In the present paper we consider problems related to propagation of plane time-harmonic coupled waves of temperature increment, translational and spinor displacements in an ultrahemitropic micropolar thermoelastic solid and investigation their wavenumbers. The ultrahemitropic model is derived from hemitropic. A closed coupled partial differential equations for the temperature increment and displacements are discussed. Terms of the partial differential equations of coupled micropolar thermoelasticity are compared with respect to micropolar characteristic length scale. The characteristic equations for the wavenumbers of plane harmonic coupled thermoelastic longitudinal (bicubic equation) and transverse (biquadratic equation) waves are found and solved. For a longitudinal wave the complex amplitudes of the temperature increment, translational and spinor displacements are also coupled, contrary to an athermal (or cold) transverse wave. The thermal part can not be eliminated from a thermoelastic longitudinal wave, whereas the transverse wave is intrinsically athermal and is called as cold. Algebraic radical expressions for the roots of the characteristic equations are obtained and normal wavenumbers with a positive real parts are discriminated.

В настоящей работе рассматривается мультивесовая теория слабых разрывов температурного инкремента, трансляционных и спинорных перемещений в полуизотропной термоупругой микрополярной среде. Предлагаемая к рассмотрению математическая теория существенным образом опирается на достижения современного псевдотензорного исчисления. Приводятся определения мультивесовых псевдотензорных элементов площади и объема. Выводится общая мультивесовая форма псевдотензорного соотношения на волновой поверхности, распространяющейся в полуизотропной термоупругой микрополярной среде. Исследуются слабые разрывы решений связанной системы псевдовекторных дифференциальных уравнений в частных производных полуизотропной микрополярной термоупругости. С этой целью геометрические и кинематические условия Адамара–Томаса обобщены на псевдотензорный случай с учетом псевдотензорной геометрии распространяющейся поверхности слабых разрывов. Найдены скорости распространения волновых поверхностей трансляционных и спинорных перемещений. Проведена классификация слабых разрывов температуры, трансляционных и спинорных перемещений и исследованы их пространственные поляризации. Установлены условия атермичности, распространяющихся волновых поверхностей слабых разрывов. In this paper, we consider a multiweight theory of weak discontinuities of the temperature increment, translational and spinor displacements in a semi-isotropic thermoelastic micropolar medium. The mathematical theory proposed for consideration is substantially based on the achievements of modern pseudotensor calculus. Definitions of multiweight pseudotensor elements of area and volume are revisited. A general multiweight form of a pseudotensor equation on a wave surface propagating in a semi-isotropic thermoelastic micropolar medium is derived. Weak discontinuities of solutions of the coupled system of pseudovector partial differential equations of semi-isotropic micropolar thermoelasticity are studied. For this aim the geometric and kinematic Hadamard–Thomas conditions are generalized to the pseudotensor case taking account of the pseudotensor geometry of the propagating surface of weak discontinuities. The propagation velocities of wave surfaces of translational and spinor displacements are found. A classification of weak discontinuities of temperature, translational and spinor displacements is carried out and their spatial polarizations are studied. The conditions of athermality of propagating wave surfaces of weak discontinuities are established.

In continuum mechanics (especially in hydroaeromechanics), methods of modeling flow (deformation) by characteristic numbers are widely used. The present study is devoted to the search for characteristic combinations of constitutive thermoelastic modules, geometric and thermomechanical parameters of the boundary value problem. Modeling the micropolar solids deformation by characteristic numbers is characterized by a sufficiently large number (13) of constitutive modules. The constitutive equations, the dynamic equations and the heat conduction equation for a semi-isotropic micropolar thermoelastic continuum are derived in a linear approximation. A dimensional analysis of the governing system of differential equations is carried out. A physically consistent sets (9 primary and several arbitrary) of dimensionless characteristic combinations of constitutive constants is proposed. The characteristic numbers for harmonic waves propagating along the axis of a stress free thermally insulated long cylindrical semi-isotropic thermoelastic waveguide are obtained and discussed.

The paper is devoted to problems related to two-dimensional Nye figures for micropolar continua. The representation technique known from studies on crystallography for 4th, 3rd and 2nd rank tensors is employed by two-dimensional matrices, supplied by relationships between their elements. Such representations are commonly used to simplify a script of the equations of athermic micropolar elastic solids. This method allows us to graphically represent micropolar constitutive tensors and pseudotensors in the form of specific two-dimensional blocks. The Nye figures for the micropolar elastic solids are obtained. The Nye figures enables us to discriminate anisotropic solids and figure out general anisotropic, hemitropic, isotropic, and ultraisotropic continua.

The present paper deals with problems of propagation of plane harmonic coupled waves of temperature increment, translational and spinor displacements in a semi-isotropic thermoelastic solid. The requisite constitutive and differential equations of semi-isotropic solids are revisited. Dispersion equation for the wavenumbers of plane harmonic coupled thermoelastic longitudinal waves (bicubic algebraic equation) are obtained and analyzed. Dispersion equations for the transverse waves (equation of the 8th algebraic degree) are splitted into two algebraic quartic equations and then solved. For a longitudinal wave, the complex amplitudes of temperature increment, translational displacements and spin–vector are coupled, unlike a transverse wave. The roots of mentioned algebraic equations are calculated by using the Wolfram Mathematica 13 symbolic computing system. The normal wavenumbers with positive real part are discriminated.

The present paper deals with the triple weights pseudotensor formulation of multivariant thermoelasticity for acentric isotropic micropolar solids. The fundamental concepts of pseudoinvariant volume/area/arc elements of odd integer weights in three-dimensional spaces are discussed. The developed theory of acentric isotropic micropolar thermoelasticity is formulated in terms of contravariant pseudovector of spinor displacements having positive odd algebraic weight and covariant absolute vector of translational displacements. Three energetic forms (H), (E), and (A) of thermoelasticity potential are proposed. The latter is derived from the irreducible system of algebraic invarinats/pseudoinvariants being actually their linear span with coefficients thus allowing us to introduce the conventional thermoelasticity moduli (shear modulus of elasticity, Poisson's ratio, characteristic nano/microlength, etc.). For others energetic forms thermoelasticity anisotropic micropolar (E) moduli are determined via (A) moduli and then (H) moduli are found in terms of (A) moduli. The triple weights formulation of multivariant constitutive equations for acentric isotropic thermoelastic solid are obtained and analyzed. A comparison of proposed multivariant constitutive equations elucidates the absolute invariance of Poisson's ratio, i.e., it insensibility to mirror reflections and prohibition of assigning any algebraic weight to this constitutive scalar.

In present paper the propagation of plane harmonic coupled waves of temperature increment, translational and spinor displacements in a semiisotropic thermo elastic solid is discussed. Characteristic equations for the wave numbers of plane harmonic coupled thermo elastic longitudinal (bicubic equation) and transverse waves (biquartic equation that naturally splits into two quartic algebraic equations) are obtained and analyzed. For a longitudinal wave, the complex amplitudes of the temperature increment, translational and spinor displacements are also coupled, contrary to a transverse wave. Algebraic forms containing multivalued complex square and cubic radicals for the wave numbers of transverse waves are derived by using the Wolfram Mathematica 13 symbolic computing system.

The paper deals with a method of the Nye figures construction for micropolar elastic solids. The method of tensors of the 4th and 3rd ranks representations by means of blocks of two-dimensional matrices and relationships between their elements is widely known in crystallography. Such approach makes it possible to simply determine the number of independent constitutive constants for micropolar elastic solids and guarantee the absence of relationships between them. In frameworks of the present study, the two-dimensional Nye figures for an ultraisotropic micropolar elastic solid were figured out based on the corresponding figures for hemitropic and isotropic micropolar elastic solids. It is shown that the constitutive tensors of ultraisotropic material characterized by only 4 independent constitutive constants: shear modulus of elasticity, Poisson’s ratio, characteristic nano/microlength and another dimensionless constant.

In the present paper triple formulation of the thermomechanics of hemitropic micropolar solids are proposed and then reduced to the three positive, negative and zero weights variants. The fundamental concepts of pseudoinvariant volume and area elements of odd integer weights in three-dimensional space are discussed. The developed theory of hemitropic micropolar thermoelasticity is formulated in terms of a contravariant pseudovector of a positive odd weight representing spinor displacements, subject to the principle of absolute invariance of absolute thermodynamic temperature, mass and specific entropy, specific internal energy, specific Helmholtz free energy, specific controllable and uncontrollable entropy production. triple weights pseudotensor formulations of the principle of virtual displacements and the reduced energy balance equation are discussed. Corresponding differential equations of statics and dynamics of a hemitropic thermoelastic solid are obtained and analyzed. The sensibility of shear modulus of elasticity and characteristic microlength to re-orientations of three-dimensional space are revisited.

Работа посвящена проблемам, связанным с распространением плоских гармонических связанных волн температурного инкремента, трансляционных и спинорных перемещений в ультрагемитропном микрополярном термоупругом теле. Приводится замкнутая система дифференциальных уравнений в частных производных второго порядка относительно температурного инкремента и перемещений (трансляционных и спинорных). Найдены и проанализированы характеристические уравнения для волновых чисел плоских гармонических связанных термоупругих волн. Получены алгебраические выражения для корней характеристических уравнений и отделены нормальные волновые числа с положительной действительной частью. Для продольной волны комплексные амплитуды температурного инкремента, трансляционных и спинорных перемещений оказываются связанными, в отличие от атермической по своей природе поперечной волны. The study is devoted to problems associated with propagation of plane harmonic coupled waves of temperature increment, translational and spinor displacements in an ultrahemitropic micropolar thermoelastic solid. A closed system of differential equations in partial derivatives of the second order with respect to the temperature increment and displacements (translational and spinor) is revisited. Characteristic equations for wavenumbers of plane harmonic coupled thermoelastic waves are found and analyzed. Algebraic expressions for the roots of the characteristic equations are obtained and normal wavenumbers with a positive real part are separated. For a longitudinal wave, complex amplitudes of the temperature increment, translational and spinor displacements are coupled, in contrast to a transverse wave, which is athermal in nature.

Статья посвящена вопросам детерминирования алгебраических весов микрои наномасштабных мультивесовых характеристик термомеханики гемитропных микрополярных тел. Обсуждаются фундаментальные понятия псевдоинвариантых элементов объема и площади нечетных целых весов в трехмерном пространстве. Развиваемая теория гемитропной микрополярной термоупругости формулируется в терминах контравариантного псевдовектора спинорных перемещений положительного нечетного веса при условии постулирования абсолютной инвариантности абсолютной термодинамической температуры, массы и массовых плотностей: энтропии, внутренней энергии, свободной энергии Гельмгольца, контролируемого и неконтролируемого производства энтропии. Предложены мультивесовые псевдотензорные формулировки принципа виртуальных перемещений и приведенного уравнения баланса энергии. Получены и проанализированы мультивесовые формулировки псевдовекторных дифференциальных уравнений статики и динамики гемитропного термоупругого тела. Обсуждаются вопросы взаимовлияния алгебраических весов определяющих псевдоскаляров с целью учета их трансформации в результате преобразования трехмерного пространства, меняющего ориентацию координатного базиса на противоположную. The article is devoted to the problems of determining the algebraic weights of microand nanoscale multi-weight characteristics of the hemitropic micropolar thermomechanics. The fundamental concepts of pseudoinvariant volume and area elements of odd integer weights in threedimensional space are discussed. The developing theory of hemitropic micropolar thermoelasticity is formulated in terms of a contravariant pseudovector of spinor displacements of a positive odd weight with the fundamental principle of the absolute invariance of absolute thermodynamic temperature, mass and mass densities of: entropy, internal energy, Helmholtz free energy, controlled and uncontrolled entropy production. Multi-weight pseudotensor formulations of wireless transmission principles and a reduced energy balance equation are proposed. Multiweights formulas for pseudovector differential equations of statics and dynamics of a hemitropic thermoelastic body are obtained and analyzed. The problem of mutual influence of algebraic weights of constitutive pseudoscalars are discussed in order to take into account their transformations as a result of the transformation of three-dimensional space, changing the orientation of the coordinate basis to the opposite.

The present paper deals with the problem of deriving the constitutive equations for the micropolar thermoelastic continuum GN-I in terms of the standard pseudotensor formalism. In most cases, the pseudotensor approach is justified in modeling hemitropic micropolar solids, the thermomechanical properties of which are sensitive to mirror reflections of three-dimensional space. The requisite equations and notions from the theory of pseudotensors are revisited. General thermodynamic approaches are used, entropy and energy balance equations are discussed. The weights of the main thermomechanical pseudotensors are established. In a linear approximation, the constitutive equations of the hemitropic micropolar thermoelastic continuum (GN-I) of the first type are derived. A coupled system of differential equations of heat conduction and dynamic equations of the micropolar thermoelastic continuum GN-I is obtained.

The paper deals with the force stress pseudotensor as proposed by Schouten and a derivation of equilibrium equations in terms of the Schouten stress pseudotensor of positive weight. The definition of Schouten's force stress pseudotensor is based on the notion of a pseudoinvariant area element. Conventional and unconventional definitions of the force stress tensor are recalled and discussed. The tensor area elements of a M-manifold immersed in N-dimensional parent (outer) plane space are revisited. The notions of vector, pseudovector, invariant and pseudo-invariant elementary areas in three-dimensional plane space are thoroughly discriminated. The usability of pseudotensor elementary volume of any given integer weight is discussed. Three realizations of the covariant differentiation of pseudotensor fields are considered and compared. Equations of equilibrium and dynamics are derived in terms of the Schouten force stress pseudotensor from the virtual displacements principle.

This article considers a variant of the heat conduction theory of thermal conductivity, in which the heat flux pseudovector has a weight of 1. The pseudoinvariants associated to the heat flux pseudovector are sensitive to mirror reflections and inversions of threedimensional space. The primary purpose of the study was to find a heat flux vector that is algebraically equivalent to the microrotation pseudovector and to measure elementary volumes and areas using pseudoinvariants that are sensitive to mirror reflections. To represent spinor displacements, a contravariant microrotation pseudovector with a weight of +1 was selected. Thus, the heat flux and mass density were expressed as odd-weight pseudotensors. The Helmholtz free energy per unit doublet pseudoinvariant volume was employed as the thermodynamic state potential of the following functional arguments: absolute temperature, symmetric parts, and accompanying vectors for the linear asymmetric strain tensor and the wryness pseudotensor. The results obtained show that the thermal conductivity coefficient and heat capacity of elastic micropolar solids are pseudoscalars of odd weight, indicating their sensitivity to mirror reflections.

Статья посвящена исследованию поливариантности динамических уравнений теории полуизотропной микрополярной термоупругости. Рассмотрены и проанализированы различные варианты присвоения целых весов полевым переменным с последующим детерминированием алгебраических весов псевдовекторных уравнений динамики полуизотропного термоупругого тела. Этих целей удается достичь, используя псевдоинвариантые элементы объема и площади нечетных целых весов. Кроме того, показано, что нечетный вес может быть приписан псевдовектору спинорных перемещений. В результате чего, тепловой поток, тензор силовых напряжений, массовая плотность, теплоемкость, модуль сдвига также оказываются псевдотензорными величинами нечетного веса, т.е. чувствительны к зеркальным отражениям и инверсиям трехмерного пространства. Обсуждается постулат абсолютной инвариантности абсолютной термодинамической температуры. Получены различные варианты связанной системы дифференциальных уравнений динамики и уравнения теплопроводности для полуизотропного микрополярного термоупругого тела. Обсуждаются вопросы взаимовлияния алгебраических весов определяющих псевдоскаляров с целью учета их реакции на преобразования трехмерного пространства, меняющих его ориентацию на противоположную. The paper is devoted to the study of dynamic equations polyvariance of the theory of semiisotropic micropolar thermoelasticity. Several variants for assigning integer weights to field variables with subsequent determination of algebraic weights of pseudo-vector equations for the dynamics of a semiisotropic thermoelastic solid are considered and analyzed. For this aim elementary volumes and areas assumed as pseudoinvariants of odd integer weights. In addition, it is shown that odd weights can be assigned to the pseudovector of spinor displacements. As a result, heat flux, force stress tensor, mass density, heat capacity, and shear modulus also can be treated as pseudotensor quantities of odd weights, i.e. manifest itself sensitivity to mirror reflections and inversions of three-dimensional spaces. The fundamental principle of absolute invariance of absolute thermodynamic temperature is discussed. Some variants of the coupled system of differential equations of dynamics and heat conduction equations for a semiisotropic micropolar thermoelastic solid are obtained. The problems of mutual influence of algebraic weights of constitutive pseudoscalars are discussed in order to taking account of their response to transformations of three-dimensional space that change its orientation to the opposite.

The paper is devoted to the theory of an anisotropic micropolar thermoelastic solid. The requisite equations and notions from pseudotensors algebra and multidimensional geometry are revisited. From the beginning we treat translational displacements as an absolute covariant fields whereas spinor displacements as a contravariant pseudovector. The Helmholtz free energy is employed as a thermodynamic state potential of the following functional arguments: absolute temperature, symmetric parts and accompanying vectors of the linear asymmetric strain tensor and the wryness pseudotensor. The constitutive equations for a general anisotropic micropolar thermoelastic solid including gyrotropic one are derived. That means heat flux vector can be treated as a pseudovector of weight + 1 (or - 1 ) algebraically consistent to spinor displacements pseudovector. Nonlinear heat conduction equation and its linearized form are obtained.

The paper is devoted to some problems concerning modeling semi-isotropic elastic media. Several quadratic energy forms of a thermodynamic state potential are introduced in terms of pseudotensors. These energy forms are assumed to be absolute invariants with respect to arbitrary transformations of the three-dimensional Euclidean space (including mirror reflections). As a result of applying special coordinate representations of semi-isotropic (semi-isotropic) pseudotensors of the fourth rank, it is possible to determine 9 covariantly constant constitutive pseudoscalars characterizing a semi-isotropic elastic medium. The Neuber’s, conventional, first and second base natural energy forms are compared and equations are derived for constitutive scalars and pseudoscalars, including the conventional semi-isotropic pseudoscalars: shear modulus, Poisson’s ratio, characteristic microlength (a pseudoscalar of negative weight, sensitive to reflections of three-dimensional space), and six dimensionless pseudoscalars. В работе обсуждаются некоторые вопросы моделирования полуизотропных упругих сред. Вводятся квадратичные энергетические формы термодинамического потенциала состояния. Исследуемые энергетические формы полагаются абсолютными инвариантами по отношению к произвольным преобразованиям трехмерного Евклидова пространства (в том числе, при зеркальных отражениях). В результате применения специальных координатных представлений полуизотропных псевдотензоров четвертого ранга можно определить все 9 ковариантно постоянных определяющих псевдоскаляров, характеризующих полуизотропную упругую среду. Выполнено сравнение и получены соотношения, связывающие определяющие скаляры и псевдоскаляры нейберовской, конвенциональной, первой и второй основных естественных энергетических форм, в том числе, с конвенционально используемыми полуизотропными псевдоскалярами: модулем сдвига, коэффициентом Пуассона, характерной микродлиной (являющейся псевдоскаляром отрицательного веса, чувствительным к отражениям трехмерного пространства), и шестью безразмерными псевдоскалярами.

Настоящая статья посвящена вопросам распространения плоских термоупругих гармонических волн в ацентрическом изотропном микрополярном теле. С этой целью сначала рассматриваются динамические уравнения ацентрического изотропного тела. Определяются пространственные поляризации плоских волн трансляционных и спинорных перемещений. Обсуждается качественный характер возможных волновых решений уравнений связанной микрополярной термоупругости. Отдельно рассматривается случай атермической волны. The present paper is devoted to the propagation of plane thermoelastic harmonic waves in an acentric isotropic micropolar solid. For this aim, the dynamic equations of an acentric isotropic solid are considered. The spatial polarizations of plane waves of translational and spinor displacements are determined. The qualitative nature of possible wave solutions to the equations of coupled micropolar thermoelasticity is discussed. The case of an athermic wave is considered separately.

Originating micropolar microrotation vectors and pseudovectors of presumably integer weights is discussed. Overwhelming majority of publications on micropolar thermoelasticity deals with absolute tensors formulations, minority is devoted to applications of positive weight microrotation contravariant pseudovector to formulations of micropolar thermoelasticity. A negative weight microrotation pseudovector is not actually employed for derivation of hemitropic thermoelastic equations as it may be concluded from a literary search. The present paper is devoted to this issue. Thermodynamic state potentials for hemitropic thermoelastic continuum are proposed in terms of asymmetric strain tensor, wryness pseudotensor of negative weight and the entropy. In virtue of contravariant algebraic treatment of translations and microrotations general algebraically consistent coordinate representations for constitutive fourth rank tensors and pseudotensors are considered in terms of the metric tensor and the constitutive hemitropic pseudoinvariants, then replaced by the conventional constitutive pseudoscalars. Energy and entropy balance equations are obtained to provide thermodynamic consistency of the proposed model. Coupled hemitropic thermoelastic equations are represented in terms of contravariant displacement vector and negative weight covariant microratation pseudovector.

Статья посвящена вопросам моделирования процессов теплопроводности в микрополярных телах, термомеханические состояния которых реагируют на зеркальные отражения трехмерного пространства. Построен новый вариант теории теплопроводности, в рамках которого тепловой поток оказывается псевдовектором алгебраического веса \(+1\), подобным псевдовектору спинорных перемещений. С этим вариантом теории связаны определяющие псевдоинварианты нечетного отрицательного веса (например, коэффициент теплопроводности и теплоемкость). Этой цели удалось достичь, выбрав естественные элементы объема и площади в виде псевдоинвариантов веса \(-1\). Для представления трансляционных перемещений использовался абсолютный контравариантный вектор, а для спинорных перемещений фиксировался контравариантный псевдовектор веса \(+1\). В результате тепловой поток, тензор силовых напряжений, плотность массы и теплоемкость оказываются псевдотензорными величинами нечетного веса. В качестве термодинамического потенциала используется свободная энергия Гельмгольца, отнесенная к единице естественного элемента объема, а функциональными аргументами выступают: температура, симметричные части и сопутствующие векторы линейного асимметричного тензора деформаций и псевдотензора изгиба-кручения. Обсуждается принцип абсолютной инвариантности абсолютной термодинамической температуры. Получено нелинейное уравнение теплопроводности и выполнена его линеаризация.