# action

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

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.

## Rotational Invariance of Non-Linear Lagrangians of Type-II Micropolar Thermoelastic Continuum

The paper contains new results related to extension of the field theoretical approach and its formalism to non-linear coupled micropolar thermoelastic media. A mathematical model of micropolar (MP) type-II (GNII) thermoelastic (TE) continuum is considered. A formulation of the least thermoelastic action principle is discussed. Partial differential equations subsequent to the least action principle are derived. The translational symmetries of non-linear Lagrangians are adopted. Those include an additional symmetry: translations of the thermal displacement.

## On precisely conserved quantities of coupled micropolar thermoelastic field

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).

## On a form of the first variation of the action integral over a varied domain

Field theories of the continuum mechanics and physics based on the least action principle are considered in a unified framework. Variation of the action integral in the least action principle corresponds variations of physical fields while space-time coordinates are not varied. However notion of the action invariance, theory of variational symmetries of action and conservation laws require a wider variation procedure including variations of the space-time coordinates.

## Micropolar Thermoelastic Continuum Models with Constrained Microstructural Parameters

A new micropolar thermoelastic continuum model forrmulated by microstructural d-vectors and d-tensors of an arbitrary ranks is proposed. The microstructural vectorial and tensorial extra-field variables are restricted by holonomic or non-holonomic (differential) constraints. The study is carried out in the framework of the Lagrange field formalism as a 4covariant field theory.

## On Rationally Complete Algebraic Systems of Finite Strain Tensors of Complex Continua

The paper is devoted to the mathematical description of complex continua and the systematic derivation of strain tensors by the notion of isometric immersion of complex continuum in a plane space of higher dimension. Problem of establishing of complete systems of irreducible objective strain and extra-strain tensors for complex continuum immersed in an external plane space is considered. The solution to the problem is given by methods of the field theory and the theory of algebraic invariants.