Thermal Effects by Means of Two-Component Cosserat Continuum

Synonyms

Thermal conduction and thermal radiation by means of generalized continuum; Thermal phenomena by means of continuum with microstructure; Thermal processes by means of micropolar continuum; Thermodynamic processes by means of continuum with rotational degrees of freedom

Definition

A model of thermal effects in a conventional material is based on an idea to introduce two-component Cosserat continuum and to interpret characteristics of motion and interactions associated with the rotational degrees of freedom as mechanical analogies of thermodynamic quantities. The material under consideration has no microstructure, inclusions, etc. Motion associated with the rotational degrees of freedom has no relation to the real motion of the material particles. This model is not similar to the models used in kinetic theories, and it is not based on statistical methods. This model is constructed within the framework of continuum mechanics and by using the methods of continuum mechanics.

Introduction

A mechanical model, which can be a basis for description of thermal processes within the framework of continuum mechanics and by using the methods of continuum mechanics, is developed since 2010. It is discussed in a number of publications (see Ivanova 2010, 2011, 2012, 2013, 2017). This model is based on an idea to introduce a continuum with an internal structure and additional degrees of freedom and to interpret characteristics of motion and interactions associated with the internal structure as mechanical analogies of temperature and other thermodynamic quantities. It is important to note that a physical object under consideration is a conventional material without microstructure, inclusions, etc. This material has elastic and thermodynamic properties. The internal structure and internal rotational degrees of freedom inherent in the model are used for simulating thermal processes in the material. Motion associated with the internal degrees of freedom has no relation to the real motion of the material particles. Characteristics of motion associated with the internal rotational degrees of freedom, as well as characteristics of interactions associated with the internal rotational degrees of freedom, should be considered as analogies of thermodynamic quantities.

There are the kinetic theories that include rotational degrees of freedom (see, e.g., Warner and Harold 1972; Giesekus 1985; Bird et al. 1987) as well as the kinetic theories that take into account internal degrees of freedom (see, e.g., Jehring 1984). The model of thermal processes under discussion is not similar to these models. It is based on quite different ideas and approaches without using statistical methods and concepts of kinetic theories.

The idea of mathematical description of various physical phenomena in microcosm by using models based on a continuum with rotational degrees of freedom was repeatedly asserted by Pavel Andreevich Zhilin (1942–2005), and this idea was realized by him as applied to the description of electromagnetic and quantum mechanical phenomena (see Zhilin 2001, 2003, 2006a,b, 2012, 2015). The model considered in the present paper is a realization of Zhilin’s ideas as applied to the description of thermal phenomena.

Different Views on the Nature of Heat

Starting from antiquity, there exist different viewpoints on the nature of heat (see Rosenberger 1887; Whittaker 1910; Gliozzi 1965; Müller 2007). According to one point of view, heat is a state of a body. For example, Roger Bacon (1214–1292) and Johannes Kepler (1571–1630) adhered to this opinion. In accordance with another point of view, heat is a substance. Galileo Galilei (1564–1642) formulated the hypothesis of existence of the imponderable fluid accounting for heat. Afterward, this imponderable fluid was called the caloric fluid. Antoine Laurent de Lavoisier (1743–1794), Pierre Simon de Laplace (1749–1827), and Jean Baptiste Joseph Fourier (1768–1830) were adherents of the caloric fluid theory. The success and popularity of the caloric fluid in the seventeenth to eighteenth centuries was caused by the fact that predictions of the theory were verified by the experiments carried out at that time. The caloric fluid theory was recognized to be erroneous only in the nineteenth century when, due to the works by Julius Robert Mayer (1814–1878), James Prescott Joule (1818–1889), Hermann Helmholtz (1821–1894), and William Thomson, Lord Kelvin (1824–1907), the principle of equivalence of heat and energy became firmly established and the heat conservation law, which had dominated earlier, was completely replaced by the energy balance equation (the first law of thermodynamics). Robert Boyle (1627–1691) assumed heat to be associated with the molecular motion. In fact, his assumption was the start of the kinetic theory, which was further developed by Rudolf Clausius (1822–1888), James Clerk Maxwell (1831–1879), Ludwig Boltzmann (1844–1906), and Josiah Willard Gibbs (1839–1903). Besides the caloric fluid theory and the kinetic theory of gas, a number of different mechanical models of thermal processes were suggested by outstanding scientists of past centuries, namely, Leonhard Euler (1707–1783), Mikhail V. Lomonosov (1711–1765), Benjamin Thompson (1753–1814), Humphry Davy (1778–1829), Thomas Young (1773–1829), and Augustin Louis Cauchy (1789–1857). Nevertheless, the interpretation of temperature as the average kinetic energy of the chaotic motion of atoms and molecules has been generally accepted up to now. The question whether this interpretation is a reflection of some physical reality can be clarified by Maxwell’s remark toward the kinetic theory (see Maxwell 1860, p. 378):

…If the properties of such a system of bodies are found to correspond to those of gases, an important physical analogy will be established, which may lead to more accurate knowledge of the properties of matter. If experiments on gases are inconsistent with the hypothesis of these propositions, then our theory, though consistent with itself, is proved to be incapable of explaining the phenomena of gases. In either case it is necessary to follow out the consequences of the hypothesis.

It is important to note that temperature cannot be measured directly. In order to measure temperature, we have to measure a physical quantity, a change of which is a sign of the change in temperature. Then, we have to calculate the value of temperature taking into account the fixed points of temperature scale and using a formula that relates the change in chosen physical quantity and the change in temperature. Thus, there is no reason to believe that measuring temperature we measure the average kinetic energy of the chaotic motion of atoms and molecules. Thus, the interpretation of temperature adopted in the kinetic theory is rather a mathematical model than a physical reality. That is why any alternative model of thermal processes, the mathematical description of which is reduced to the known equations, is of interest from a theoretical point of view.

The Physical Meaning of the Model

According to the concepts of modern physics, atoms have a very complex internal structure. For example, they can be in different energy states and possess the ability to radiate and absorb the energy quanta and elementary particles. These facts should be taken into account when the properties of a single atom or a molecule consisting of several atoms are studied. When modeling a medium which consists of millions of atoms, many properties of atoms can be ignored or taken into account integrally, and it should be done so. For example, when modeling crystal lattices, very simple models of atoms are used, namely, atoms are assumed to be mass points or infinitesimal rigid bodies. For modeling a medium with a combination of various physical properties (mechanical, thermal, electric, and magnetic), more complicated models of atoms should be used. These are the models considering atoms as complex particles with internal structure and internal degrees of freedom. There are two different types of particles with an internal structure: particles with internal translational degrees of freedom (deformable particles) and particles with internal rotational degrees of freedom (multi-spin particles). Continua consisting of particles of the first type are called micromorphic continua. Continua consisting of particles of the second type are called micropolar continua. In principle, both deformable particles and multi-spin ones can be used to model atoms, and, consequently, both the micromorphic and micropolar continua can be used to model a medium with some nonmechanical properties.

In the model under consideration, atoms are assumed to be multi-spin particles like a quasi-rigid body (see Fig. 1a). The quasi-rigid body is a rigid body in the sense that the distances between any two points of this particle are kept unchanged under arbitrary motions of the quasi-rigid body. However, unlike the standard rigid body, the quasi-rigid body contains several rotors inside. Each rotor can rotate independently, and the rotation of rotors does not change the inertia tensor of the quasi-rigid body. In fact, the quasi-rigid body is a multi-rotor gyrostat that consists of a carrier body and a number of rotors rotating independently relative to the carrier body. For the first time, the multi-spin particles were introduced in Zhilin (2001, 2003, 2006a,b, 2012, 2015). The idea to consider the multi-spin particles (the multi-rotor gyrostats) as models of atoms was also stated in the cited works.

Fig. 1

A quasi-rigid body and its approximate model (the one-rotor gyrostat), which are equivalent in a first approximation

When modeling atoms by the multi-rotor gyrostats, the motion of carrier bodies characterizes the motion of atoms as rigid bodies. It is the motion of atoms that causes mechanical strains and mechanical stresses in the material medium. The rotors simulate elementary particles constituting the atoms. Pursuant to this model, the motion of rotors simulates the change in the internal state of atoms, and the internal state of atoms determines all the physical processes occurring in the material medium, namely, electrical, magnetic, and thermal. The multi-rotor gyrostat is rather complicated model with a large number of parameters. Therefore, for modeling a heat-conducting elastic medium, a simpler model of atom, namely, a one-rotor gyrostat (see Fig. 1b) is used instead of the multi-rotor gyrostat. It is important to note that the one-rotor gyrostat retains key features of the multi-rotor gyrostat since the expressions for the kinetic energy, the linear momentum, and the angular momentum of the one-rotor gyrostat coincide with those of the multi-rotor gyrostat in a first approximation. In a continuum theory, the physical characteristics averaged over a representative volume are used. The dynamic properties of a representative volume of the continuous medium have no qualitative difference from the dynamic properties of particles in the representative volume.

The Basic Ideas of the Approach

The one-rotor gyrostat continuum is considered. The one-rotor gyrostat is a particle that consists of the carrier body and the rotor (see Fig. 2). The rotor can rotate independently of the carrier body rotation, but it cannot execute translatory motion relative to the carrier body. Thus, the one-rotor gyrostat has nine degrees of freedom, three translational ones, and six rotational ones. Free space between the gyrostats is filled up by the ether. The ether is shown in Fig. 2 as the body-points in the space between the gyrostats.

Fig. 2

An elementary volume of the continuum of one-rotor gyrostats together with the continuum of body-points in the space between the gyrostats (on the left-hand side) and the one-rotor gyrostat (on the right-hand side)

The main ideas of the description of thermoelastic processes by means of the mechanical model with internal rotational degrees of freedom consist in the following:

• The one-rotor gyrostat continuum is used for modeling solids, liquids, and gases. This continuum is considered to be elastic. The interaction of carrier bodies of gyrostats is attributed to the mechanical processes. The interaction of rotors of gyrostats models thermal processes. The interaction of the carrier bodies and the rotors provides the interplay of mechanical and thermal processes.

• The gyrostats (which model material particles) are considered to be embedded into some medium occupying the whole infinite space. This medium represents the physical vacuum, the field, the ether, or something like that.

• It is assumed that all gyrostats interact with the ether by means of elastic moments associated with the rotational degrees of freedom. Due to the fact that the ether fills the whole infinite space and interacts with all gyrostats, it plays a double role in the model.

• On the one hand, it is assumed that all interactions of gyrostats with each other are performed by the instrumentality of the ether. To be exact, the carrier bodies of different gyrostats interact through the agency of the ether, and the rotors belonging to different gyrostats interact also via the ether. From a mathematical point of view, this means that the constitutive equations for all quantities characterizing the stress state of the one-rotor gyrostat continuum depend not only on the properties of the carrier bodies and the rotors of the gyrostats but also on the elastic properties and the stress–strain state of the ether filling the space between the gyrostats.

• On the other hand, it is assumed that the ether provides a dissipation of gyrostats energy. Since the gyrostats interact with the ether, their motion causes appearance of waves in the ether. As a result, certain part of the gyrostats energy is spent on formation of the waves. Since the ether is considered to be infinite, waves carrying away the gyrostats energy do not come back. The result is the dissipation of the gyrostats energy into the ether.

• The dissipation of the gyrostats energy into the ether becomes apparent in the material medium in the form of the heat conduction and the internal damping. The heat conduction mechanism is supposed to be provided only by the moment interactions between the rotors and the ether. The internal damping mechanism can be provided in different ways, both due to the kinematic connection between the rotors and the carrier bodies and thanks to the interaction of the carrier bodies with the ether.

A Model of the Ether

Initially, a two-component medium that consists of the one-rotor gyrostat continuum simulating the conventional substance and the body-point continuum simulating the ether is considered. This two-component medium is assumed to be conservative. The following assumptions are made with respect to the ether:

• The ether particles are much smaller than elementary particles of the conventional substance. The structure of the ether particles coincides with the structure of the rotors that belong to the gyrostats.

• The ether is assumed to be a medium that is less dense than the conventional substance. The ether particles fill the space between elementary particles of the conventional substance, and the elementary particles interact with each other via the ether particles.

• The interactions of ether particles with each other and the interactions of ether particles with the elementary particles of the conventional substance are based only on the rotational degrees of freedom and the principle of moment interactions. There are no interactions between these particles by means of forces. Thus, from a continuum mechanics point of view, the model of ether is the special case of the Cosserat continuum.

• The ether is an infinite medium, i.e., it occupies the whole space. The ether is assumed to be an elastic medium. However, due to its infinite extent, the ether carries away the energy of rotational motion of material particles located in it. When the particles interact with the ether, their motion disturbs the ether and causes appearance of waves in it. Since the ether is infinite, the waves cannot be reflected from the boundaries, and hence, they cannot come back. Thus, the part of the material particles’ energy, which is spent on formation of the waves in the ether, is irretrievably lost.

The Interaction of the Ether with Material Particles

Interacting with the material particles via the rotational degrees of freedom, the boundless ether creates a moment of viscous damping acting on the material particles. A structure of the moment of viscous damping is chosen in accordance with the results obtained by solving two model problems (see Ivanova 2011, 2012). The model considered in Ivanova (2011) consists of the semi-infinite inertial elastic rod (a one-dimensional model of the ether) that is connected with the rotor by means of the inertialess spring working in torsion (rotation about the axis of the rod) see Fig. 3, on the left-hand side). The rotation of the rotor disturbs the elastic rod and causes the torsion waves in it. If the rod had a limited size, the waves would be reflected from the boundary and come back. In this case, the system would be conservative. The dissipation of the rotor energy occurs only due to the infinite length of the rod and the absence of sources at infinity. As shown in Ivanova (2011), after eliminating the variables that characterize the rod motion, the problem is reduced to the set of equations describing the rotor motion. In the set of equations, there is the equation that contains the moment of viscous damping characterizing the energy radiation in the ambient medium. It is proved that the moment of viscous damping is proportional to the angular momentum of the rotor, and the coefficient of damping depends on the parameters of the rod and the torsional stiffness of the spring connecting the rotor and the rod. The spherically symmetric three-dimensional model is considered in Ivanova (2012). It consists of the spherical source (the spherical surface each point of which is the rotor) and the infinite inertial elastic continuum modeling the ether (see Fig. 3, on the right-hand side). All rotors of the spherical source are connected with the continuum by means of the inertialess springs working in torsion (rotation about a radius of the spherical source). As shown in Ivanova (2012), after eliminating the variables characterizing the ether motion, the problem is reduced to the set of equations that contain the dissipative term proportional to the angular momentum. The coefficient of viscous damping has the same dependence on the model parameters as in the one-dimensional case discussed above.

Fig. 3

Two models of interaction between the rotors and the ether: the 1D model on the left-hand side and the 3D model on the right-hand side

It is important to note that, in essence, the model problems considered above demonstrate the description of a heat wave radiation. The heat transfer by radiation can be represented in different ways. On the one hand, it is generally accepted that only electromagnetic waves propagate and it is precisely these waves that cause heating of matter when interacting with it. On the other hand, one can assume that waves of different nature, namely, heat waves, propagate together with electromagnetic waves. The model problems considered above are the realization of the latter viewpoint. A similar approach is developed in Ivanova et al. (2007) where it is suggested an original method of description of the rotational molecular spectra lying in the infrared range and associated with the thermal radiation. This method is based on the continuum mechanics model with the rotational degrees of freedom.

The One-Rotor Gyrostat Continuum

Further, only the continuum of one-rotor gyrostats is considered. In this case, the ether plays the role of an external factor with respect to the continuum under study. The influence of the ether on the one-rotor gyrostats is modeled by the moment of viscous damping the structure of which is chosen in accordance with the results obtained by solving the model problems considered above. Thus, by eliminating the ether, a nonconservative model of the one-rotor gyrostat continuum is obtained.

Below the following notations are used: r is a position vector of some point in space; ρ_∗ is the mass density at the reference configuration; I = IE and J = JE are the mass densities of inertia tensors of the carrier bodies and the rotors, respectively, where E is the unit tensor; u(r, t) is the displacement vector; v(r, t) is the velocity vector; φ(r, t) and $\tilde {\boldsymbol {\omega }}(\mathbf {r}, t)$ are the rotation vector and the angular velocity vector of the carrier bodies; and θ(r, t) and ω(r, t) are the rotation vector and the angular velocity vector of the rotors. The linear theory is considered. Therefore, the kinematic relations have the form

$$\displaystyle \begin{aligned} \mathbf{v} = \frac{d \mathbf{u}}{d t}, \qquad \tilde{\boldsymbol{\omega}} = \frac{d \boldsymbol{\varphi}}{d t}, \qquad \boldsymbol{\omega} = \frac{d \boldsymbol{\theta}}{d t}.\end{aligned} $$

(1)

The balance equations of the linear momentum for the gyrostats and of the angular momentum for the carrier bodies of gyrostats are

$$\displaystyle \begin{aligned} & \nabla \cdot \boldsymbol{\tau} + \rho_* \mathbf{f} = \rho_* \frac{d \mathbf{v}}{d t}\,,\\ &\nabla \cdot \boldsymbol{\mu} + \boldsymbol{\tau}_{\times} + \rho_* \mathbf{m} = \rho_* I \frac{d \tilde{\boldsymbol{\omega}}}{d t}.\end{aligned} $$

(2)

Here ∇ is the gradient operator; τ is the stress tensor; μ is the moment stress tensor modeling the interaction of the carrier bodies of gyrostats; f is the mass density of external forces; and m is the mass density of external moments acting on the carrier bodies of gyrostats. The balance equation of the angular momentum for the rotors of gyrostats has the form

$$\displaystyle \begin{aligned} \nabla \cdot \mathbf{T} + \rho_* \mathbf{L} = \rho_* J \frac{d \boldsymbol{\omega}}{d t},\end{aligned} $$

(3)

where T is the moment stress tensor modeling the interaction of the rotors of gyrostats and L is the mass density of external moments acting on the rotors.

In view of Eqs. (2) and (3), the energy balance equation is written as

$$\displaystyle \begin{aligned} \frac{d (\rho_* U)}{d t} = \boldsymbol{\tau}^T \cdot \cdot \frac{d \boldsymbol{\varepsilon}}{d t} + \boldsymbol{\mu}^T \cdot \cdot \frac{d \boldsymbol{\kappa}}{d t} + {\mathbf{T}}^T \cdot \cdot\frac{d \boldsymbol{\vartheta}}{d t},\end{aligned} $$

(4)

where U is the internal energy density per unit mass and the double scalar product is defined as ab ⋅⋅cd = (b ⋅c)(a ⋅d). The strain tensors ε, κ, and 𝜗 are determined by the formulas

$$\displaystyle \begin{aligned} \boldsymbol{\varepsilon} = \nabla \mathbf{u} + \mathbf{E} \times \boldsymbol{\varphi}, \qquad \boldsymbol{\kappa} = \nabla \boldsymbol{\varphi}, \qquad \boldsymbol{\vartheta} = \nabla \boldsymbol{\theta}. \end{aligned} $$

(5)

The Main Hypotheses

Further a special case of the theory of one-rotor gyrostat continuum based on two hypotheses is considered.

Hypothesis 1. Vector L is a sum of the moment L_h characterizing external actions of all sorts and the moment of linear viscous damping

$$\displaystyle \begin{aligned} {\mathbf{L}}_f = - \beta J \boldsymbol{\omega}, \end{aligned} $$

(6)

where β is the coefficient of damping. The moment L_f models the influence of the ether (the body-points positioned in the space between the gyrostats) that causes the dissipation of the rotors energy. The moment L_h models the influence of external ponderable bodies that is passed by means of the ether. It can model actions of various physical nature, e.g., heat supply, electromagnetic excitation, or some kind of radiation. The main difference between the moment L_h and the moment L_f is the fact that the moment L_h occurs only when there are some ponderable bodies, whereas the moment L_f occurs regardless of the presence or absence of other bodies. The structure of moment (6) is chosen in accordance with the results obtained by solving the model problems considered above.

Hypothesis 2. The moment stress tensor T characterizing the interactions between the rotors is the spherical part of tensor

$$\displaystyle \begin{aligned} \mathbf{T} = T\, \mathbf{E}. \end{aligned} $$

(7)

Assumption (7) is based on the following interpretations. The interaction of the carrier bodies of gyrostats is attributed to the mechanical processes. The interaction of the rotors of gyrostats models thermal processes, and the interaction of the carrier bodies and the rotors provides the interplay of mechanical and thermal processes. The moment interaction between the rotors is considered to be analogy of temperature. Since temperature is a scalar, the moment stress tensor T must be characterized by one scalar quantity. Hence, it must be the spherical part of tensor (see Eq. (7)).

Thermodynamic Analogies

In view of Eq. (7), the energy balance equation (4) is reduced to the form

$$\displaystyle \begin{aligned} \frac{d (\rho_* U)}{d t} &= \boldsymbol{\tau}^T \cdot \cdot \frac{d \boldsymbol{\varepsilon}}{d t} + \boldsymbol{\mu}^T \cdot \cdot \frac{d \boldsymbol{\kappa}}{d t} + T\,\frac{d \vartheta}{d t},\\ \vartheta &= \mathrm{tr}\,\boldsymbol{\vartheta}. \end{aligned} $$

(8)

If Eq. (8) is considered to be the energy balance equation for the classical medium, then the last term on the right-hand side of this equation should be interpreted as the thermodynamic one. Since the quantity T has the sense of temperature analogy, the quantity 𝜗 acquires the meaning of volume density of entropy analogy. The units of measurement of the temperature analogy and the entropy analogy that are introduced within the framework of the considered model are different from the standard units of measurement of temperature and entropy. Indeed, the unit of measurement of T is N∕m, whereas the unit of measurement of temperature is kelvin; the unit of measurement of 𝜗 is 1/m, whereas the unit of measurement of volume density of entropy is J∕(m³ K). This obstacle can be overcome by introducing a normalization factor a and changing the variables:

(9)

Here T_a is the temperature that can be measured by a thermometer. Its unit of measurement is kelvin. Correspondingly, 𝜗_a is the volume density of entropy. Its unit of measurement is J∕(m³ K). In the case of the linear theory, the normalization factor a can be eliminated from all equations.

A Model of Heat-Conductive Elastic Material

Now the model of continuum satisfying the hypotheses stated above is considered. In view of the hypotheses, the balance equation of the angular momentum for rotors takes the form

$$\displaystyle \begin{aligned} \nabla T_a - \rho_* \beta J_a \boldsymbol{\omega}_a + \rho_* {\mathbf{L}}_h^a = \rho_* J_a\, \frac{d \boldsymbol{\omega}_a}{d t}. \end{aligned} $$

(10)

In view of Eqs. (1), (5), (8), and (9), from Eq. (10), it follows one of the forms of the heat conduction equation, namely,

$$\displaystyle \begin{aligned} \varDelta T_a - \rho_* \beta J_a\, \frac{d \vartheta_a}{d t} - \rho_* J_a\, \frac{d^2 \vartheta_a}{d t^2} = - \rho_* \nabla \cdot {\mathbf{L}}_h^a,\end{aligned} $$

(11)

where the term $\rho _* \nabla \cdot {\mathbf {L}}_h^a$ plays the role of a heat supply. The heat conduction Eq. (11) can be reduced to the conventional form. In order to do this, it is necessary to express 𝜗_a in terms of temperature and strain tensors by using the constitutive equations that are given below.

According to the energy balance equation (8), the internal energy density is a function of the strain tensors ε, κ and the scalar strain measure 𝜗 (or the volume density of entropy 𝜗_a that is the same thing). In the linear theory, the internal energy density ρ_∗U is assumed to be a quadratic form of the quantities listed above. In this case, the constitutive equations are written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \boldsymbol{\tau}^T &\displaystyle =&\displaystyle \boldsymbol{\tau}^T_0 + {\ }^4{\mathbf{C}}_1 \cdot \cdot \boldsymbol{\varepsilon} + {\ }^4{\mathbf{C}}_2 \cdot \cdot \boldsymbol{\kappa} \\&\displaystyle +&\displaystyle {\mathbf{C}}_4\, (\vartheta_a - \vartheta_a^*), \boldsymbol{\mu}^T = \boldsymbol{\mu}^T_0 + \boldsymbol{\varepsilon} \cdot \cdot {\ }^4{\mathbf{C}}_2\\ &\displaystyle +&\displaystyle {\ }^4{\mathbf{C}}_3 \cdot \cdot \boldsymbol{\kappa} + {\mathbf{C}}_5\, (\vartheta_a - \vartheta_a^*),\\ T_a &\displaystyle =&\displaystyle T_a^* + \boldsymbol{\varepsilon} \cdot \cdot {\mathbf{C}}_4 + \boldsymbol{\kappa} \cdot \cdot {\mathbf{C}}_5 + C_6\,(\vartheta_a - \vartheta_a^*).\\ \end{array} \end{aligned} $$

(12)

Here τ₀ and μ₀ are the initial stresses; $T_a^*$ is the value of absolute temperature at which the thermodynamic parameters are determined; $\vartheta _a^*$ is the corresponding value of volume density of entropy; ⁴C₁, ⁴C₂, and ⁴C₃ are the fourth-rank stiffness tensors; C₄ and C₅ are the second-rank tensors characterizing the interplay of mechanical and thermodynamic processes; and C₆ is the scalar quantity characterizing the specific heat.

In view of the foregoing analogies between the mechanical and thermodynamic quantities, the set of Eqs. (1), (2), (5), (9), (11), and (12) can be considered as the mathematical description of a conventional material which possesses elastic and thermodynamic properties.

An Isotropic Chiral Medium

The model of continuum discussed above contains both polar and axial material tensors. The fourth-rank tensors ⁴C₁ and ⁴C₃ and the second-rank tensor C₅ are polar. The fourth-rank tensor ⁴C₂ and the second-rank tensor C₄ are axial. In the case of an isotropic chiral medium, polar and axial tensors have the same structure. To be exact, the fourth-rank tensors and the second-rank ones have the form

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} {\ }^4{\mathbf{C}}_1 &\displaystyle =&\displaystyle C_1 \mathbf{E} \mathbf{E} + C_2 \sum_{i=1}^3 {\mathbf{e}}_i \mathbf{E}\, {\mathbf{e}}_i \\&\displaystyle +&\displaystyle C_3 \sum_{i=1}^3 \sum_{j=1}^3 {\mathbf{e}}_i {\mathbf{e}}_j {\mathbf{e}}_i {\mathbf{e}}_j,\\ {\mathbf{C}}_4 &\displaystyle =&\displaystyle C_4 \mathbf{E}, \end{array} \end{aligned} $$

(13)

where e₁, e₂, and e₃ are the mutually orthogonal unit vectors. In the case of an isotropic non-chiral medium, the polar tensors ⁴C₁, ⁴C₃, and C₅ have the form (13), and the axial tensors ⁴C₂ and C₄ are equal to zero. At the same time, it is known that tensor C₄ characterizing the thermal expansion is not equal to zero. That is why, the isotropic media that are chiral with respect to the microstructure are considered in the presented theory.

The question is, what sorts of engineering materials can be qualified as chiral media? It is obvious, it depends on what properties of a material and what processes in the material are that we want to study. If we want to study only the mechanical properties and processes, then almost all materials can be qualified as non-chiral media. The only exceptions are materials consisting of sufficiently large particles that do not have a mirror symmetry, such as materials consisting of large polymer molecules having a helical structure or materials containing the DNA molecules. If we want to model a conventional material taking into account not only its mechanical properties but in addition some other its physical properties, then a representative volume of the continuum must reflect the properties of the material at the microlevel, i.e., the properties of the material that are conditioned by the state of its atoms. Atoms consist of elementary particles with spin. The presence of spin eliminates the mirror symmetry. That is why, in order to model a conventional material taking into account not only its mechanical properties but in addition some other its physical properties, we should consider this material as a chiral medium. Certainly, the foregoing assertion concerns only the method of modeling that is based on using the Cosserat continuum with the microstructure.

The Hyperbolic Thermoelasticity and the Classical One

It is known that when describing mechanical processes in three-dimensional media, the moment interactions and the rotation inertia can be neglected. In accordance with this fact, it is assumed that μ = 0, m = 0, I = 0, and hence, τ = τ^T. For the medium that is chiral with respect to the microstructure, in view of Eq. (13), the constitutive equations (12) take the form

(14)

The parameters of the model are chosen as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} C_1 &\displaystyle =&\displaystyle K_{ad} - \frac{2}{3}\,G, \quad C_2 + C_3 = G, \\ C_4 &\displaystyle =&\displaystyle - \frac{\alpha K_{iz} T_a^*}{\rho_* c_v}, \quad C_6 = \frac{T_a^*}{\rho_* c_v}, \\ \beta J_a &\displaystyle =&\displaystyle \frac{T_a^*}{\rho_* \lambda}. {} \end{array} \end{aligned} $$

(15)

where K_iz and $K_{ad} = K_{iz} + \alpha ^2 K_{iz}^2 T_a^* / (\rho _* c_v)$ are the isothermal and adiabatic modules of compression, G is the shear modulus, α is the volume coefficient of thermal expansion, c_v is the specific heat at constant volume, and λ is the heat conduction coefficient. It is easy to see that the inverse coefficient of heat conduction is directly proportional to the dynamic coefficient of damping ρ_∗βJ_a, the inverse specific heat is directly proportional to the angular stiffness C₆ characterizing the moment interaction between the rotors, and the volume coefficient of thermal expansion is directly proportional to the stiffness C₄ characterizing the dependence of the stress tensor on the angular strains and the dependence of the moment stress tensor on the linear strains.

In view of Eq. (15), the set of Eqs. (2), (11), and (14) can be reduced to the form

$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla \cdot \boldsymbol{\tau} + \rho_* \mathbf{f} = \rho_* \frac{d^2 \mathbf{u}}{d t^2}, \quad \\ \boldsymbol{\tau} = \boldsymbol{\tau}_0 + \Bigl(K_{iz} - \frac{2}{3}\, G\Bigr) \varepsilon \mathbf{E} + 2 G\, \boldsymbol{\varepsilon}^s - \alpha K_{iz} ({T}_a - {T}_a^*)\, \mathbf{E}, {} \quad \\ \varDelta {T}_a - \frac{\rho_* c_v}{\lambda} \left( \frac{d {T}_a}{d t} + \frac{1}{\beta}\, \frac{d^2 {T}_a}{d t^2} \right) = \frac{\alpha K_{iz} T_a^*}{\lambda} \left( \frac{d \varepsilon}{d t} + \frac{1}{\beta}\, \frac{d^2 \varepsilon}{d t^2} \right) - \rho_* \nabla \cdot {\mathbf{L}}_h^a. \quad \end{array} \end{aligned} $$

(16)

The parameter β⁻¹ is usually called the heat flux relaxation time. If the parameter β⁻¹ becomes zero on conditions that the product βJ_a remains finite, then the set of Eqs. (16) is equivalent to the classical statement of coupled problem of thermoelasticity (see, e.g., Nowacki 1976). If the parameter β⁻¹ is not equal to zero, then Eq. (16) is the statement of problem of the hyperbolic type thermoelasticity (see Lord and Shulman 1967).

Nonlinear Models of Heat Transfer: State of the Art

Nonlinear thermal processes are actively studied and discussed in the modern literature. Without claiming to be an exhaustive literature review, we indicate the main research areas in the field of nonlinear thermal conductivity and denote a place of the presented theory among the other models.

Many papers covering only mathematical questions are regularly published for several decades. Various aspects of constructing analytical, semi-analytic, and numerical solutions of the nonlinear heat conduction equations are discussed in such papers see, e.g., Campo (1982), Jordan et al. (1987), Polyanin et al. (2000), Ebadian and Darania (2008), and Habibi et al. (2015). Such works usually deal with the simplest nonlinear heat conduction equations. The nonlinearity of these equations consists in the fact that the material constants of the linear equations are replaced by some functions of temperature (more often by polynomials). Among the mathematical works, it is worth mentioning the papers where the authors consider laser heat sources (see, e.g., Fong et al. 2010), as this type of thermal influences is most often found in the modern literature. Another large group of publications consists of applied works, which are devoted to modeling nonlinear thermal processes in technical devices (see, e.g., Grudinin et al. 2011; Chaibi et al. 2012; Huang et al. 2012; Markides et al. 2013). In such works, the mutual influence of thermal processes and processes of other physical nature (optical, electrical, magnetic) is usually taken into account. Other distinctive features of applied works are the use of numerical methods, the use of parameters of specific technical devices in calculations, and the comparison of modeling results with the experimental data. The models of nonlinear thermal processes in the widest scale range, from geophysical processes (see, e.g., Mottaghy and Rath 2006), up to biological processes at the molecular level (see, e.g., LeMesurier 2008), are presented in the modern literature. There are a large number of papers devoted to studying nonlinear effects associated with thermal radiation (see, e.g., Khandekar et al. 2015; Ananth et al. 2015). There exist a variety of mathematical models used to describe various thermal processes. Some of them are based on classical concepts, and others are based on quantum-mechanical concepts. However, purely empirical relations, which are not based on any models, play an important role in the majority of nonlinear theories.

The main feature of the linear theory presented in this paper and its nonlinear analogue presented in Ivanova (2017) is that these theories provide the description of two fundamentally different processes of heat transfer (heat conduction and thermal radiation) within the framework of one model. Another feature of the present approach is the fact that it is based on the mechanical model different from those used in statistical physics and quantum mechanics.

Cross-References

• Cosserat Media

• Dispersion Relations for the Coupled Hyperbolic Thermoelasticity

• Thermoelastic Waves in a Medium with Heat-flux Relaxation

• Zhilin, Pavel Andreevich