required in order to formulate conservation laws particularly the conservation of energy. The latter may be represented as path- or surface-independent integrals known from the continuum mechanics and often used in applied problems. Regular explicit covariant formulae for the field current are obtained provided the symmetry group of the variational action functional is known. Explicit covariant formulae for the canonical energy-momentum and angular momentum tensors are also given. Precisely conserved quantities (among them the total canonical angular momentum) for a micropolar thermoelastic field are discussed.