Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables

The classical irreversible thermodynamic with internal variables (CIT-IV) means a general approach to the study of different interactions among irreversible thermodynamic processes using classical exstensive variables. After the pioneering works of Lars Onsager [1, 2], many excellent scientists: Eckart [3,4,5,6], Prigogine [7], Groot et al. [8], Meixner et al. [9], Colemann et al. [10], Truesdell [11], Kluitenberg [12,13,14,15,16], Maugin et al. [17, 18] developed the basics of the theory CIT-IV characterized by the introduction of thermodynamic variables, called internal (or hidden) variables.

The flexibility of the methodologies used in CIT-IV consists in the fact that “a priori” the physical meaning of the internal thermodynamical variables is not specified but only their influence on particular types of occurring phenomena in the considered medium is assumed. For this reason, in the following discussion, we will call these thermodynamical variables: dynamical variables since they have also been used, successfully, for the study of problems in dielectric and magnetic relaxation phenomena [19,20,21,22], diffusion and anelastic deformation [23,24,25,26,27,28,29,30,31,32]. It’s important to note that the many theoretical results have been confirmed by the experimental data [33,34,35,36,37,38,39,40].

Parallel approaches were worked out in the frame of Extended Irreversible Thermodynamics (EIT) [41] and in Rational Extended Thermodynamics (RET) [42, 43] which, in consideration of suggestions from the kinetic theory, considers an entropy depending on the fluxes as well as on the classical variables. It was shown [29,30,31] that by using the usual procedure of CIV-IV it is possible to describe the relaxation of thermal phenomena thus obtaining results which are generally justified by additional hypothesis suggested by kinetic theory.

In this paper, we show that thermoconduction phenomena in visco-anelastic media may be studied by using the systematic procedure of CIT-IV. In order to study the behavior of rheological media, in the presence of viscous and anelastic phenomena, besides determining the stress-strain relationships it is important to obtain the heat propagation equation. For this purpose the tensor $ε_{α β}^{(1)}$ \varepsilon _{\alpha \beta }^{(1)} (inelastic strain) and the vector ξ are introduced as internal variables which charaterize both stress-strain relationships and which generalize the Fourier heat equation (parabolic type equation) [44] and Maxwell-Cattaneo-Vernotte (hyperbolic type equation) [45,46,47].

In Section 2, introducing a vectorial dynamical variable j and a stress field $τ_{α β}^{(eq)}$ \tau _{\alpha \beta }^{(eq)} which is of a thermoelastic nature, an explicit form for the entropy production is derived.

In Sections 3–7, the phenomenological equations, which generalize the Kelvin, Jeffreys and Poynting-Thomson bodies are obtained.

In Section 8, by virtue of the dynamical variable, influencing the thermal transport phenomena, the heat equation is obtained. In particular, in the isotropic case, when the medium has symmetry properties that, under orthogonal transformations, are invariant with respect to all rotations and inversions of the frame of axes, it is obtained that the heat flux can be split in two parts: a first contribution J⁽⁰⁾, governed by Fourier law and a second contribution J⁽¹⁾, obeying the Maxwell-Cattaneo-Vernotte equation (MCV) [45,46,47,48] in which a relaxation time is present.

Finally, in Section 9, we obtain a general temperature equation which generalizes the analogous equations of Fourier and MCV.

The balance equation of entropy

In the contest of irreversible processes an important role is played by the flow of heat which, classically, is not considered to be a state variable. Therefore, we will suppose that the specific entropy s, depends not only on the specific internal energy u and the total strain ɛ_αβ, the tensor $ε_{α β}^{(1)}$ \varepsilon _{\alpha \beta }^{(1)} describing the inelastic strain and also on a vectorial dynamic variable, ξ that is an odd function of microscopic particles velocities that have influence on the propagation phenomena which occur in the medium. (1) $s = s (u, ε_{α β}, ε_{α β}^{(1)}, ξ_{α}),$ s = s(u,{\varepsilon _{\alpha \beta }},\varepsilon _{\alpha \beta }^{(1)},{\xi _\alpha }){\kern 1pt} , where ξ_α (α = 1,2,3) is the α-component of the vector ξ.

Theorem 1 (Gibbs relation)

By using (1) we can obtain the following Gibbs relation (2) $Tds = du - ν τ_{α β}^{(eq)} d ε_{α β} + ν τ_{α β}^{(1)} d ε_{α β}^{(1)} + ν j_{α} d ξ_{α},$ Tds = du - \nu {\kern 1pt} \tau _{\alpha \beta }^{(eq)}d{\varepsilon _{\alpha \beta }} + \nu {\kern 1pt} \tau _{\alpha \beta }^{(1)}d\varepsilon _{\alpha \beta }^{(1)} + \nu {\kern 1pt} {j_\alpha }{\kern 1pt} d{\xi _\alpha }{\kern 1pt} , where ν is the specific volume, $τ_{α β}^{(eq)}$ \tau _{\alpha \beta }^{(eq)} is the equilibrium stress tensor, $τ_{α β}^{(1)}$ \tau _{\alpha \beta }^{(1)} is the affinity-stress conjugate to $ε_{α β}^{(1)}$ \varepsilon _{\alpha \beta }^{(1)} , T is the absolute temperature and j_α is the α-component of the vector j conjugate of dξ_α /dt.

Proof

We define the absolute temperature (3) $T^{- 1} \overset{def}{=} \frac{\partial}{\partial u} s (u, ε_{α β}, ε_{α β}^{(1)}, ξ_{α}),$ {T^{ - 1}}{\kern 1pt} \mathop = \limits^{\rm def} {\kern 1pt} \frac{\partial }{{\partial u}}s(u,{\varepsilon _{\alpha \beta }},\varepsilon _{\alpha \beta }^{(1)},{\xi _\alpha }){\kern 1pt} , the equilibrium-stress tensor (4) $τ_{α β}^{(eq)} \overset{def}{=} - ρ T \frac{\partial}{\partial ε_{α β}} s (u, ε_{α β}, ε_{α β}^{(1)}, ξ_{α}),$ \tau _{\alpha \beta }^{(eq)}{\kern 1pt} \mathop = \limits^{\rm def} {\kern 1pt} - \rho T\frac{\partial }{{\partial {\varepsilon _{\alpha \beta }}}}s(u,{\varepsilon _{\alpha \beta }},\varepsilon _{\alpha \beta }^{(1)},{\xi _\alpha }){\kern 1pt} , the affinity-stress conjugate to $ε_{α β}^{(1)}$ \varepsilon _{\alpha \beta }^{(1)} (5) $τ_{α β}^{(1)} \overset{def}{=} ρ T \frac{\partial}{\partial ε_{α β}^{(1)}} s (u, ε_{α β}, ε_{α β}^{(1)}, ξ_{α}),$ \tau _{\alpha \beta }^{(1)}{\kern 1pt} \mathop = \limits^{\rm def} {\kern 1pt} \rho T\frac{\partial }{{\partial \varepsilon _{\alpha \beta }^{(1)}}}s(u,{\varepsilon _{\alpha \beta }},\varepsilon _{\alpha \beta }^{(1)},{\xi _\alpha }){\kern 1pt} , and the vector j conjugate to the internal vector variable ξ (6) $j_{α} \overset{def}{=} ρ T \frac{\partial}{\partial ξ_{α}} s (u, ε_{α β}, ε_{α β}^{(1)}, ξ_{α}),$ {j_\alpha }{\kern 1pt} \mathop = \limits^{\rm def} {\kern 1pt} \rho T\frac{\partial }{{\partial {\xi _\alpha }}}s(u,{\varepsilon _{\alpha \beta }},\varepsilon _{\alpha \beta }^{(1)},{\xi _\alpha }){\kern 1pt} , where $ρ \overset{def}{=} ν^{- 1}$ \rho {\kern 1pt} \mathop = \limits^{\rm def} {\kern 1pt} {\nu ^{ - 1}} is the mass density. By using equations (3)–(6) from (2), we obtain the differential ds of s: (7) $Tds = du - ν τ_{α β}^{(eq)} d ε_{α β} + ν τ_{α β}^{(1)} d ε_{α β}^{(1)} + ν j_{α} d ξ_{α} .$ Tds = du - \nu {\kern 1pt} \tau _{\alpha \beta }^{(eq)}d{\varepsilon _{\alpha \beta }} + \nu {\kern 1pt} \tau _{\alpha \beta }^{(1)}{\kern 1pt} d\varepsilon _{\alpha \beta }^{(1)} + \nu {\kern 1pt} {j_\alpha }{\kern 1pt} d{\xi _\alpha }{\kern 1pt} . The relation (7) is usually called Gibbs relation, in which the usual summation convection for dummy is used.

From (7), we have: (8) $ρ T \frac{ds}{dt} = ρ \frac{du}{dt} - τ_{α β}^{(eq)} \frac{d ε_{α β}}{dt} + τ_{α β}^{(1)} \frac{d ε_{α β}^{(1)}}{dt} + j_{α} \frac{d ξ_{α}}{dt},$ \rho T\frac{{ds}}{{dt}} = \rho \frac{{du}}{{dt}} - {\kern 1pt} \tau _{\alpha \beta }^{(eq)}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}} + \tau _{\alpha \beta }^{(1)}\frac{{d\varepsilon _{\alpha \beta }^{(1)}}}{{dt}} + {j_\alpha }{\kern 1pt} \frac{{d{\xi _\alpha }}}{{dt}}{\kern 1pt} , where (9) $\frac{d}{dt} = \frac{\partial}{\partial t} + v \cdot \nabla,$ \frac{d}{{dt}} = \frac{\partial }{{\partial t}} + {\bf v} \cdot \nabla , is the substantial derivative respect to time and v is the velocity field and ∇ is the gradient.

To analyze phenomena due to viscous flows (analogous to those which occur during flows in ordinary viscous liquids and gases) we introduce the following viscous stress tensor (10) $τ_{α β}^{(vi)} \overset{def}{=} τ_{α β} - τ_{α β}^{(eq)},$ \tau _{\alpha \beta }^{(vi)}{\kern 1pt} \mathop = \limits^{\rm def} {\kern 1pt} {\tau _{\alpha \beta }} - \tau _{\alpha \beta }^{(eq)}{\kern 1pt} , where τ_αβ is the mechanical stress tensor which occurs in the equation of motion (11) $ρ \frac{{d v}_{α}}{dt} = ρ F_{α} + \frac{\partial τ_{α β}}{\partial x^{β}},$ \rho \frac{{{d{\rm v}_\alpha }}}{{dt}} = \rho {\kern 1pt} {F_\alpha } + \frac{{\partial {\tau _{\alpha \beta }}}}{{\partial {x^\beta }}}, and in the first law of thermodynamics (12) $ρ \frac{du}{dt} = - \nabla \cdot J^{(q)} + τ_{α β} \frac{d ε_{α β}}{dt} .$ \rho \frac{{du}}{{dt}} = - \nabla \cdot {{\bf J}^{(q)}} + {\tau _{\alpha \beta }}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}}. In (11) the force F_α is the volume force per unit of mass and in (12) the vector J^(q) is the heat flux.

Theorem 2 (Entropy production)

By using the first law of thermodynamics (12) the balance equation of the entropy can be obtained (13) $ρ \frac{ds}{dt} = - \nabla \cdot (\frac{J^{(q)}}{T}) + σ^{(s)},$ \rho \frac{{ds}}{{dt}} = - \nabla \cdot \left( {\frac{{{{\boldsymbol J}^{(q)}}}}{T}} \right) + {\sigma ^{(s)}}, and the entropy production (14) $σ^{(s)} = T^{- 1} [J^{(q)} \cdot (- T^{- 1} \nabla T) + τ_{α β}^{(vi)} \frac{d ε_{α β}}{dt} + τ_{α β}^{(1)} \frac{d ε_{α β}^{(1)}}{dt} + j_{α} \frac{d ξ_{α}}{dt}] \geq 0 .$ {\sigma ^{(s)}} = {T^{ - 1}}\left[ {{{\boldsymbol J}^{(q)}} \cdot \left( { - {T^{ - 1}}\nabla {\kern 1pt} T} \right) + \tau _{\alpha \beta }^{(vi)}{\kern 1pt} \frac{{d{\kern 1pt} {\varepsilon _{\alpha \beta }}}}{{dt}}{\kern 1pt} + {\kern 1pt} \tau _{\alpha \beta }^{(1)}{\kern 1pt} \frac{{d\varepsilon _{\alpha \beta }^{(1)}}}{{dt}}{\kern 1pt} + {\kern 1pt} {j_\alpha }{\kern 1pt} \frac{{d{\xi _\alpha }}}{{dt}}} \right] \ge 0.

Proof

The equation (12), by virtue of (10) becomes (15) $ρ \frac{du}{dt} = - \nabla \cdot J^{(q)} + (τ_{α β}^{(eq)} + τ_{α β}^{(vi)}) \frac{d ε_{α β}}{dt},$ \rho \frac{{du}}{{dt}} = - \nabla \cdot {{\bf J}^{(q)}} + \left( {\tau _{\alpha \beta }^{(eq)} + \tau _{\alpha \beta }^{(vi)}} \right)\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}}, which substituted in the equation (8) gives (16) $ρ \frac{ds}{dt} = T^{- 1} (- \nabla \cdot J^{(q)} + τ_{α β}^{(vi)} \frac{d ε_{α β}}{dt} + τ_{α β}^{(1)} \frac{d ε_{α β}}{dt} + j_{α} \frac{d ξ_{α}}{dt}) .$ \rho {\kern 1pt} \frac{{ds}}{{dt}} = {T^{ - 1}}{\kern 1pt} \left( { - \nabla \cdot {{\bf J}^{(q)}} + \tau _{\alpha \beta }^{(vi)}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}} + \tau _{\alpha \beta }^{(1)}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}} + {j_\alpha }\frac{{d{\xi _\alpha }}}{{dt}}} \right){\kern 1pt} . Using the following identity (17) $T^{- 1} \nabla \cdot J^{(q)} = \nabla \cdot (\frac{J^{(q)}}{T}) + T^{- 2} J^{(q)} \cdot \nabla T,$ {T^{ - 1}}\nabla \cdot {{\bf J}^{(q)}} = \nabla \cdot \left( {\frac{{{{\bf J}^{(q)}}}}{T}} \right) + {T^{ - 2}}{{\bf J}^{(q)}} \cdot \nabla T, the equation (16) gives (13) (balance equation of the entropy) and (14) (entropy production of entropy).

It is observed that the entropy production is due to three types of phenomena: the first term on the right-hand of (14) gives the contribution of the heat conduction phenomena, the second sum is the contribution of viscous phenomena, the last sum is the contribution of the variation of the dynamical variable. Each term is a production of flux (J^(q), $τ_{α β}^{(vi)}$ \tau _{\alpha \beta }^{(vi)} , $d ε_{α β}^{(1)} / dt$ d\varepsilon _{\alpha \beta }^{(1)}/dt , j_α of the process and affinities conjugated to them: T⁻¹∇T, d ɛ_αβ /dt, $τ_{α β}^{(1)}$ \tau _{\alpha \beta }^{(1)} , dξ_α /dt, respectively.

Phenomenological equations

According to the usual procedure of non-equilibrium thermodynamics, by virtue of the form (14) for the entropy production, as anisotropic media, we have the following phenomenological equations: (18) $J_{α}^{(q)} = L_{α β}^{(q) (q)} (- T^{- 1} \frac{\partial T}{\partial x^{β}}) + L_{α (μ ν)}^{(q) (0)} \frac{d ε_{μ ν}}{dt} + L_{α (μ ν)}^{(q) (1)} τ_{μ ν}^{(1)} + L_{α μ}^{(q) (ξ)} \frac{d ξ_{μ}}{dt},$ {J}_\alpha ^{(q)} = L_{\alpha \beta }^{(q)(q)}{\kern 1pt} \left( { - {T^{ - 1}}\frac{{\partial T}}{{\partial {x^\beta }}}} \right) + L_{\alpha (\mu \nu )}^{(q)(0)}{\kern 1pt} \frac{{d{\varepsilon _{\mu \nu }}}}{{dt}}{\kern 1pt} + L_{\alpha (\mu \nu )}^{(q)(1)}\tau _{\mu \nu }^{(1)}{\kern 1pt} + {\kern 1pt} L_{\alpha \mu }^{(q)(\xi )}{\kern 1pt} \frac{{d{\xi _\mu }}}{{dt}}{\kern 1pt} , (19) $τ_{α β}^{(vi)} = L_{(α β) μ}^{(0) (q)} (- T^{- 1} \frac{\partial T}{\partial x^{μ}}) + L_{(α β) (μ ν)}^{(0) (0)} \frac{d ε_{μ ν}}{dt} + L_{(α β) (μ ν)}^{(0) (1)} τ_{μ ν}^{(1)} + L_{(α β) μ}^{(0) (ξ)} \frac{d ξ_{μ}}{dt},$ \tau _{\alpha \beta }^{(vi)} = L_{(\alpha \beta )\mu }^{(0)(q)}{\kern 1pt} \left( { - {T^{ - 1}}\frac{{\partial T}}{{\partial {x^\mu }}}} \right) + L_{(\alpha \beta )(\mu \nu )}^{(0)(0)}{\kern 1pt} \frac{{d{\varepsilon _{\mu \nu }}}}{{dt}}{\kern 1pt} + L_{(\alpha \beta )(\mu \nu )}^{(0)(1)}\tau _{\mu \nu }^{(1)} + {\kern 1pt} L_{(\alpha \beta )\mu }^{(0)(\xi )}{\kern 1pt} \frac{{d{\xi _\mu }}}{{dt}}{\kern 1pt} , (20) $\frac{d ε_{α β}^{(1)}}{dt} = L_{(α β) μ}^{(1) (q)} (- T^{- 1} \frac{\partial T}{\partial x^{μ}}) + L_{(α β) (μ ν)}^{(1) (0)} \frac{d ε_{μ ν}}{dt} + L_{(α β) (μ ν)}^{(1) (1)} τ_{μ ν}^{(1)} + L_{(α β) μ}^{(1) (ξ)} \frac{d ξ_{μ}}{dt},$ \frac{{d\varepsilon _{\alpha \beta }^{(1)}}}{{dt}} = L_{(\alpha \beta )\mu }^{(1)(q)}{\kern 1pt} \left( { - {T^{ - 1}}\frac{{\partial T}}{{\partial {x^\mu }}}} \right) + L_{(\alpha \beta )(\mu \nu )}^{(1)(0)}{\kern 1pt} \frac{{d{\varepsilon _{\mu \nu }}}}{{dt}}{\kern 1pt} + L_{(\alpha \beta )(\mu \nu )}^{(1)(1)}\tau _{\mu \nu }^{(1)} + {\kern 1pt} L_{(\alpha \beta )\mu }^{(1)(\xi )}{\kern 1pt} \frac{{d{\xi _\mu }}}{{dt}}{\kern 1pt} , (21) $j_{α} = L_{α β}^{(ξ) (q)} (- T^{- 1} \frac{\partial T}{\partial x^{β}}) + L_{α (μ ν)}^{(ξ) (0)} \frac{d ε_{μ ν}}{dt} + L_{α (μ ν)}^{(ξ) (1)} τ_{μ ν}^{(ξ) (1)} + L_{α β}^{(ξ) (ξ)} \frac{d ξ_{β}}{dt} .$ {j_\alpha } = L_{\alpha \beta }^{(\xi )(q)}{\kern 1pt} \left( { - {T^{ - 1}}\frac{{\partial T}}{{\partial {x^\beta }}}} \right) + L_{\alpha (\mu \nu )}^{(\xi )(0)}{\kern 1pt} \frac{{d{\varepsilon _{\mu \nu }}}}{{dt}}{\kern 1pt} + L_{\alpha (\mu \nu )}^{(\xi )(1)}{\kern 1pt} \tau _{\mu \nu }^{(\xi )(1)}{\kern 1pt} + L_{\alpha \beta }^{(\xi )(\xi )}{\kern 1pt} \frac{{d{\xi _\beta }}}{{dt}}{\kern 1pt} . The tensors L are called phenomenological tensors and the indices of these tensors enclosed in round brackets mean that they are symmetrical because the tensors ɛ_μν, $ε_{α β}^{(1)}$ \varepsilon _{\alpha \beta }^{(1)} , $τ_{α β}^{(1)}$ \tau _{\alpha \beta }^{(1)} and $τ_{α β}^{(vi)}$ \tau _{\alpha \beta }^{(vi)} are symmetric. The first of these equations may be regarded as a generalization of Fourier’s law. Equation (19) describes the viscous flow phenomenon and it may be considered to be a generalization of Stockes-Navier’s law. Finally, the equations (20) and (21) are the phenomenological equations for the irreversible process of the dynamic degrees of freedom.

Substituting (18)–(21) into (14) one has (22) $\begin{array}{l} T σ^{(s)} = & L_{α β}^{(q) (q)} (T^{- 2} \frac{\partial T}{\partial x^{α}} \frac{\partial T}{\partial x^{β}}) + [L_{α β}^{(ξ) (q)} + L_{α β}^{(q) (ξ)}] \frac{d ξ_{α}}{dt} (- T^{- 1} \frac{\partial T}{\partial x^{β}}) + \\ + L_{(α β) (μ ν)}^{(0) (0)} \frac{d ε_{α β}}{dt} \frac{d ε_{μ ν}}{dt} + \\ + [L_{(α β) (μ ν)}^{(1) (0)} + L_{(μ ν) (α β)}^{(0) (1)}] τ_{α β}^{(1)} \frac{d ε_{μ ν}}{dt} + \\ + L_{(α β) (μ ν)}^{(1) (1)} τ_{α β}^{(1)} τ_{μ ν}^{(1)} + \\ + L_{α β}^{(ξ) (ξ)} \frac{d ξ_{α}}{dt} \frac{d ξ_{β}}{dt} . \end{array}$ \begin{array}{*{20}{l}}{T{\sigma ^{(s)}} = }&{L_{\alpha \beta }^{(q)(q)}\left( {{T^{ - 2}}\frac{{\partial T}}{{\partial {x^\alpha }}}\frac{{\partial T}}{{\partial {x^\beta }}}} \right) + \left[ {L_{\alpha \beta }^{(\xi )(q)} + L_{\alpha \beta }^{(q)(\xi )}} \right]\frac{{d{\xi _\alpha }}}{{dt}}\left( { - {T^{ - 1}}\frac{{\partial T}}{{\partial {x^\beta }}}} \right) + }\\{}&{ + L_{(\alpha \beta )(\mu \nu )}^{(0)(0)}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}}{\kern 1pt} \frac{{d{\varepsilon _{\mu \nu }}}}{{dt}} + }\\{}&{ + \left[ {L_{(\alpha \beta )(\mu \nu )}^{(1)(0)} + L_{(\mu \nu )(\alpha \beta )}^{(0)(1)}} \right]{\kern 1pt} \tau _{\alpha \beta }^{(1)}{\kern 1pt} \frac{{d{\varepsilon _{\mu \nu }}}}{{dt}} + }\\{}&{ + L_{(\alpha \beta )(\mu \nu )}^{(1)(1)}\tau _{\alpha \beta }^{(1)}{\kern 1pt} \tau _{\mu \nu }^{(1)} + }\\{}&{ + L_{\alpha \beta }^{(\xi )(\xi )}\frac{{d{\xi _\alpha }}}{{dt}}{\kern 1pt} \frac{{d{\xi _\beta }}}{{dt}}{\kern 1pt} .}\end{array} This form for the entropy production (s^(s) ≥ 0) is a useful detail if isotropic media are considered in Section 5.

Symmetric relations

Since the time derivatives dɛ_αβ /dt, $d ε_{α β}^{(1)} / dt$ d\varepsilon _{\alpha \beta }^{(1)}/dt , the heat flow J^(q) and j vector conjugate to the internal variable ξ_α are odd functions of the microscopic particle velocities and the stress $τ_{α β}^{(vi)}$ \tau _{\alpha \beta }^{(vi)} , $τ_{α β}^{(1)}$ \tau _{\alpha \beta }^{(1)} , dξ_α /dt and the temperature gradient ∂T /∂t are even functions of these velocities, the Onsager-Casimir reciprocity relation reads [8]: (23) $L_{α β}^{(q) (q)} = L_{β α}^{(q) (q)}, L_{α β}^{(ξ) (q)} = L_{β α}^{(ξ) (q)}, L_{α β}^{(q) (ξ)} = L_{β α}^{(q) (ξ)}, L_{α β}^{(ξ) (ξ)} = L_{β α}^{(ξ) (ξ)},$ L_{\alpha \beta }^{(q)(q)} = L_{\beta \alpha }^{(q)(q)}{\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} L_{\alpha \beta }^{(\xi )(q)} = L_{\beta \alpha }^{(\xi )(q)}{\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} L_{\alpha \beta }^{(q)(\xi )} = L_{\beta \alpha }^{(q)(\xi )}{\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} L_{\alpha \beta }^{(\xi )(\xi )} = L_{\beta \alpha }^{(\xi )(\xi )}{\kern 1pt} , (24) $L_{(α β) (μ ν)}^{(0) (0)} = L_{(μ ν) (α β)}^{(0) (0)}, L_{(α β) (μ ν)}^{(1) (1)} = L_{(μ ν) (α β)}^{(1) (1)},$ L_{(\alpha \beta )(\mu \nu )}^{(0)(0)} = L_{(\mu \nu )(\alpha \beta )}^{(0)(0)}{\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} L_{(\alpha \beta )(\mu \nu )}^{(1)(1)} = L_{(\mu \nu )(\alpha \beta )}^{(1)(1)}{\kern 1pt} , (25) $L_{(α β) (μ ν)}^{(1) (0)} = L_{(μ ν) (α β)}^{(1) (0)}, L_{(α β) (μ ν)}^{(0) (1)} = L_{(μ ν) (α β)}^{(0) (1)},$ L_{(\alpha \beta )(\mu \nu )}^{(1)(0)} = L_{(\mu \nu )(\alpha \beta )}^{(1)(0)}{\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} L_{(\alpha \beta )(\mu \nu )}^{(0)(1)} = L_{(\mu \nu )(\alpha \beta )}^{(0)(1)}{\kern 1pt} , (26) $L_{(α β) (μ ν)}^{(1) (0)} = - L_{(μ ν) (α β)}^{(0) (1)} .$ L_{(\alpha \beta )(\mu \nu )}^{(1)(0)} = - L_{(\mu \nu )(\alpha \beta )}^{(0)(1)}{\kern 1pt} . The equations (23)–(26) reduce the number of indipendent components of the phenomenological tensors.

Phenomenological equations for isotropic media

In this section, we consider perfect isotropic media for which the symmetry properties are invariant under orthogonal transformations with respect to all rotations and to inversion of the frame of axes.

In this case, it can be shown [8, 49] that the phenomenological tensors of odd order vanish while even tensors can have the following forms: (27) $L_{α β}^{(q) (q)} = T λ^{(q, q)} δ_{α β}, L_{α β}^{(q) (ξ)} = λ^{(q, ξ)} δ_{α β},$ L_{\alpha \beta }^{(q)(q)} = T{\lambda ^{(q,q)}}{\delta _{\alpha \beta }}{\kern 1pt} \quad ,\quad L_{\alpha \beta }^{(q)(\xi )} = {\lambda ^{(q,\xi )}}{\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} , (28) $L_{α β}^{(ξ) (q)} = T λ^{(ξ, q)} δ_{α β}, L^{(ξ) (ξ)} = λ^{(ξ, ξ)} δ_{α β},$ L_{\alpha \beta }^{(\xi )(q)} = T{\lambda ^{(\xi ,q)}}{\delta _{\alpha \beta }}{\kern 1pt} \quad ,\quad {L^{(\xi )(\xi )}} = {\lambda ^{(\xi ,\xi )}}{\delta _{\alpha \beta }}{\kern 1pt} , (29) $L_{(α β) (μ ν)}^{(i) (k)} = \frac{1}{2} η_{s}^{(i, k)} (δ_{α μ} δ_{β ν} + δ_{α ν} δ_{β μ}) + \frac{1}{3} (η_{v}^{(i, k)} - η_{s}^{(i . k)}) δ_{α β} δ_{μ ν} (i, k = 0, 1) .$ L_{(\alpha \beta )(\mu \nu )}^{(i)(k)} = \frac{1}{2}\eta _s^{(i,k)}({\delta _{\alpha \mu }}{\delta _{\beta \nu }} + {\delta _{\alpha \nu }}{\delta _{\beta \mu }}) + \frac{1}{3}(\eta _v^{(i,k)} - \eta _s^{(i.k)}){\delta _{\alpha \beta }}{\delta _{\mu \nu }}{\kern 1pt} \quad (i,k = 0,1). By using (27)–(29), the equations (18)–(21) become (30) $J_{α}^{(q)} = - λ^{(q, q)} \frac{\partial T}{\partial x^{α}} + λ^{(q, ξ)} \frac{d ξ_{α}}{dt},$ {J}_\alpha ^{(q)} = - {\lambda ^{(q,q)}}\frac{{\partial T}}{{\partial {x^\alpha }}} + {\lambda ^{(q,\xi )}}{\kern 1pt} \frac{{d{\xi _\alpha }}}{{dt}}{\kern 1pt} , (31) $τ_{α β}^{(vi)} = η_{s}^{(0, 0)} \frac{d ε_{α β}}{dt} + (η_{v}^{(0, 0)} - η_{s}^{(0, 0)}) \frac{d ε}{dt} δ_{α β} + η_{s}^{(0, 1)} τ_{α β}^{(1)} + (η_{v}^{(0, 1)} - η_{s}^{(0, 1)}) τ^{(1)} δ_{α β},$ \tau _{\alpha \beta }^{(vi)} = \eta _s^{(0,0)}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}} + (\eta _v^{(0,0)} - \eta _s^{(0,0)}){\kern 1pt} \frac{{d\varepsilon }}{{dt}}{\kern 1pt} {\delta _{\alpha \beta }} + \eta _s^{(0,1)}\tau _{\alpha \beta }^{(1)} + (\eta _v^{(0,1)} - \eta _s^{(0,1)}){\kern 1pt} {\tau ^{(1)}}{\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} , (32) $\frac{d ε_{α β}^{(1)}}{dt} = η_{s}^{(1, 0)} \frac{d ε_{α β}}{dt} + (η_{v}^{(1, 0)} - η_{s}^{(1, 0)}) \frac{d ε}{dt} δ_{α β} + η_{s}^{(1, 1)} τ_{α β}^{(1)} + (η_{v}^{(1, 1)} - η_{s}^{(1, 1)}) τ^{(1)} δ_{α β},$ \frac{{d\varepsilon _{\alpha \beta }^{(1)}}}{{dt}} = \eta _s^{(1,0)}\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}} + (\eta _v^{(1,0)} - \eta _s^{(1,0)}){\kern 1pt} \frac{{d\varepsilon }}{{dt}}{\kern 1pt} {\delta _{\alpha \beta }} + \eta _s^{(1,1)}\tau _{\alpha \beta }^{(1)} + (\eta _v^{(1,1)} - \eta _s^{(1,1)}){\kern 1pt} {\tau ^{(1)}}{\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} , (33) $j_{α} = - λ^{(ξ, q)} \frac{\partial T}{\partial x^{α}} + λ^{(ξ, ξ)} \frac{d ξ_{α}}{dt} .$ {j_\alpha } = - {\lambda ^{(\xi ,q)}}{\kern 1pt} \frac{{\partial T}}{{\partial {x^\alpha }}} + {\lambda ^{(\xi ,\xi )}}{\kern 1pt} \frac{{d{\xi _\alpha }}}{{dt}}{\kern 1pt} . The scalar quantities λ, $η_{s}^{(0, 0)}$ \eta _s^{(0,0)} and $η_{v}^{(0, 0)}$ \eta _v^{(0,0)} are called phenomenological coefficients. In particular $η_{s}^{(0, 0)}$ \eta _s^{(0,0)} and $η_{v}^{(0, 0)}$ \eta _v^{(0,0)} may be called the shear viscosity and the volume viscosity, respectively. These coefficients also occur in the theory of ordinary (Stokes-Navier) viscous fluids. Equation (22), by virtue of (30)–(33), becomes (34) $\begin{array}{l} T σ^{(s)} = & λ^{(q, q)} {(\frac{\partial T}{\partial x^{α}})}^{2} + η_{s}^{(0, 0)} {(\frac{d ε_{α β}}{dt})}^{2} + 3 (η_{v}^{(0, 0)} - η_{s}^{(0, 0)}) {(\frac{d ε}{dt})}^{2} + \\ + η_{s}^{(1, 1)} {(τ^{(1)})}^{2} + 3 (η_{v}^{(1, 1)} - η_{s}^{(1, 1)}) {(τ^{(1)})}^{2} + λ^{(ξ, ξ)} {(\frac{d ξ_{α}}{dt})}^{2} . \end{array}$ \begin{array}{*{20}{l}}{T{\kern 1pt} {\sigma ^{(s)}} = }&{{\kern 1pt} {\lambda ^{(q,q)}}{\kern 1pt} {{(\frac{{\partial T}}{{\partial {x^\alpha }}})}^2} + \eta _s^{(0,0)}{{(\frac{{d{\varepsilon _{\alpha \beta }}}}{{dt}})}^2}{\kern 1pt} + 3(\eta _v^{(0,0)} - \eta _s^{(0,0)}){{(\frac{{d\varepsilon }}{{dt}})}^2} + }\\{}&{ + \eta _s^{(1,1)}{{({\tau ^{(1)}})}^2}{\kern 1pt} + 3(\eta _v^{(1,1)} - \eta _s^{(1,1)}){{({\tau ^{(1)}})}^2} + {\lambda ^{(\xi ,\xi )}}{{(\frac{{d{\xi _\alpha }}}{{dt}})}^2}{\kern 1pt} .}\end{array} By virtue of the positive definite character of the entropy production, several inequalities for the phenomenological coefficients may be derived.

For example, we have (35) $λ^{(q, q)} \geq 0, η_{s}^{(0, 0)} \geq 0, η_{v}^{(0, 0)} \geq η_{s}^{(0, 0)},$ {\lambda ^{(q,q)}} \ge 0\quad ,\quad \eta _s^{(0,0)} \ge 0\quad ,\quad \eta _v^{(0,0)} \ge \eta _s^{(0,0)}{\kern 1pt} , (36) $η_{s}^{(1, 1)} \geq 0, η_{v}^{(1, 1)} \geq η_{s}^{(1, 1)}, λ^{(ξ, ξ)} \geq 0 .$ \eta _s^{(1,1)} \ge 0\quad ,\quad \eta _v^{(1,1)} \ge \eta _s^{(1,1)}\quad ,\quad {\lambda ^{(\xi ,\xi )}} \ge 0{\kern 1pt} . The coefficients λ^(q,ξ) and λ^(ξ,q) represent possible cross effects between the heat flux and the irreversible process of the dynamic degrees of freedom.

Linear equations of state

Let f the specific free energy of the medium (37) $f = u - T s .$ f = u - T{\kern 1pt} s. With the aid of Gibbs relation (7), we have: (38) $df = ν τ_{α β}^{(eq)} d ε_{α β} - ν τ_{α β}^{(1)} d ε_{α β}^{(1)} - ν j_{α} d ξ_{α} - sdT,$ df = \nu {\kern 1pt} \tau _{\alpha \beta }^{(eq)}d{\varepsilon _{\alpha \beta }} - \nu \tau _{\alpha \beta }^{(1)}d\varepsilon _{\alpha \beta }^{(1)} - \nu {\kern 1pt} {j_\alpha }\;d{\xi _\alpha } - sdT{\kern 1pt} , where (39) $ν τ_{α β}^{(eq)} = \frac{\partial f}{\partial ε_{α β}}, ν τ_{α β}^{(1)} = - \frac{\partial f}{\partial ε_{α β}^{(1)}}, ν j_{α} = - \frac{\partial f}{\partial ξ_{α}}, s = - \frac{\partial f}{\partial T} .$ \nu {\kern 1pt} \tau _{\alpha \beta }^{(eq)} = \frac{{\partial f}}{{\partial {\varepsilon _{\alpha \beta }}}},\quad {\kern 1pt} \nu {\kern 1pt} \tau _{\alpha \beta }^{(1)} = - \frac{{\partial f}}{{\partial \varepsilon _{\alpha \beta }^{(1)}}},\quad \nu {\kern 1pt} {j_\alpha } = - {\kern 1pt} \frac{{\partial f}}{{\partial {\xi _\alpha }}},\quad s = - {\kern 1pt} \frac{{\partial f}}{{\partial T}}. Let us choose a configuration Σ⁽⁰⁾ with uniform temperature T₀ and in which s₀ is the specific entropy, ν₀ is the specific volume and ${(τ_{α β}^{(eq)})}_{0}$ {(\tau _{\alpha \beta }^{(eq)})_0} , ${(τ_{α β}^{(1)})}_{0}$ {(\tau _{\alpha \beta }^{(1)})_0} , ${(τ_{α β})}_{0}$ {({\tau _{\alpha \beta }})_0} , ${(ε_{α β})}_{0}$ {({\varepsilon _{\alpha \beta }})_0} , ${(ε_{α β}^{(1)})}_{0}$ {(\varepsilon _{\alpha \beta }^{(1)})_0} and (j_α)₀ are zero.

Let us suppose that the deviation from Σ⁽⁰⁾ is sufficiently small (with ρ ≈ ρ₀) and the free energy for isotropic medium can be written in the following form (40) $\begin{array}{l} f = ν_{0} & \{\frac{1}{2} [a_{α β μ ν}^{(0) (0)} ε_{α β} ε_{μ ν} + a_{α β μ ν}^{(1) (1)} ε_{α β}^{(1)} ε_{μ ν}^{(1)} + a_{α β}^{(ξ) (ξ)} ξ_{α} ξ_{β}] \\ + a_{α β μ ν}^{(0) (1)} ε_{α β} ε_{μ ν}^{(1)} + a_{α β μ}^{(0) (ξ)} ε_{α β}^{(1)} ξ_{μ} + a_{α β μ}^{(1) (ξ)} ε_{α β}^{(1)} ξ_{μ} \\ + (T - T_{0}) [a_{α β}^{(0) (T)} ε_{α β} + a_{α β}^{(1) (T)} ε_{α β}^{(1)} + a_{α}^{(ξ) (T)} ξ_{α}]\} - Ψ (T) . \end{array}$ \begin{array}{*{20}{l}}{f = {\kern 1pt} {\nu _0}{\kern 1pt} }&{\left\{ {\frac{1}{2}\left[ {a_{\alpha \beta \mu \nu }^{(0)(0)}{\kern 1pt} {\varepsilon _{\alpha \beta }}{\kern 1pt} {\varepsilon _{\mu \nu }}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu \nu }^{(1)(1)}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} \varepsilon _{\mu \nu }^{(1)} + a_{\alpha \beta }^{(\xi )(\xi )}{\xi _\alpha }{\kern 1pt} {\xi _\beta }} \right]} \right.}\\{}&{ + a_{\alpha \beta \mu \nu }^{(0)(1)}{\kern 1pt} {\varepsilon _{\alpha \beta }}{\kern 1pt} \varepsilon _{\mu \nu }^{(1)}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu }^{(0)(\xi )}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} {\xi _\mu }{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu }^{(1)(\xi )}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} {\xi _\mu }}\\{}&{\left. { + (T - {T_0})\left[ {a_{\alpha \beta }^{(0)(T)}{\varepsilon _{\alpha \beta }}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta }^{(1)(T)}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} + {\kern 1pt} a_\alpha ^{(\xi )(T)}{\xi _\alpha }} \right]} \right\} - \Psi (T){\kern 1pt} .}\end{array} where Ψ is some function of the temperature. In (40), the tensors “a” and the vector $a_{α}^{(ξ) (T)}$ a_\alpha ^{(\xi )(T)} , are constants (i.e. they do not depend on temperature and the strains) and are determined by the physical properties of the medium in the reference state.

From (39)₄ we have (41) $s = - ν_{0} (a_{α β}^{(0) (T)} ε_{α β} + a_{α β}^{(1) (T)} ε_{α β}^{(1)} + a_{α}^{(ξ) (T)} ξ_{α}) + \frac{d Ψ}{dT},$ s{\kern 1pt} = {\kern 1pt} - {\kern 1pt} {\nu _0}\left( {a_{\alpha \beta }^{(0)(T)}{\varepsilon _{\alpha \beta }} + a_{\alpha \beta }^{(1)(T)}\varepsilon _{\alpha \beta }^{(1)} + a_\alpha ^{(\xi )(T)}{\kern 1pt} {\xi _\alpha }} \right){\kern 1pt} + {\kern 1pt} \frac{{d{\kern 1pt} \Psi }}{{dT}}{\kern 1pt} , and as u = f + T s we obtain (42) $\begin{array}{l} u = & ν_{0} \{\frac{1}{2} [a_{α β μ ν}^{(0) (0)} ε_{α β} ε_{μ ν} + a_{α β μ ν}^{(1) (1)} ε_{α β}^{(1)} ε_{μ ν}^{(1)} + a_{α β}^{(ξ) (ξ)} ξ_{α} ξ_{β}] \\ - T_{0} [a_{α β}^{(0) (T)} ε_{α β} + a_{α β}^{(1) (T)} ε_{α β}^{(1)} + a_{α}^{(ξ) (T)} ξ_{α}]\} + T \frac{d Ψ}{dT} - Ψ (T) . \end{array}$ \begin{array}{*{20}{l}}{u{\kern 1pt} = {\kern 1pt} }&{{\nu _0}\left\{ {\frac{1}{2}\left[ {a_{\alpha \beta \mu \nu }^{(0)(0)}{\kern 1pt} {\varepsilon _{\alpha \beta }}{\kern 1pt} {\varepsilon _{\mu \nu }}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu \nu }^{(1)(1)}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} \varepsilon _{\mu \nu }^{(1)}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta }^{(\xi )(\xi )}{\kern 1pt} {\xi _\alpha }{\kern 1pt} {\xi _\beta }} \right]} \right.}\\{\kern 1pt} &{\left. { - {T_0}\left[ {a_{\alpha \beta }^{(0)(T)}{\varepsilon _{\alpha \beta }} + a_{\alpha \beta }^{(1)(T)}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} + {\kern 1pt} a_\alpha ^{(\xi )(T)}{\kern 1pt} {\xi _\alpha }} \right]} \right\}{\kern 1pt} + {\kern 1pt} T{\kern 1pt} \frac{{d\Psi }}{{dT}}{\kern 1pt} - {\kern 1pt} \Psi (T){\kern 1pt} .}\end{array} The specific heat at constant deformation, c_(ɛ), may be defined by (43) $c_{(ε)} = \frac{\partial u}{\partial T},$ {c_{(\varepsilon )}}{\kern 1pt} = {\kern 1pt} \frac{{\partial {\kern 1pt} u}}{{\partial T}}{\kern 1pt} , and from (42) it obtains (44) $c_{(ε)} = T \frac{d^{2} Ψ}{d T^{2}} .$ {c_{(\varepsilon )}}{\kern 1pt} = {\kern 1pt} T{\kern 1pt} \frac{{{d^2}{\kern 1pt} \Psi }}{{d{T^2}}}{\kern 1pt} . By integrating the equation (44), one has (45) $Ψ = c_{(ε)} T log (\frac{T}{T_{0}}) + s_{0}, T - c_{(ε)} (T - T_{0}) - u_{0} .$ \Psi {\kern 1pt} = {\kern 1pt} {c_{(\varepsilon )}}T{\kern 1pt} \log {\kern 1pt} {\kern 1pt} \left( {\frac{T}{{{T_0}}}} \right){\kern 1pt} + {\kern 1pt} {s_0},T - {c_{(\varepsilon )}}(T - {T_0}) - {u_0}{\kern 1pt} . If c_(ɛ) is constant where s₀ and u₀ are integration constants that are the specific entropy and the specific energy in the reference state, respectively.

From (39)_1,2,3 we obtain (46) $τ_{α β}^{(eq)} = a_{α β μ ν}^{(0) (0)} ε_{μ ν} + a_{α β μ ν}^{(0) (1)} ε_{μ ν}^{(1)} + a_{α β μ}^{(0) (ξ)} ξ_{μ} + a_{α β}^{(0) (T)} (T - T_{0}),$ \tau _{\alpha \beta }^{(eq)} = {\kern 1pt} a_{\alpha \beta \mu \nu }^{(0)(0)}{\kern 1pt} {\varepsilon _{\mu \nu }}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu \nu }^{(0)(1)}{\kern 1pt} {\kern 1pt} \varepsilon _{\mu \nu }^{(1)}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu }^{(0)(\xi )}{\kern 1pt} {\xi _\mu }{\kern 1pt} + {\kern 1pt} a_{\alpha \beta }^{(0)(T)}{\kern 1pt} (T - {T_0}){\kern 1pt} , (47) $- τ_{α β}^{(1)} = a_{α β μ ν}^{(1) (1)} ε_{μ ν}^{(1)} + a_{μ ν α β}^{(0) (1)} ε_{μ ν} + a_{α β μ}^{(1) (ξ)} ξ_{μ} + a_{α β}^{(1) (T)} (T - T_{0}),$ - {\kern 1pt} \tau _{\alpha \beta }^{(1)} = {\kern 1pt} a_{\alpha \beta \mu \nu }^{(1)(1)}{\kern 1pt} {\kern 1pt} \varepsilon _{\mu \nu }^{(1)}{\kern 1pt} + {\kern 1pt} a_{\mu \nu \alpha \beta }^{(0)(1)}{\kern 1pt} {\varepsilon _{\mu \nu }}{\kern 1pt} + {\kern 1pt} a_{\alpha \beta \mu }^{(1)(\xi )}{\kern 1pt} {\xi _\mu }{\kern 1pt} + {\kern 1pt} a_{\alpha \beta }^{(1)(T)}{\kern 1pt} (T - {T_0}), (48) $- j_{α} = a_{α β}^{(ξ) (ξ)} ξ_{β} + a_{α μ ν}^{(0) (ξ)} ε_{μ ν} + a_{α μ ν}^{(1) (ξ)} ε_{μ ν}^{(1)} + a_{α}^{(ξ) (T)} (T - T_{0}) .$ - {j_\alpha } = {\kern 1pt} a_{\alpha \beta }^{(\xi )(\xi )}{\kern 1pt} {\xi _\beta }{\kern 1pt} + {\kern 1pt} a_{\alpha \mu \nu }^{(0)(\xi )}{\kern 1pt} {\varepsilon _{\mu \nu }}{\kern 1pt} + {\kern 1pt} a_{\alpha \mu \nu }^{(1)(\xi )}{\kern 1pt} \varepsilon _{\mu \nu }^{(1)}{\kern 1pt} + {\kern 1pt} a_\alpha ^{(\xi )(T)}{\kern 1pt} (T - {T_0}){\kern 1pt} . We consider that the medium is isotropic and the even order tensors have the following forms (49) $a_{α β μ ν}^{(0) (0)} = \frac{1}{2} a^{(0, 0)} (δ_{α μ} δ_{β ν} + δ_{α ν} δ_{β μ}) + \frac{1}{3} (b^{(0, 0)} - a^{(0, 0)}) δ_{α β} δ_{μ ν},$ a_{\alpha \beta \mu \nu }^{(0)(0)} = \frac{1}{2}{\kern 1pt} {a^{(0,0)}}({\delta _{\alpha \mu }}{\kern 1pt} {\delta _{\beta \nu }} + {\delta _{\alpha \nu }}{\kern 1pt} {\delta _{\beta \mu }}) + \frac{1}{3}({b^{(0,0)}} - {a^{(0,0)}}){\delta _{\alpha \beta }}{\kern 1pt} {\delta _{\mu \nu }}, (50) $a_{α β μ ν}^{(0) (1)} = \frac{1}{2} a^{(0, 1)} (δ_{α μ} δ_{β ν} + δ_{α ν} δ_{β μ}) + \frac{1}{3} (b^{(0, 1)} - a^{(0, 1)}) δ_{α β} δ_{μ ν},$ a_{\alpha \beta \mu \nu }^{(0)(1)} = \frac{1}{2}{\kern 1pt} {a^{(0,1)}}({\delta _{\alpha \mu }}{\kern 1pt} {\delta _{\beta \nu }} + {\delta _{\alpha \nu }}{\kern 1pt} {\delta _{\beta \mu }}) + \frac{1}{3}({b^{(0,1)}} - {a^{(0,1)}}){\delta _{\alpha \beta }}{\kern 1pt} {\delta _{\mu \nu }}, (51) $a_{α β μ ν}^{(1) (1)} = \frac{1}{2} a^{(1, 1)} (δ_{α μ} δ_{β ν} + δ_{α ν} δ_{β μ}) + \frac{1}{3} (b^{(1, 1)} - a^{(1, 1)}) δ_{α β} δ_{μ ν},$ a_{\alpha \beta \mu \nu }^{(1)(1)} = \frac{1}{2}{\kern 1pt} {a^{(1,1)}}({\delta _{\alpha \mu }}{\kern 1pt} {\delta _{\beta \nu }} + {\delta _{\alpha \nu }}{\kern 1pt} {\delta _{\beta \mu }}) + \frac{1}{3}({b^{(1,1)}} - {a^{(1,1)}}){\delta _{\alpha \beta }}{\kern 1pt} {\delta _{\mu \nu }}, (52) $a_{μ ν α β}^{(0) (1)} = \frac{1}{2} a^{(0, 1)} (δ_{μ α} δ_{ν β} + δ_{μ β} δ_{ν α}) + \frac{1}{3} (b^{(0, 1)} - a^{(0, 1)}) δ_{μ ν} δ_{α β},$ a_{\mu \nu \alpha \beta }^{(0)(1)} = \frac{1}{2}{\kern 1pt} {a^{(0,1)}}({\delta _{\mu \alpha }}{\kern 1pt} {\delta _{\nu \beta }} + {\delta _{\mu \beta }}{\kern 1pt} {\delta _{\nu \alpha }}) + \frac{1}{3}({b^{(0,1)}} - {a^{(0,1)}}){\delta _{\mu \nu }}{\kern 1pt} {\delta _{\alpha \beta }}, (53) $a_{α β}^{(ξ) (ξ)} = a^{(ξ, ξ)} δ_{α β}, a_{α β}^{(0) (T)} = a^{(0, T)} δ_{α β}, a_{α β}^{(1) (T)} = a^{(1, T)} δ_{α β},$ a_{\alpha \beta }^{(\xi )(\xi )} = {\kern 1pt} {a^{(\xi ,\xi )}}{\delta _{\alpha \beta }},\quad a_{\alpha \beta }^{(0)(T)} = {\kern 1pt} {a^{(0,T)}}{\delta _{\alpha \beta }},\quad a_{\alpha \beta }^{(1)(T)} = {a^{(1,T)}}{\delta _{\alpha \beta }}{\kern 1pt} , and the odd tensors are zero. Using (49)–(53), the equations (46)–(48) become (54) $τ_{α β}^{(eq)} = a^{(0, 0)} ε_{α β} + (b^{(0, 0)} - a^{(0, 0)}) ε δ_{α β} + a^{(0, 1)} ε_{α β}^{(1)} + (b^{(0, 1)} - a^{(0, 1)}) ε^{(1)} δ_{α β} + a^{(0, T)} (T - T_{0}) δ_{α β} .$ \tau _{\alpha \beta }^{(eq)} = {\kern 1pt} {a^{(0,0)}}{\kern 1pt} {\varepsilon _{\alpha \beta }}{\kern 1pt} + ({b^{(0,0)}} - {a^{(0,0)}}){\kern 1pt} \varepsilon {\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} + {a^{(0,1)}}\varepsilon _{\alpha \beta }^{(1)} + ({b^{(0,1)}} - {a^{(0,1)}}){\kern 1pt} {\varepsilon ^{(1)}}{\delta _{\alpha \beta }} + {\kern 1pt} {a^{(0,T)}}{\kern 1pt} (T - {T_0}){\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} . (55) $τ_{α β}^{(1)} = a^{(1, 1)} ε_{α β}^{(1)} + (b^{(1, 1)} - a^{(1, 1)}) ε^{(1)} δ_{α β} + a^{(0, 1)} ε_{α β} + (b^{(0, 1)} - a^{(0, 1)}) ε δ_{α β} + a^{(1, T)} (T - T_{0}) δ_{α β},$ \tau _{\alpha \beta }^{(1)} = {\kern 1pt} {a^{(1,1)}}{\kern 1pt} \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} + ({b^{(1,1)}} - {a^{(1,1)}}){\kern 1pt} {\varepsilon ^{(1)}}{\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} + {a^{(0,1)}}{\varepsilon _{\alpha \beta }} + ({b^{(0,1)}} - {a^{(0,1)}}){\kern 1pt} \varepsilon {\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} + {\kern 1pt} {a^{(1,T)}}{\kern 1pt} (T - {T_0}){\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} , (56) $j_{α} = - a^{(ξ, ξ)} ξ_{α} .$ {j_\alpha } = - {a^{(\xi ,\xi )}}{\kern 1pt} {\xi _\alpha }{\kern 1pt} . The equation (54) is the generalization of the Duhamel-Neumann law for thermoelastic phenomena in isotropic media. If a^(0,T) vanishes or if one considers isothermal processes at the temperature T₀ is analogous to Hooke’s law for isotropic media [50]. The equation of state (55) and (56) are specific for the theory developed in this paper.

Stress-strain equations

In the following the deviator Ã_αβ and the scalar part A of arbitrary tensor field A_αβ are defined by (57) $\{\begin{array}{l} {\tilde{A}}_{α β} = A_{α β} - A δ_{α β}, \\ A = \frac{1}{3} Σ_{γ = 1}^{3} A_{γ γ} . \end{array}$ \left\{ {\begin{array}{*{20}{l}}{{{\tilde A}_{\alpha \beta }}{\kern 1pt} = {\kern 1pt} {A_{\alpha \beta }} - A{\kern 1pt} {\delta _{\alpha \beta }}{\kern 1pt} ,}\\{A = \frac{1}{3}{\kern 1pt} \Sigma _{\gamma = 1}^3{\kern 1pt} {A_{\gamma \gamma }}{\kern 1pt} .}\end{array}} \right. Using (54), (55), (31) and (32), we have (58) ${\tilde{τ}}_{α β}^{(eq)} = a^{(0, 0)} {\tilde{ε}}_{α β} + a^{(0, 1)} {\tilde{ε}}^{(1)}_{α β},$ \tilde \tau _{\alpha \beta }^{(eq)}{\kern 1pt} = {\kern 1pt} {a^{(0,0)}}{\kern 1pt} {\tilde \varepsilon _{\alpha \beta }}{\kern 1pt} + {\kern 1pt} {a^{(0,1)}}{\kern 1pt} {\tilde \varepsilon ^{(1)}}_{\alpha \beta }{\kern 1pt} , (59) ${\tilde{τ}}_{α β}^{(1)} = a^{(1, 1)} {\tilde{ε}}_{α β}^{(1)} + a^{(0, 1)} {\tilde{ε}}_{α β} .$ \tilde \tau _{\alpha \beta }^{(1)}{\kern 1pt} = {\kern 1pt} {a^{(1,1)}}{\kern 1pt} \tilde \varepsilon _{\alpha \beta }^{(1)}{\kern 1pt} + {\kern 1pt} {a^{(0,1)}}{\kern 1pt} {\tilde \varepsilon _{\alpha \beta }}{\kern 1pt} . (60) ${\tilde{τ}}_{α β}^{(vi)} = η_{s}^{(0, 0)} \frac{d {\tilde{ε}}_{α β}}{dt} + η_{s}^{(0, 1)} {\tilde{τ}}_{α β}^{(1)},$ \tilde \tau _{\alpha \beta }^{(vi)}{\kern 1pt} = {\kern 1pt} \eta _s^{(0,0)}{\kern 1pt} \frac{{d{\kern 1pt} {{\tilde \varepsilon }_{\alpha \beta }}}}{{dt}}{\kern 1pt} + {\kern 1pt} \eta _s^{(0,1)}{\kern 1pt} \tilde \tau _{\alpha \beta }^{(1)}{\kern 1pt} , (61) $\frac{d {\tilde{ε}}^{(1)}_{α β}}{dt} = η_{s}^{(1, 0)} \frac{d {\tilde{ε}}_{α β}}{dt} + η_{s}^{(1, 1)} {\tilde{τ}}_{α β}^{(1)} .$ \frac{{d{{\tilde \varepsilon }^{(1)}}_{\alpha \beta }}}{{dt}}{\kern 1pt} = {\kern 1pt} \eta _s^{(1,0)}{\kern 1pt} \frac{{d{{\tilde \varepsilon }_{\alpha \beta }}}}{{dt}}{\kern 1pt} + {\kern 1pt} \eta _s^{(1,1)}{\kern 1pt} \tilde \tau _{\alpha \beta }^{(1)}{\kern 1pt} . The equations (58) and (59) are generalization of the Duhamel-Neuman law for thermoviscoanelastic phenomena in isotropic media. From (10), using (58) and (60), we have (62) ${\tilde{τ}}_{α β} = χ_{s}^{(1)} {\tilde{ε}}_{α β} + χ_{s}^{(2)} {\tilde{ε}}^{(1)}_{α β} + η_{s}^{(0, 0)} \frac{d {\tilde{ε}}_{α β}}{dt},$ {\tilde \tau _{\alpha \beta }}{\kern 1pt} = {\kern 1pt} \chi _s^{(1)}{\kern 1pt} {\tilde \varepsilon _{\alpha \beta }}{\kern 1pt} + {\kern 1pt} \chi _s^{(2)}{\kern 1pt} {\tilde \varepsilon ^{(1)}}_{\alpha \beta }{\kern 1pt} + {\kern 1pt} \eta _s^{(0,0)}{\kern 1pt} \frac{{d{{\tilde \varepsilon }_{\alpha \beta }}}}{{dt}}{\kern 1pt} , where (63) $χ_{s}^{(1)} = a^{(0, 0)} - a^{(0, 1)} η_{s}^{(0, 1)},$ \chi _s^{(1)}{\kern 1pt} = {\kern 1pt} {a^{(0,0)}}{\kern 1pt} - {\kern 1pt} {a^{(0,1)}}{\kern 1pt} \eta _s^{(0,1)}{\kern 1pt} , (64) $χ_{s}^{(2)} = a^{(0, 1)} - a^{(1, 1)} η_{s}^{(0, 1)} .$ \chi _s^{(2)}{\kern 1pt} = {\kern 1pt} {a^{(0,1)}}{\kern 1pt} - {\kern 1pt} {a^{(1,1)}}{\kern 1pt} \eta _s^{(0,1)}{\kern 1pt} . Eliminating $τ_{α β}^{(1)}$ \tau _{\alpha \beta }^{(1)} between (61) and (59), we obtain (65) $\frac{d {\tilde{ε}}_{α β}^{(1)}}{dt} = η_{s}^{(1, 0)} \frac{d {\tilde{ε}}_{α β}}{dt} - η_{s}^{(1, 1)} a^{(1, 1)} {\tilde{ε}}_{α β}^{(1)} - η_{s}^{(1, 1)} a^{(0, 1)} {\tilde{ε}}_{α β},$ \frac{{d\tilde \varepsilon _{\alpha \beta }^{(1)}}}{{dt}}{\kern 1pt} = {\kern 1pt} \eta _s^{(1,0)}{\kern 1pt} \frac{{d{{\tilde \varepsilon }_{\alpha \beta }}}}{{dt}}{\kern 1pt} - {\kern 1pt} \eta _s^{(1,1)}{\kern 1pt} {a^{(1,1)}}{\kern 1pt} \tilde \varepsilon _{\alpha \beta }^{(1)} - {\kern 1pt} \eta _s^{(1,1)}{\kern 1pt} {a^{(0,1)}}{\kern 1pt} {\tilde \varepsilon _{\alpha \beta }}{\kern 1pt} , The equation (65), using ${\tilde{ε}}_{α β}^{(1)}$ \tilde \varepsilon _{\alpha \beta }^{(1)} from (62), becomes (66) $\frac{d {\tilde{τ}}_{α β}}{dt} + R_{(d) 0}^{(τ)} {\tilde{τ}}_{α β} = R_{(d) 2}^{(ε)} \frac{d^{2} {\tilde{ε}}_{α β}}{d t^{2}} + R_{(d) 1}^{(ε)} \frac{d {\tilde{ε}}_{α β}}{dt} + R_{(d) 0}^{(ε)} {\tilde{ε}}_{α β},$ \frac{{d{{\tilde \tau }_{\alpha \beta }}}}{{dt}}{\kern 1pt} + {\kern 1pt} R_{(d)0}^{(\tau )}{\kern 1pt} {\tilde \tau _{\alpha \beta }}{\kern 1pt} = {\kern 1pt} R_{(d)2}^{(\varepsilon )}{\kern 1pt} \frac{{{d^2}{{\tilde \varepsilon }_{\alpha \beta }}}}{{d{t^2}}}{\kern 1pt} + {\kern 1pt} R_{(d)1}^{(\varepsilon )}{\kern 1pt} \frac{{d{{\tilde \varepsilon }_{\alpha \beta }}}}{{dt}}{\kern 1pt} + {\kern 1pt} R_{(d)0}^{(\varepsilon )}{\kern 1pt} {\tilde \varepsilon _{\alpha \beta }}{\kern 1pt} , where (67) $R_{(d) 0}^{(τ)} = η_{s}^{(1, 1)} a^{(1, 1)},$ R_{(d)0}^{(\tau )}{\kern 1pt} = {\kern 1pt} \eta _s^{(1,1)}{\kern 1pt} {a^{(1,1)}}, (68) $R_{(d) 2}^{(ε)} = η_{s}^{(0, 0)},$ R_{(d)2}^{(\varepsilon )}{\kern 1pt} = {\kern 1pt} \eta _s^{(0,0)}, (69) $R_{(d) 1}^{(ε)} = χ_{s}^{(1)} + η_{s}^{(1, 0)} χ_{s}^{(2)} + η_{s}^{(0, 0)} η_{s}^{(1, 1)} a^{(1, 1)},$ R_{(d)1}^{(\varepsilon )}{\kern 1pt} = {\kern 1pt} \chi _s^{(1)}{\kern 1pt} + \eta _s^{(1,0)}{\kern 1pt} \chi _s^{(2)}{\kern 1pt} + {\kern 1pt} \eta _s^{(0,0)}{\kern 1pt} \eta _s^{(1,1)}{\kern 1pt} {a^{(1,1)}}{\kern 1pt} , (70) $R_{(d) 0}^{(ε)} = η_{s}^{(1, 1)} (χ_{s}^{(1)} a^{(1, 1)} - a^{(0, 1)} χ_{s}^{(2)}) .$ R_{(d)0}^{(\varepsilon )}{\kern 1pt} = {\kern 1pt} \eta _s^{(1,1)}(\chi _s^{(1)}{\kern 1pt} {a^{(1,1)}}{\kern 1pt} - {a^{(0,1)}}{\kern 1pt} \chi _s^{(2)}). Let’s examine the scalar parts of the tensors. Using (54), (55), (31) and (32) we have (71) $τ^{(eq)} = b^{(0, 0)} ε + b^{(0, 1)} ε^{(1)} + a^{(0, T)} (T - T_{0})$ {\tau ^{(eq)}} = {\kern 1pt} {b^{(0,0)}}{\kern 1pt} \varepsilon {\kern 1pt} + {\kern 1pt} {b^{(0,1)}}{\kern 1pt} {\varepsilon ^{(1)}} + {a^{(0,T)}}(T - {T_0}) (72) $- τ^{(1)} = b^{(1, 1)} ε^{(1)} + b^{(0, 1)} ε + a^{(1, T)} (T - T_{0}),$ - {\tau ^{(1)}}{\kern 1pt} = {\kern 1pt} {b^{(1,1)}}{\kern 1pt} {\varepsilon ^{(1)}}{\kern 1pt} + {\kern 1pt} {b^{(0,1)}}{\kern 1pt} \varepsilon {\kern 1pt} + {\kern 1pt} {a^{(1,T)}}{\kern 1pt} (T - {T_0}){\kern 1pt} , (73) $τ^{(vi)} = η_{v}^{(0, 0)} \frac{d ε}{dt} + η_{v}^{(0, 1)} τ^{(1)},$ {\tau ^{(vi)}}{\kern 1pt} = {\kern 1pt} \eta _v^{(0,0)}{\kern 1pt} \frac{{d{\kern 1pt} \varepsilon }}{{dt}}{\kern 1pt} + {\kern 1pt} \eta _v^{(0,1)}{\kern 1pt} {\tau ^{(1)}}{\kern 1pt} , (74) $\frac{d ε^{(1)}}{dt} = η_{v}^{(1, 0)} \frac{d ε}{dt} + η_{v}^{(1, 1)} τ^{(1)} .$ \frac{{d{\varepsilon ^{(1)}}}}{{dt}}{\kern 1pt} = {\kern 1pt} \eta _v^{(1,0)}{\kern 1pt} \frac{{d\varepsilon }}{{dt}}{\kern 1pt} + {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} {\tau ^{(1)}}{\kern 1pt} . As τ = τ^(eq) + τ^(vi), using (71), (73) and (72) we have: (75) $τ = χ_{v}^{(1)} ε + χ_{v}^{(2)} ε^{(1)} + η_{v}^{(0, 0)} \frac{d ε}{dt} + χ_{v}^{(T)} (T - T_{0}),$ \tau {\kern 1pt} = {\kern 1pt} \chi _v^{(1)}{\kern 1pt} \varepsilon {\kern 1pt} + {\kern 1pt} \chi _v^{(2)}{\kern 1pt} {\varepsilon ^{(1)}}{\kern 1pt} + {\kern 1pt} \eta _v^{(0,0)}{\kern 1pt} \frac{{d\varepsilon }}{{dt}}{\kern 1pt} + {\kern 1pt} \chi _v^{(T)}{\kern 1pt} (T - {T_0}){\kern 1pt} , with (76) $χ_{v}^{(1)} = b^{(0, 0)} - η_{v}^{(0, 1)} b^{(0, 1)},$ \chi _v^{(1)}{\kern 1pt} = {\kern 1pt} {b^{(0,0)}}{\kern 1pt} - {\kern 1pt} \eta _v^{(0,1)}{\kern 1pt} {b^{(0,1)}}{\kern 1pt} , (77) $χ_{v}^{(2)} = b^{(0, 1)} - η_{v}^{(0, 1)} b^{(1, 1)},$ \chi _v^{(2)}{\kern 1pt} = {\kern 1pt} {b^{(0,1)}}{\kern 1pt} - {\kern 1pt} \eta _v^{(0,1)}{\kern 1pt} {b^{(1,1)}}{\kern 1pt} , (78) $χ_{v}^{(T)} = a^{(0, T)} - η_{v}^{(0, 1)} a^{(1, T)} .$ \chi _v^{(T)}{\kern 1pt} = {\kern 1pt} {a^{(0,T)}}{\kern 1pt} - {\kern 1pt} \eta _v^{(0,1)}{\kern 1pt} {a^{(1,T)}}{\kern 1pt} . Using (72), the equation (74) becomes (79) $\frac{d ε^{(1)}}{dt} = η_{v}^{(1, 0)} \frac{d ε}{dt} - η_{v}^{(1, 1)} b^{(1, 1)} ε^{(1)} - η_{v}^{(1, 1)} b^{(0, 1)} ε - η_{v}^{(1, 1)} a^{(0, T)} (T - T_{0}) .$ \frac{{d{\kern 1pt} {\varepsilon ^{(1)}}}}{{dt}}{\kern 1pt} = {\kern 1pt} \eta _v^{(1,0)}{\kern 1pt} \frac{{d{\kern 1pt} \varepsilon }}{{dt}}{\kern 1pt} - {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} {b^{(1,1)}}{\kern 1pt} {\varepsilon ^{(1)}}{\kern 1pt} - {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} {b^{(0,1)}}{\kern 1pt} \varepsilon {\kern 1pt} - {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} {a^{(0,T)}}{\kern 1pt} (T - {T_0}){\kern 1pt} . From (75), we get (80) $η_{v}^{(2)} ε^{(1)} = τ - χ_{v}^{(1)} ε - η_{v}^{(0, 0)} \frac{d ε}{dt} - χ_{v}^{(T)} (T - T_{0}),$ \eta _v^{(2)}{\kern 1pt} {\varepsilon ^{(1)}}{\kern 1pt} = {\kern 1pt} \tau {\kern 1pt} - {\kern 1pt} \chi _v^{(1)}{\kern 1pt} \varepsilon {\kern 1pt} - {\kern 1pt} \eta _v^{(0,0)}{\kern 1pt} \frac{{d\varepsilon }}{{dt}}{\kern 1pt} - {\kern 1pt} \chi _v^{(T)}{\kern 1pt} (T - {T_0}){\kern 1pt} , and replacing in (79) we have (81) $\frac{d τ}{dt} + R_{(v) 0}^{(τ)} τ = R_{(v) 2}^{(ε)} \frac{d^{2} ε}{d t^{2}} + R_{(v) 1}^{(ε)} \frac{d ε}{dt} + R_{(v) 0}^{(ε)} ε + R_{(v) 1}^{(T)} \frac{d T}{dt} + R_{(v) 0}^{(T)} (T - T_{0})$ \frac{{d{\kern 1pt} \tau }}{{dt}}{\kern 1pt} + {\kern 1pt} R_{(v)0}^{(\tau )}{\kern 1pt} \tau {\kern 1pt} = {\kern 1pt} R_{(v)2}^{(\varepsilon )}{\kern 1pt} \frac{{{d^2}\varepsilon }}{{d{t^2}}}{\kern 1pt} + {\kern 1pt} R_{(v)1}^{(\varepsilon )}{\kern 1pt} \frac{{d\varepsilon }}{{dt}}{\kern 1pt} + {\kern 1pt} R_{(v)0}^{(\varepsilon )}{\kern 1pt} \varepsilon {\kern 1pt} + {\kern 1pt} R_{(v)1}^{(T)}{\kern 1pt} \frac{{d{\kern 1pt} T}}{{dt}}{\kern 1pt} + {\kern 1pt} {\kern 1pt} R_{(v)0}^{(T)}{\kern 1pt} (T - {T_0}){\kern 1pt} where (82) $R_{(v) 0}^{(τ)} = b^{(1, 1)} η_{v}^{(1, 1)},$ R_{(v)0}^{(\tau )}{\kern 1pt} = {\kern 1pt} {b^{(1,1)}}{\kern 1pt} \eta _v^{(1,1)}, (83) $R_{(v) 2}^{(ε)} = η_{v}^{(0, 0)},$ R_{(v)2}^{(\varepsilon )}{\kern 1pt} = {\kern 1pt} \eta _v^{(0,0)}, (84) $R_{(v) 1}^{(ε)} = χ_{v}^{(1)} + η_{v}^{(1, 0)} χ_{v}^{(2)} + η_{v}^{(1, 1)} b^{(1, 1)} η_{v}^{(0, 0)},$ R_{(v)1}^{(\varepsilon )}{\kern 1pt} = {\kern 1pt} \chi _v^{(1)}{\kern 1pt} + {\kern 1pt} \eta _v^{(1,0)}{\kern 1pt} \chi _v^{(2)}{\kern 1pt} + {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} {b^{(1,1)}}{\kern 1pt} \eta _v^{(0,0)}{\kern 1pt} , (85) $R_{(v) 0}^{(ε)} = η_{v}^{(1, 1)} (b^{(1, 1)} χ_{v}^{(1)} - b^{(0, 0)} χ_{v}^{(2)}),$ R_{(v)0}^{(\varepsilon )}{\kern 1pt} = {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} ({b^{(1,1)}}{\kern 1pt} \chi _v^{(1)}{\kern 1pt} - {\kern 1pt} {b^{(0,0)}}{\kern 1pt} \chi _v^{(2)}), (86) $R_{(v) 1}^{(T)} = χ_{v}^{(T)},$ R_{(v)1}^{(T)}{\kern 1pt} = {\kern 1pt} \chi _v^{(T)}{\kern 1pt} , (87) $R_{(v) 0}^{(T)} = η_{v}^{(1, 1)} (b^{(1, 1)} χ_{v}^{(T)} - a^{(1, T)} χ_{v}^{(2)}) .$ R_{(v)0}^{(T)}{\kern 1pt} = {\kern 1pt} \eta _v^{(1,1)}{\kern 1pt} ({b^{(1,1)}}\chi _v^{(T)} - {a^{(1,T)}}\chi _v^{(2)}). The equations (66) and (81) generalize the stress-strain relations for ordinary viscous fluids, for elastic media end for Maxwell, Kelvin, Jeffreys and Poynting-Thomson bodies.

Heat flows in visco-anelastic processes

Using the phenomenological equations (Section 5) and the linear equations of state (Section 6) we obtain the following results:

Theorem 3 (Decomposition of the heat flow)

By using equation (56) and (33) the heat flow J^(q) can be split in two parts (88) $J^{(q)} = J^{(0)} + J^{(1)},$ {{\boldsymbol J}^{(q)}} = {{\boldsymbol J}^{(0)}}{\kern 1pt} + {\kern 1pt} {{\boldsymbol J}^{(1)}}{\kern 1pt} , where J⁽⁰⁾ satisfies the Fourier’s law and J⁽¹⁾ obeys the Maxwell-Cattaneo-Vernotte law.

Proof

Assuming λ^(ξ)(ξ) ≠ 0, by virtue of equation (56), from the equation (33) we obtain (89) $\frac{d ξ_{α}}{dt} = - \frac{a^{(ξ, ξ)}}{λ^{(ξ, ξ)}} ξ_{α} + \frac{λ^{(ξ, q)}}{λ^{(ξ, ξ)}} \frac{\partial T}{\partial x^{α}} .$ \frac{{d{\xi _\alpha }}}{{dt}} = - \frac{{{a^{(\xi ,\xi )}}}}{{{\lambda ^{(\xi ,\xi )}}}}{\kern 1pt} {\xi _\alpha }{\kern 1pt} + {\kern 1pt} \frac{{{\lambda ^{(\xi ,q)}}}}{{{\lambda ^{(\xi ,\xi )}}}}\frac{{\partial T}}{{\partial {x^\alpha }}}{\kern 1pt} . Substituting the equation (89) into equation (30), one obtains (90) $J^{(q)} = J^{(0)} + J^{(1)},$ {{\bf J}^{(q)}} = {{\bf J}^{(0)}}{\kern 1pt} + {\kern 1pt} {{\bf J}^{(1)}}{\kern 1pt} , where (91) $J^{(0)} = - λ^{(0)} \nabla T,$ {{\bf J}^{(0)}} = {\kern 1pt} - {\lambda ^{(0)}}{\kern 1pt} \nabla T{\kern 1pt} , with (92) $λ^{(0)} = λ^{(q, q)} - \frac{λ^{(q, ξ)} λ^{(ξ, q)}}{λ^{(ξ, ξ)}},$ {\lambda ^{(0)}}{\kern 1pt} = {\kern 1pt} {\lambda ^{(q,q)}}{\kern 1pt} - {\kern 1pt} \frac{{{\lambda ^{(q,\xi )}}{\kern 1pt} {\lambda ^{(\xi ,q)}}}}{{{\lambda ^{(\xi ,\xi )}}}}{\kern 1pt} , and (93) $J^{(1)} = - \frac{λ^{(q, ξ)} a^{(ξ, ξ)}}{λ^{(ξ, ξ)}} ξ .$ {{\bf J}^{(1)}} = - \frac{{{\lambda ^{(q,\xi )}}{\kern 1pt} {a^{(\xi ,\xi )}}}}{{{\lambda ^{(\xi ,\xi )}}}}{\kern 1pt} {\boldsymbol \xi} {\kern 1pt} . By virtue of equation (89), we see that J⁽¹⁾ satisfies the following equation: (94) $t_{r} \frac{d J^{(1)}}{dt} + J^{(1)} = - \frac{λ^{(ξ, q)} λ^{(q, ξ)}}{λ^{(ξ, ξ)}} \nabla T,$ {t_r}{\kern 1pt} \frac{{d{{\bf J}^{(1)}}}}{{dt}} + {{\bf J}^{(1)}}{\kern 1pt} = {\kern 1pt} - \frac{{{\lambda ^{(\xi ,q)}}{\kern 1pt} {\lambda ^{(q,\xi )}}}}{{{\lambda ^{(\xi ,\xi )}}}}{\kern 1pt} \nabla T{\kern 1pt} , where (95) $t_{r} = \frac{λ^{(ξ, ξ)}}{a^{(ξ, ξ)}},$ {t_r}{\kern 1pt} = {\kern 1pt} \frac{{{\lambda ^{(\xi ,\xi )}}}}{{{a^{(\xi ,\xi )}}}}{\kern 1pt} , is the relaxation time.

From (90) we can see that the heat current is split into two parts: the first one, J⁽⁰⁾, is governed by Fourier’s law (91) while the second part, J⁽¹⁾, is governed by the MCV type constitutive equation (94).

Temperature equation

We consider an undeformable medium at rest, i.e. (96) $\frac{d}{dt} = \frac{\partial}{\partial t},$ \frac{d}{{dt}} = \frac{\partial }{{\partial t}}, and we assume that the specific internal energy u is related to the temperature by (97) $du = c_{(v)} dT,$ du = {c_{(v)}}dT{\kern 1pt} , being c_(v) the heat capacity per unit of mass at constant volume.

Theorem 4 (Generalization of temperature equation)

By using (91) and the first law of thermodynamics we obtain the following temperature equation (98) $t_{r} \frac{\partial^{2} T}{\partial t^{2}} + \frac{\partial T}{\partial t} = α^{'} Δ T + t_{r} η^{'} \frac{\partial}{\partial t} Δ T,$ {t_r}\frac{{{\partial ^2}T}}{{\partial {t^2}}} + \frac{{\partial T}}{{\partial t}} = \alpha '\Delta T{\kern 1pt} + {\kern 1pt} {t_r}{\kern 1pt} \eta '\frac{\partial }{{\partial t}}\Delta T, where α′ and η′ are parameters expressed through the phenomenological and state coefficients.

Proof

By virtue of (96) and (97) the first law of thermostatic (12) becomes (99) $ρ c_{(v)} \frac{\partial T}{\partial t} + \nabla \cdot (J^{(0)} + J^{(1)}) = 0,$ \rho {\kern 1pt} {c_{(v)}}{\kern 1pt} \frac{{\partial T}}{{\partial t}}{\kern 1pt} + {\kern 1pt} \nabla \cdot ({{\bf J}^{(0)}} + {{\bf J}^{(1)}}) = 0, and from (99) we obtain (100) $ρ c_{(v)} t_{r} \frac{\partial^{2} T}{\partial t^{2}} + t_{r} \nabla \cdot [\frac{\partial J^{(0)}}{\partial t} + \frac{\partial J^{(1)}}{\partial t}] = 0 .$ \rho {\kern 1pt} {c_{(v)}}{\kern 1pt} {t_r}{\kern 1pt} \frac{{{\partial ^2}T}}{{\partial {t^2}}}{\kern 1pt} + {\kern 1pt} {t_r}{\kern 1pt} \nabla \cdot \left[ {\frac{{\partial {{\bf J}^{(0)}}}}{{\partial t}} + \frac{{\partial {{\bf J}^{(1)}}}}{{\partial t}}} \right] = 0{\kern 1pt} . Summing (99) and (100), by using (91) and (94), we have the following temperature equation (101) $t_{r} \frac{\partial^{2} T}{\partial t^{2}} + \frac{\partial T}{\partial t} = α^{'} Δ T + t_{r} η^{'} \frac{\partial Δ T}{\partial t},$ {t_r}{\kern 1pt} \frac{{{\partial ^2}T}}{{\partial {t^2}}}{\kern 1pt} + {\kern 1pt} \frac{{\partial T}}{{\partial t}} = {\kern 1pt} \alpha '{\kern 1pt} \Delta T{\kern 1pt} + {\kern 1pt} {t_r}{\kern 1pt} \eta '{\kern 1pt} \frac{{\partial \Delta T}}{{\partial t}}, where (102) $\{\begin{array}{l} α^{'} = \frac{λ^{(0)} λ^{(ξ, ξ)} + λ^{(ξ, q)} λ^{(q, ξ)}}{ρ c_{(v)} λ^{(ξ, ξ)}} \\ η^{'} = \frac{λ^{(0)}}{ρ c_{(v)}} . \end{array}$ \left\{ {\begin{array}{*{20}{l}}{\alpha '{\kern 1pt} = {\kern 1pt} \frac{{{\lambda ^{(0)}}{\lambda ^{(\xi ,\xi )}} + {\lambda ^{(\xi ,q)}}{\lambda ^{(q,\xi )}}}}{{\rho {\kern 1pt} {c_{(v)}}{\kern 1pt} {\lambda ^{(\xi ,\xi )}}}}}\\{\eta ' = \frac{{{\lambda ^{(0)}}}}{{\rho {\kern 1pt} {c_{(v)}}}}{\kern 1pt} .}\end{array}} \right. The equation (101) generalizes the temperature equation with analogous equations of Fourier and Maxwell-Cattaneo-Vernotte.

9.1

Special cases

If the relaxation time t_r (95) can be neglected with respect to the heat propagation time t (t_r ≪ t) the equation (101) becomes (103) $\frac{\partial T}{\partial t} = α^{'} Δ T .$ \frac{{\partial T}}{{\partial t}} = \alpha '\Delta T{\kern 1pt} . i.e. the Fourier equation.

• If the phenomenological coefficients satisfy the following condition (104) $λ^{(q, q)} = \frac{λ^{(q, ξ)}, λ^{(ξ, q)}}{λ^{(ξ, ξ)}},$ {\lambda ^{(q,q)}} = \frac{{{\lambda ^{(q,\xi )}},{\lambda ^{(\xi ,q)}}}}{{{\lambda ^{(\xi ,\xi )}}}}{\kern 1pt} , from (92) one has: λ⁽⁰⁾ = 0 and the equations (102) become (105) $\{\begin{array}{l} α^{'} = \frac{λ^{(ξ, q)} λ^{(q, ξ)}}{ρ c_{(v)} λ^{(ξ, ξ)}} \\ η^{'} = 0 \end{array},$ \left\{ {\begin{array}{*{20}{l}}{\alpha '{\kern 1pt} = {\kern 1pt} \frac{{{\lambda ^{(\xi ,q)}}{\lambda ^{(q,\xi )}}}}{{\rho {\kern 1pt} {c_{(v)}}{\kern 1pt} {\lambda ^{(\xi ,\xi )}}}}}\\{\eta ' = 0}\end{array}} \right.{\kern 1pt} , and the equation (101) becomes (106) $t_{r} \frac{\partial^{2} T}{\partial t^{2}} + \frac{\partial T}{\partial t} - α^{'} Δ T = 0,$ {t_r}{\kern 1pt} \frac{{{\partial ^2}T}}{{\partial {t^2}}}{\kern 1pt} + {\kern 1pt} \frac{{\partial T}}{{\partial t}}{\kern 1pt} - {\kern 1pt} \alpha '{\kern 1pt} \Delta T{\kern 1pt} = {\kern 1pt} 0{\kern 1pt} , being the Maxwell-Cattaneo-Vernotte.

Remarks and conclusion

The Fourier equation (103) is a parabolic equation and therefore the heat flux can be considered of infinite speed. Several scientists view this result as a paradox. In 1992, Fichera [51] states that this claim is unfounded as Fourier’s theory is not correctly interpreted. In our opinion the Fourier result is justified by the value of the phenomenological parameters of the medium and therefore it cannot be said that Fourier’s theory leads to a paradox.

The Maxwell-Cattaneo-Vernotte law leads to a hyperbolic equation obtained with intuitive hypotheses but without justifying it in the context of a general theory. Various authors, for instance [41,42,43], have proposed several theories suggested by kinetic theory in order to justify the Maxwell-Cattaneo-Vernotte equation by assuming that entropy depends on fluxes instead of state variables.

In this paper, without a priori hypotheses and the important role of phenomenological and state coefficients, we obtained the stress-strain relations for viscoanelastic media and an equation for the temperature which generalizes the Fourier and MCV temperature equations.

Declarations

11.1

Conflict of interest

There is no conflict of interests regarding the publication of this paper.

11.2

Funding

There is no funding regarding the publication of this paper.

11.3

Author’s contribution

V.C.-Writing original draft, Writing review editing, Methodology.

11.4

Acknowledgement

The author deeply appreciates the anonymous reviewers for their helpful and constructive suggestions, which can help further improve this paper.

11.5

Data availability statement

All data that support the findings of this study are included within the article.

11.6

Using of AI tools

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

eISSN:: 2956-7068
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Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables

Article Category: Original Study

Publié en ligne: 10 janv. 2024

Pages: 141 - 154

Reçu: 27 sept. 2023

Accepté: 03 nov. 2023

DOI: https://doi.org/10.2478/ijmce-2024-0011

Mots clésNon-equilibrium thermodynamic, rheology, viscoanelastic media, heat flow

© 2024 Vincenzo Ciancio, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
Non-equilibrium thermodynamic, rheology, viscoanelastic media, heat flow