Heat-Flux Relaxation and the Possibility of Spatial Interactions in Higher-Grade Materials

Abstract

We investigate the thermodynamic compatibility of weakly nonlocal materials with constitutive equations depending on the third spatial gradient of the deformation and the heat flux ruled by an independent balance law. In such materials, the molecules experience long-range interactions. Examples of biological systems undergoing nonlocal interactions are given. Under the hypothesis of weak nonlocality (constitutive equations depending on the gradients of the unknown fields), we exploit the second law of thermodynamics by considering the spatial differential consequences (gradients) of the balance laws as additional equations to be substituted into the entropy inequality, up to the order of the gradients entering the state space. As a consequence of such a procedure, we obtain generalized constitutive laws for the stress tensor and the specific entropy, as well as new forms of the balance equations. Such equations are, in general, parabolic, although hyperbolic situations are also possible. For small deformations of homogeneous and isotropic bodies, under the validity of a generalized Maxwell–Cattaneo equation for the heat flux, which depends on the deformation too, we study the propagation of small-amplitude thermomechanical waves, proving that mechanical, thermal and thermomechanical waves are possible.

1. Introduction

An elastic material of grade N is a deformable continuum in which long-range spatial interactions are modeled by permitting the constitutive functionals to depend not only on the first gradient of the deformation, the strain, but also on all gradients of the deformation less than or equal to the integer N. An elastic material of grade N with heat conduction and viscosity is said to be thermoviscoelastic material of grade N [1]. Several different approaches to the problem of heat conduction in thermoelastic systems can be found in the literature, especially those devoted to the propagation of thermomechanical disturbances with finite speed [2]. The latter are generally known as models of generalized thermoelasticity [3,4,5,6].

The propagation of temperature waves with finite speed (second sound) is one of the fundamental problems that motivated the foundation and development of Extended Irreversible Thermodynamics (EIT), i.e., the thermodynamic theory which upgrades the dissipative fluxes (as, for example, the heat flux) to the role of independent thermodynamic variables [2,7,8]. Such a theory is capable of modeling both parabolic and hyperbolic theories of heat conduction such as Fourier theory [9], Maxwell–Cattaneo theory [10], and Guyer–Krumhansl theory [11], in which the heat flux is ruled by the parabolic equation

$${\tau }_R\dot{q_I}+q_I=-\kappa {\vartheta }_{,I}+{\displaystyle \frac{9}{5}}{\displaystyle \frac{\kappa {\tau }_N}{c}}\left(q_{I,KK}+2q_{K,KI}\right).$$

(1)

In Equation (1), $\vartheta$ denotes the absolute temperature, $q_I,I=1,2,3,$ are the components of the heat-flux vector, $\kappa$ is the thermal conductivity, c is the specific heat, and ${\tau }_N$ and ${\tau }_R$ are the relaxation times related to normal and resistive phonon scatterings [2,11], respectively. Finally, the symbols $\dot{f}$ and $f_{,J}$ denote the partial derivatives of function f with respect to time t and spatial coordinates $X_J$, respectively, and the Einstein convention of summation over repeated indices has been adopted. It is worth noticing that $\ell =\sqrt{{\displaystyle \frac{9}{5}}{\displaystyle \frac{\kappa {\tau }_N}{c}}}$ denotes the mean free path of the phonons, which in solids are the heat carriers. When ℓ is negligible, Equation (1) reduces to the Maxwell–Cattaneo equation

$${\tau }_R\dot{q_I}+q_I=-\kappa {\vartheta }_{,I},$$

(2)

which allows for heat propagation with finite speed [10]. Equation (2), in turn, yields the classical Fourier law

$$q_I=-\kappa {\vartheta }_{,I},$$

(3)

when ${\tau }_R$ is negligible [9].

It is worth noticing that Christov and Jordan [12] have shown that the Maxwell–Cattaneo equation governing the propagation of second sound should involve a material time derivative of the heat flux instead of a partial time derivative. One could wonder if it is possible to set up such a Maxwell–Cattaneo equation from the free energy and the dissipation function. Such a problem has been considered by Ostoja-Starzewski in [13], who proved that this is possible within a more general interpretation of Edelen representation theory [14]. All the equations above can be coupled with the classical equations of viscoelasticity in order to model thermoviscoelastic behavior. Thermoviscoelastic behaviors are observed in biological networks and gels, viscoelastic phase separation, formations of biomembranes and soft biomaterials, colloidal and protein systems [15,16]. In all such cases, the Maxwell–Cattaneo-type evolution equation for the heat flux can also be written by employing different types of time derivatives [15,16] related to the stress relaxation in complex thermoviscoelastic circumstances, for instance, the fractional derivative [15,16].

Within the framework of Rational Thermodynamics (RT) [17], with the heat flux as a constitutive quantity, thermoviscoelastic solids have been studied extensively by Dunn and Serrin [1]. These authors postulated the following state space (see Equation (1.17) in [1]):

$$\overline{\Sigma }=\left\{\vartheta ,{\vartheta }_{,K},F_{iL},F_{iL,K},F_{iL,KM},{\dot{F}}_{iK}\right\}$$

(4)

with $\vartheta$ as the absolute temperature of a continuous body $\mathcal{B}$ in a given reference configuration ${\mathcal{C}}^{*}$, $F_{iL}\equiv \partial {\chi }_i/\partial X_L$ as the components of the deformation tensor, where ${\chi }_i\left(X_L\right)\equiv x_i\left(X_L\right)$ are the components of the transplacement of the points of ${\mathcal{C}}^{*}$ from their reference places $X_L$ to their actual places $x_i\left(X_L\right)$, in the actual configuration $\mathcal{C}$, [18]. The state space (4) may represent materials such that their molecules experience long-range interactions as, for instance, nanosystems [2]. In such a case, due to the reduced dimension, any material point can interact with all the other points of the system. According to the classical tenets of RT, the materials with the state space given by Equation (4) are, in general, incompatible with the second law of thermodynamics. To overcome that problem, Dunn and Serrin postulated a new form of the local balance of energy given by

$$\varrho \dot{\epsilon }+q_{I,I}-T_{iL}{\dot{F}}_{iL}-\varrho r=u_{I,I},$$

(5)

wherein $\varrho$ is the referential mass density, $T_{iL}$ are the components of the first Piola–Kirchhoff stress tensor, $\epsilon$ is the referential specific internal energy, $q_I$ are the components of the referential heat flux, r is the local heat supply, and $u_I$ are the components of an additional flux of mechanical energy, the interstitial working, engendered by long-range spatial interactions [1]. In the classical form of the energy balance, the flux $u_I$ is not present [19]. The local balances of linear momentum and entropy conserve, instead, their classical form, namely

$$\varrho {\ddot{x}}_i-T_{iL,L}=\varrho b_i,$$

(6)

$$\varrho \dot{s}+{(q_I/\vartheta )}_{,I}\ge \varrho r/\vartheta ,$$

(7)

with $b_i$ as the components of the body force, and s as the specific entropy [19].

In recent years, we proposed an alternative approach to the above problem that generalizes the classical exploitation method of the entropy inequality developed by Coleman and Noll in 1963 [19], which does not need to modify the fundamental thermodynamic laws. The basic idea of this methodology is to consider the spatial differential consequences (gradients) of the balance laws as additional equations to be substituted into the entropy inequality, up to the order of the gradients entering the state space [20,21,22].

With the help of such a methodology, in the present paper, we explore the thermodynamic compatibility of one-dimensional thermoviscoelastic solids of grade 3, with the heat flux obeying an additional balance law. In fact, several systems at the nanometric scale, such as nanowires and nanotubes, can be represented as one-dimensional systems. Miniaturized structures such as nanowires may deform under applied mechanical stress and at this length scale, long-range interactions and nonlocal effects arise. Note that in the one-dimensional case, it is easier to obtain complete solutions of the set of thermodynamic restrictions imposed by the second law of thermodynamics, pointing out the essential properties of the material. The dependency of the constitutive quantities on the gradients of the unknown fields is examined in detail. As a consequence of such a dependency, we obtain new forms of the system of balance laws which are, in general, parabolic, although hyperbolic situations are also possible. For homogeneous and isotropic bodies, under the validity of a generalized Maxwell–Cattaneo equation which depends on the deformation too, we study the propagation of small-amplitude thermomechanical waves.

The paper has the following layout.

In Section 2, we provide a short overview on the generalized Coleman–Noll procedure for the exploitation of the entropy inequality.

In Section 3, we explore the thermodynamic compatibility of a one-dimensional thermoviscoelastic system, with the heat flux as an independent thermodynamic variable ruled by an additional balance law.

In Section 4, after providing a particular solution of the system of thermodynamic restrictions, we analyze the propagation of thermomechanical disturbances under the validity of a generalized Maxwell–Cattaneo equation.

In Section 5, we overview the results obtained and review some possible applications.

2. Thermoviscoelastic Solids with Heat-Flux Relaxation

In this section, we consider a thermoviscoelastic solid of grade 3, in the absence of body force and heat supply, with the heat flux ruled by an additional balance law. For such a system, the field equations read

$$\varrho {\ddot{x}}_i-T_{iL,L}=0,$$

(8)

$$\varrho \dot{\epsilon }+q_{L,L}-T_{iL}{\dot{F}}_{iL}=0,$$

(9)

$${\dot{q}}_I-{\Phi }_{IL,L}=r_I,$$

(10)

wherein $x_i(X_L,t)$ are the coordinates of the points of a continuous body $\mathcal{B}$ in the actual configuration $\mathcal{C}$, $\varrho$ is the referential mass density, $T_{iL}$ are the components of the first Piola–Kirchhoff stress tensor, $\epsilon$ is the referential specific internal energy, $q_I$ are the components of the referential heat flux, $-{\Phi }_{IL}$ are the components of the referential flux of heat flux, and $r_I$ the components of the referential production of heat flux.

Equations (1)–(3) constitute particular cases of the general balance law (10). The system (8)–(10) can be closed by assigning constitutive equations for the 18 constitutive functionals $\{T_{iL},{\Phi }_{IL},r_I\}$. Note that, if accordingly to Grad theory [23], ${\Phi }_{IL}$ is supposed to be symmetric, the number of constitutive equations to be assigned reduces to 15. We assign such equations on the state space

$$\overline{\Sigma }=\left\{\epsilon ,{\epsilon }_{,K},q_I,q_{I,L},F_{iL},F_{iL,K},F_{iL,KM},{\dot{F}}_{iK}\right\}.$$

(11)

Note also that, although the one-to-one correspondence between $\epsilon$ and $\vartheta$, $\overline{\Sigma }$ does not coincide with the state space considered in [1] (see Equation (4) above), since in [1] the heat flux is regarded as a constitutive quantity and, as a consequence, it does not enter the state space. The entropy inequality now takes the form

$$\varrho \dot{s}+J_{L,L}\ge 0,$$

(12)

wherein s is the referential specific entropy and $J_I$ are the components of the referential entropy flux which, in general, is different from that postulated in [19], namely $q_I/\vartheta$ (see Equation (7)). One could observe that, from the mathematical point of view, the decomposition $J_I=q_I/\vartheta +K_I$, where $K_I$ is the so-called Müller entropy extraflux [24], is always possible. Indeed, from the physical point of view, we cannot be sure that the entropy flux can be always decomposed as the sum of a purely thermal contribution $(q_I/\vartheta )$ and a non-thermal contribution $K_I$. Thus, in the following, we do not use such a decomposition, and leave the vector $J_I$ in its general form, without any further assumption on it.

In order to exploit the inequality (12), additional constitutive equations for the functions s and $J_I$ must be assigned as well.

On the state space $\overline{\Sigma }$, Equation (8) yields

$$\varrho \ddot{x_i}-{\displaystyle \frac{\partial T_{iL}}{\partial \epsilon }}{\epsilon }_{,L}-{\displaystyle \frac{\partial T_{iL}}{\partial {\epsilon }_{,K}}}{\epsilon }_{,KL}-{\displaystyle \frac{\partial T_{iL}}{\partial q_I}}{q}_{I,L}-{\displaystyle \frac{\partial T_{iL}}{\partial q_{I,K}}}{q}_{I,KL}-{\displaystyle \frac{\partial T_{iL}}{\partial F_{jK}}}{F}_{jK,L}-{\displaystyle \frac{\partial T_{iL}}{\partial F_{jK,M}}}{F}_{jK,ML}$$

(13)

$$-{\displaystyle \frac{\partial T_{iL}}{\partial F_{jK,MN}}}{F}_{jK,MNL}-{\displaystyle \frac{\partial T_{iL}}{\partial {\dot{F}}_{jK}}}{\dot{F}}_{jK,L}=0,$$

while Equation (9) remains unchanged. Finally, Equation (10) yields

$${\dot{q}}_I-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial \epsilon }}{\epsilon }_{,L}-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial {\epsilon }_{,K}}}{\epsilon }_{,KL}-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial q_I}}{q}_{I,L}-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial q_{I,K}}}{q}_{I,KL}-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial F_{jK}}}{F}_{jK,L}-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial F_{jK,M}}}{F}_{jK,ML}$$

(14)

$$-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial F_{jK,MN}}}{F}_{jK,MNL}-{\displaystyle \frac{\partial {\Phi }_{iL}}{\partial {\dot{F}}_{jK}}}{\dot{F}}_{jK,L}=r_I.$$

Then, the inequality (12) takes the form

$$\varrho {\displaystyle \frac{\partial s}{\partial \epsilon }}\dot{\epsilon }+\varrho {\displaystyle \frac{\partial s}{\partial {\epsilon }_{,K}}}{\dot{\epsilon }}_{,K}+\varrho {\displaystyle \frac{\partial s}{\partial q_I}}{\dot{q}}_I+\varrho {\displaystyle \frac{\partial s}{\partial q_{I,K}}}{\dot{q}}_{I,K}+\varrho {\displaystyle \frac{\partial s}{\partial F_{jK}}}{\dot{F}}_{jK}+\varrho {\displaystyle \frac{\partial s}{\partial F_{jK,M}}}{\dot{F}}_{jK,M}+\varrho {\displaystyle \frac{\partial s}{\partial F_{jK,MN}}}{\dot{F}}_{jK,MN}+\varrho {\displaystyle \frac{\partial s}{\partial {\dot{F}}_{jK}}}{\ddot{F}}_{jK}$$

$$+{\displaystyle \frac{\partial J_L}{\partial \epsilon }}{\epsilon }_{,L}+{\displaystyle \frac{\partial J_L}{\partial {\epsilon }_{,K}}}{\epsilon }_{,KL}+{\displaystyle \frac{\partial J_L}{\partial q_I}}{q}_{I,L}+{\displaystyle \frac{\partial J_L}{\partial q_{I,K}}}{\dot{q}}_{I,KL}+{\displaystyle \frac{\partial J_L}{\partial F_{jK}}}{F}_{jK,L}+{\displaystyle \frac{\partial J_L}{\partial F_{jK,M}}}{F}_{jK,ML}$$

(15)

$$+{\displaystyle \frac{\partial J_L}{\partial F_{jK,MN}}}{F}_{jK,MNL}+{\displaystyle \frac{\partial J_L}{\partial {\dot{F}}_{jK}}}{\dot{F}}_{jK,L}\ge 0.$$

Once Equations (9) and (10) and their gradient extensions, together with the gradient extension of Equation (8), have been substituted into the relation above, we obtain a rather long expression, which in compact form can be written as [25],

$$\mathbf{A}·\mathbf{V}+\mathbf{B}·\mathbf{W}+{\mathbf{W}}^{T}C\mathbf{W}+D\ge 0,$$

(16)

where $\mathbf{A}$ and $\mathbf{B}$ are vectors depending on the state variables, C is a symmetric matrix with entries defined on the state space, and D a scalar function defined on the state space. Notice that nothing prevents the possibility to have a thermodynamic process such that $D=0$.

The following theorem has been proved in [25].

Theorem 1. 

The inequality

$$\mathbf{A}·\mathbf{V}+\mathbf{B}·\mathbf{W}+{\mathbf{W}}^{T}C\mathbf{W}+D\ge 0,$$

(17)

where the scalar D can vanish for some thermodynamic processes, holds for arbitrary values of $\mathbf{V}$ and $\mathbf{W}$ if, and only if,

$$\begin{array}{cc}\hfill & \mathbf{A}=\mathbf{0},\hfill \\ \hfill & \mathbf{B}=\mathbf{0},\hfill \\ \hfill & C\mathit{is}\mathrm{a}\mathit{positive}\text{-}\mathit{semidefinite}\mathit{matrix},\hfill \\ \hfill & D\ge 0.\hfill \end{array}$$

(18)

Theorem 1 will be applied in the next section.

3. Thermodynamic Compatibility

In the following, we apply the results reviewed in Section 3 in order to explore the thermodynamic compatibility of one-dimensional thermoviscoelastic solids of grade 3. In fact, several systems very important in the applications, such as nanowires and nanotubes, can be modeled as one-dimensional systems. Due to the reduced dimensions, nonlocal effects arise, modeled through the gradients of the unknown fields in the state space.

It is worth observing that the modelization of the physical properties of low-dimensional systems is a difficult task, since in the passage to a low-dimensional representation, non-classical effects may appear. For one-dimensional systems, for instance, starting from a classical n-dimensional dissipative aggregation of matter, a dimensional reduction towards a one-dimensional quantum-wire system must be realized. Such a situation requires the passage from a classical stochastic to a quantum stochastic description, with the consequence that even a non-drastic temperature change can eventually generate the onset of the corresponding quantum behavior [26]. At the micro/nano scale, such a passage is observed in seed mucilage, a soft material of great importance from a biological point of view. Scientists studying mucilage emphasize the importance of intermolecular interactions, as they influence many physicochemical and rheological properties. Such types of interactions strictly result from quantum mechanics. Accordingly, a classical-quantum passage arises, wherein the quantum nature of the interactions strongly influences the structure and properties of mesoscopic or even macroscopic objects, such as mucilage [27].

We denote with X and x the position of the points of the system in the reference and in the actual configuration, respectively. The first Piola–Kirchhoff stress tensor, the deformation tensor, the heat flux, the flux of heat flux, the entropy and the entropy flux will be denoted by T, F, q$\Phi$, s and J, respectively, while the symbols $f_{,X}$ and $f_{,x}$ will denote the partial derivative of the function f with respect to the spatial coordinates X and x in the reference and in the actual configuration, respectively.

The state space is now

$$\overline{\Sigma }=\left\{\epsilon ,{\epsilon }_{,X},q,q_{,X},F,F_{,X},F_{,XX},\dot{F}\right\}.$$

(19)

Note that, since we are interested to the relaxation of the heat flux, such a quantity was included in the state space as an independent thermodynamic variable. On $\overline{\Sigma }$, the balance of linear momentum reads

$$\varrho \ddot{u}-{\displaystyle \frac{\partial T}{\partial \epsilon }}{\epsilon }_{,X}-{\displaystyle \frac{\partial T}{\partial {\epsilon }_{,X}}}{\epsilon }_{,XX}-{\displaystyle \frac{\partial T}{\partial q}}{q}_{,X}-{\displaystyle \frac{\partial T}{\partial q_{,X}}}{q}_{,XX}-{\displaystyle \frac{\partial T}{\partial F}}{F}_{,X}-{\displaystyle \frac{\partial T}{\partial F_{,X}}}{F}_{,XX}-{\displaystyle \frac{\partial T}{\partial F_{,XX}}}{F}_{,XXX}-{\displaystyle \frac{\partial T}{\partial \dot{F}}}{\dot{F}}_{,X}=0,$$

(20)

with

$$u=x(X,t)-X,$$

(21)

as the displacement of the points of the system. The local balance of energy takes the form

$$\varrho \dot{\epsilon }+q_{,X}-T\dot{F}=0.$$

(22)

Besides the balance equations above, we also postulate a balance equation for the heat flux which, in the present approach, according to the general tenets of EIT [7,8], is regarded as an independent thermodynamic variable, ruled by its own governing equation. Such an equation is supposed to have the form

$$\dot{q}-{\Phi }_{,X}=r,$$

(23)

with $-\Phi$ as the flux and r as the production of heat flux, respectively. On the state space, it yields

$$\dot{q}-{\displaystyle \frac{\partial \Phi }{\partial \epsilon }}{\epsilon }_{,X}-{\displaystyle \frac{\partial \Phi }{\partial {\epsilon }_{,X}}}{\epsilon }_{,XX}-{\displaystyle \frac{\partial \Phi }{\partial q}}{q}_{,X}-{\displaystyle \frac{\partial \Phi }{\partial q_{,X}}}{q}_{,XX}-{\displaystyle \frac{\partial \Phi }{\partial F}}{F}_{,X}-{\displaystyle \frac{\partial \Phi }{\partial F_{,X}}}{F}_{,XX}-{\displaystyle \frac{\partial \Phi }{\partial F_{,XX}}}{F}_{,XXX}-{\displaystyle \frac{\partial \Phi }{\partial \dot{F}}}{\dot{F}}_{,X}=r,$$

(24)

Finally, the entropy inequality becomes

$$\varrho {\displaystyle \frac{\partial s}{\partial \epsilon }}\dot{\epsilon }+\varrho {\displaystyle \frac{\partial s}{\partial {\epsilon }_{,X}}}{\dot{\epsilon }}_{,X}+\varrho {\displaystyle \frac{\partial s}{\partial q}}\dot{q}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,X}}}{\dot{q}}_{,X}+\varrho {\displaystyle \frac{\partial s}{\partial F}}\dot{F}+\varrho {\displaystyle \frac{\partial s}{\partial F_{,X}}}{\dot{F}}_{,X}+\varrho {\displaystyle \frac{\partial s}{\partial F_{,XX}}}{\dot{F}}_{,XX}+\varrho {\displaystyle \frac{\partial s}{\partial \dot{F}}}\ddot{F}$$

(25)

$$+{\displaystyle \frac{\partial J}{\partial \epsilon }}{\epsilon }_{,X}+{\displaystyle \frac{\partial J}{\partial {\epsilon }_{,X}}}{\epsilon }_{,XX}+{\displaystyle \frac{\partial J}{\partial q}}{q}_{,X}+{\displaystyle \frac{\partial J}{\partial q_{,X}}}{q}_{,XX}+{\displaystyle \frac{\partial J}{\partial F}}{F}_{,X}+{\displaystyle \frac{\partial J}{\partial F_{,X}}}{F}_{,XX}+{\displaystyle \frac{\partial J}{\partial F_{,XX}}}{F}_{,XXX}+{\displaystyle \frac{\partial J}{\partial \dot{F}}}{\dot{F}}_{,X}\ge 0.$$

According to the generalized Coleman–Noll procedure, Equations (22) and (24), together with their gradient extensions and the gradient extension of Equation (20), namely,

$${\left[\varrho \dot{\epsilon }\right]}_{,X}+q_{,XX}-T_{,X}\dot{F}-T{\dot{F}}_{,X}=0,$$

(26)

$${\dot{q}}_{,X}-{\left[{\displaystyle \frac{\partial \Phi }{\partial \epsilon }}{\epsilon }_{,X}+{\displaystyle \frac{\partial \Phi }{\partial {\epsilon }_{,X}}}{\epsilon }_{,XX}+{\displaystyle \frac{\partial \Phi }{\partial q}}{q}_{,X}+{\displaystyle \frac{\partial \Phi }{\partial q_{,X}}}{q}_{,XX}+{\displaystyle \frac{\partial \Phi }{\partial F}}{F}_{,X}+{\displaystyle \frac{\partial \Phi }{\partial F_{,X}}}{F}_{,XX}+{\displaystyle \frac{\partial \Phi }{\partial F_{,XX}}}{F}_{,XXX}+{\displaystyle \frac{\partial \Phi }{\partial \dot{F}}}{\dot{F}}_{,X}\right]}_{,X}=r_{,X},$$

(27)

$${\left[\varrho \ddot{u}-{\displaystyle \frac{\partial T}{\partial \epsilon }}{\epsilon }_{,X}-{\displaystyle \frac{\partial T}{\partial {\epsilon }_{,X}}}{\epsilon }_{,XX}-{\displaystyle \frac{\partial T}{\partial F}}{F}_{,X}-{\displaystyle \frac{\partial T}{\partial F_{,X}}}{F}_{,XX}-{\displaystyle \frac{\partial T}{\partial F_{,XX}}}{F}_{,XXX}-{\displaystyle \frac{\partial T}{\partial \dot{F}}}{\dot{F}}_{,X}\right]}_{,X}=0,$$

(28)

must be substituted into the inequality (25).

Since Equations (20), (22) and (24) are linear in the spatial derivatives, which are one order higher with respect to those entering the state space (higher derivatives), their gradients will contain terms which are quadratic in such quantities. For the sake of simplicity, we pursue our analysis under the assumption that, for the system at hand, the term ${\mathbf{W}}^{T}C\mathbf{W}$ in the entropy inequality is negligible. It is immediately seen that, under the above approximation, Equations (26)–(28), as well as the entropy inequality (25), become linear in the higher derivatives $\left\{{\epsilon }_{,XX},q_{,XX}{F}_{,XXX},{\dot{F}}_{,X}\right\}$, and the highest derivatives $\left\{{\epsilon }_{,XXX},q_{,XXX}{F}_{,XXXX},{\dot{F}}_{,XX}\right\}$, i.e., the gradients which are two orders higher with respect to those entering the state space. Since such quantities may assume arbitrary values, their coefficients must vanish; otherwise, the inequality could be easily violated. Letting such coefficients be zero, a set of thermodynamic restrictions on the constitutive quantities $T,\Phi ,r,s,$ and J ensue.

It is worth observing that the previous assumption does not obscure any physical properties of the system at hand, since the restrictions on the constitutive equations are determined by the terms which are linear in the higher and highest derivatives. The quadratic terms enter the reduced entropy inequality only and contribute to determine the effective local entropy production. Thus, the effect of our assumptions consists only in reducing the number of terms entering the local entropy production.

Before we proceed further, we recall that, as F is not frame-invariant, it is not the most appropriate tensor to enter the constitutive equations. Then, it is usually substituted either by the right Cauchy–Green tensor C, or by the Green–Saint Venant deformation tensor G, yielding, in the one-dimensional case, $C=F{F}^T=F^2$ and $G=(1/2)(C-1)=(1/2)(F^2-1)$, respectively.

From now on, we pursue our analysis by restricting ourselves to homogeneous bodies undergoing small deformations. For such systems, let $H=u_{,X}=F-1$ be the gradient of the displacement. A small deformation is such that H is of the first order of magnitude and the quantities of order $H^2$ are negligible, so that $G=(1/2)[{(1+H)}^2-1]=(1/2)[H^2+2H]\simeq H\equiv E$, with $E\simeq u_{,x}$ as the strain tensor, where, as usual in linear elasticity, we have considered the difference between the Eulerian coordinate x and the Lagrangian coordinate X to be negligible. Note that in the three-dimensional case, instead, E is the symmetric part of H. Thus, for the system at hand, the constitutive quantities can be expressed as functions of $E=u_{,x}$ and the space state becomes

$$\overline{\Sigma }=\left\{\epsilon ,{\epsilon }_{,x},q,q_{,x},E,E_{,x},E_{,xx},\dot{E}\right\}.$$

(29)

By the generalized procedure illustrated above, the following theorem ensues.

Theorem 2. 

For a one-dimensional thermoviscoelastic solid undergoing small deformations, the constitutive equations for T, Φ, r, s and J are compatible with the second law of thermodynamics if, and only if, they fulfill the following thermodynamic restrictions

$${\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial {\epsilon }_{,x}}}\dot{E}+\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial {\epsilon }_{,x}}}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial {\epsilon }_{,x}\partial \epsilon }}{\epsilon }_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial \Phi }{\partial \epsilon }}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial \epsilon \partial {\epsilon }_{,x}}}{\epsilon }_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E\partial {\epsilon }_{,x}}}{E}_{,x}$$

(30)

$$+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,x}\partial {\epsilon }_{,x}}}{E}_{,xx}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q\partial {\epsilon }_{,x}}}{q}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{{\epsilon }_{,x}\partial E}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,x}\partial {\epsilon }_{,x}}}{E}_{,xx}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q\partial {\epsilon }_{,x}}}{q}_{,x}$$

$$+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{{\epsilon }_{,x}\partial E}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial {\epsilon }_{,x}\partial E_{,x}}}{E}_{,xx}+{\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial r}{\partial {\epsilon }_{,x}}}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial {\epsilon }_{,x}\partial \epsilon }}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \epsilon \partial {\epsilon }_{,x}}}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial {\epsilon }_{,x}\partial q}}{q}_{,x}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial {\epsilon }_{,x}\partial E}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial {\epsilon }_{,x}\partial E_{,x}}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial \epsilon }}+{\displaystyle \frac{\partial J}{\partial {\epsilon }_{,x}}}=0,$$

$$-{\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}+{\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial q_{,x}}}\dot{E}+\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial q_{,x}}}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q_{,x}\partial \epsilon }}{\epsilon }_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q_{,x}\partial E}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q_{,x}\partial E_{,x}}}{E}_{,xx}$$

(31)

$$+{\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial r}{\partial q_{,x}}}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q_{,x}\partial \epsilon }}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q_{,x}\partial q}}{q}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \epsilon \partial q_{,x}}}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q\partial q_{,x}}}{q}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E\partial q_{,x}}}{E}_{,x}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q_{,x}\partial E}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q_{,x}\partial E_{,x}}}{E}_{,xx}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial q}}+{\displaystyle \frac{\partial J}{\partial q_{,x}}}=0,$$

$${\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial E_{,xx}}}\dot{E}-\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial E_{,xx}}}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,xx}\partial \epsilon }}{\epsilon }_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,xx}\partial E}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,xx}\partial E_{,x}}}{E}_{,xx}$$

(32)

$$+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial \epsilon \partial E_{,xx}}}{\epsilon }_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q\partial E_{,xx}}}{q}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E\partial E_{,xx}}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,x}\partial E_{,xx}}}{E}_{,xx}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial r}{\partial E_{,xx}}}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E_{,xx}\partial \epsilon }}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E_{,xx}\partial q}}{q}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E_{,xx}\partial E_{,x}}}{E}_{,xx}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \epsilon \partial E_{,xx}}}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q\partial E_{,xx}}}{q}_{,x}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E\partial E_{,xx}}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E_{,x}\partial E_{,xx}}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial E_{,x}}}{E}_{,x}+{\displaystyle \frac{\partial J}{\partial E_{,xx}}}=0,$$

$${\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}T+{\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial \dot{E}}}\dot{E}+\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial \dot{E}}}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial \dot{E}\partial E}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial \dot{E}\partial E_{,x}}}{E}_{,xx}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial \epsilon \partial \dot{E}}}{\epsilon }_{,x}$$

(33)

$$+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial q\partial \dot{E}}}{q}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E\partial \dot{E}}}{E}_{,x}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{{\partial }^2\Phi }{\partial E_{,x}\partial \dot{E}}}{E}_{,xx}+\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial r}{\partial \dot{E}}}+\varrho {\displaystyle \frac{\partial s}{\partial E_{,x}}}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \dot{E}\partial \epsilon }}{\epsilon }_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E\partial q}}{q}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \dot{E}\partial E}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \dot{E}\partial E_{,x}}}{E}_{,xx}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial \epsilon \partial \dot{E}}}{\epsilon }_{,x}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q\partial \dot{E}}}{q}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q\partial \dot{E}}}{q}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E\partial \dot{E}}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial q\partial \dot{E}}}{q}_{,x}$$

$$+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E\partial \dot{E}}}{E}_{,x}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{{\partial }^{2}T}{\partial E_{,x}\partial \dot{E}}}{E}_{,xx}+{\displaystyle \frac{\partial J}{\partial \dot{E}}}=0,$$

$$\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial \Phi }{\partial {\epsilon }_{,x}}}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial {\epsilon }_{,x}}}=0,$$

(34)

$$\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial \Phi }{\partial E_{,xx}}}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial E_{,xx}}}=0,$$

(35)

$$\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial \Phi }{\partial \dot{E}}}+\varrho {\displaystyle \frac{\partial s}{\partial E_{,xx}}}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial \dot{E}}}=0,$$

(36)

$$\varrho {\displaystyle \frac{\partial s}{\partial q_{,x}}}{\displaystyle \frac{\partial \Phi }{\partial q_{,x}}}+{\displaystyle \frac{\partial s}{\partial \dot{E}}}{\displaystyle \frac{\partial T}{\partial q_{,x}}}=0,$$

(37)

$$g\left(\overline{\Sigma }\right)\ge 0,$$

(38)

where $g\left(\overline{\Sigma }\right)$ is a given scalar function defined on the state space (local entropy production).

We note that $g\left(\overline{\Sigma }\right)$ is the residual entropy production, once the set of thermodynamic restrictions (30)–(37) has been satisfied. We also note that nothing prevents s from depending on the quantities $\left\{{\epsilon }_{,x},q_{,x},E_{,x},E_{,xx},\dot{E}\right\}$. Such a dependency would be not allowed in the classical case, where the entropy inequality is exploited by the classical Coleman–Noll procedure [7,8]. In particular, T may depend on ${\epsilon }_{,X}$, so that the system of balance laws is parabolic even in the case of validity of the classical Maxwell–Cattaneo Equation (2). Moreover, $\Phi$ may depend on q and $q_{,X}$, so that both Equations (1) and (2) can be obtained by Equation (23). By the restrictions above, the key role played by the dependency of s on $q_{,x}$ and $\dot{E}$ is evident. If s is independent of such quantities, then Equations (34), (35) and (37) are immediately satisfied, whereas Equations (30)–(33) and (36) reduce to

$${\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial {\epsilon }_{,x}}}\dot{E}+\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial {\epsilon }_{,x}}}+{\displaystyle \frac{\partial J}{\partial {\epsilon }_{,x}}}=0,$$

(39)

$$-{\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}+{\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial q_{,x}}}\dot{E}+\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial q_{,x}}}{E}_{,xx}+{\displaystyle \frac{\partial J}{\partial q_{,x}}}=0,$$

(40)

$${\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial E_{,xx}}}\dot{E}-\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial E_{,xx}}}+{\displaystyle \frac{\partial J}{\partial E_{,xx}}}=0,$$

(41)

$${\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}T+{\displaystyle \frac{\partial s}{\partial {\epsilon }_{,x}}}{\displaystyle \frac{\partial T}{\partial \dot{E}}}\dot{E}+\varrho {\displaystyle \frac{\partial s}{\partial q}}{\displaystyle \frac{\partial \Phi }{\partial \dot{E}}}+\varrho {\displaystyle \frac{\partial s}{\partial E_{,x}}}+{\displaystyle \frac{\partial J}{\partial \dot{E}}}=0,$$

(42)

$$\varrho {\displaystyle \frac{\partial s}{\partial E_{,xx}}}=0.$$

(43)

Equations (39)–(43) show that the dependency of T and $\Phi$ on ${\epsilon }_{,x}$, $q_{,x}$, $E_{,xx}$ and $\dot{E}$ determines the dependency of the entropy flux on the same quantities. From the previous equations, one can also see that the possible constitutive equations for J are more general with respect to the equation postulated in Rational Thermodynamics, namely $J=q/\vartheta$ [19].

4. Thermomechanical Wave Propagation

In Section 3, we showed how the application of the extended Coleman–Noll procedure allows for more general constitutive equations for the entropy and the stress tensor with respect to the classical one. We can summarize the previous results by saying that owing to the extended procedure, the number of parabolic models which are compatible with the second law of thermodynamics increases, in such a way that the hyperbolic models occur only under particular conditions. For instance, the dependency of the stress or of the flux of heat flux on $E_{,x}$, $E_{,xx}$$\dot{E}$, $q_{,x}$ and ${\epsilon }_{,x}$ implies the presence of the terms $u_{,xxx}$, $u_{,xxxx}$, $u_{,xxt}$$q_{,xx}$ and ${\epsilon }_{,xx}$ in the system of balance laws (8)–(10), rendering it parabolic. Hyperbolic theories arise only if T and $\Phi$ are local, while the production of the heat flux depends $E_{,x}$, $\dot{E}$, $q_{,x}$ and ${\epsilon }_{,x}$, but is independent of $E_{,xx}$. In such a case, the propagation of thermoelastic perturbations is possible. The present section is devoted to the study of such a physical situation. Our analysis will be pursued under the following hypotheses:

• The constitutive equation for the Cauchy stress is

  $$T=(\lambda +2\mu )E-\overline{b}\epsilon +{\displaystyle \frac{s_1{\tau }_E}{{\tau }_q{A}_q}}{q}^2,$$

  (44)

  wherein $\lambda$ and $\mu$ are the Lamè coefficients, ${\displaystyle b=(3\lambda +2\mu )\frac{\alpha }{c}}$, where c is the specific heat, depending on the temperature, $\alpha$ is the coefficient of thermal expansion, $s_1$ is a material coefficient with the dimension of a force per unit surface divided a squared heat flux, $A_q$ is a non-dimensional material coefficient depending on the deformation, and ${\tau }_E$ and ${\tau }_q$ are two relaxation times depending on $\epsilon$ (or, equivalently, on the temperature);
• The flux of heat flux is

  $${\displaystyle \frac{1}{{\tau }_q{A}_q}}\overline{\kappa }{A}_K\epsilon ,$$

  (45)

  where $A_K$ is a further non-dimensional material coefficient depending on the deformation, and $\overline{\kappa }=\kappa /c$, where $\kappa$ is the temperature-dependent thermal conductivity;
• The production of the heat flux is

  $${\displaystyle \frac{1}{{\tau }_q{A}_q}}\left[-q+{\displaystyle \frac{2{\tau }_Q}{\varrho \epsilon }}q{q}_{,x}+{\tau }_{E}q\dot{E}+q_0{L}_0{E}_{,x}\right],$$

  (46)

  where ${\tau }_Q$ is a further temperature-dependent relaxation time, $q_0$ is a constant reference value of the heat flux and $L_0$ a characteristic length;
• The spatial derivatives of the material coefficients are of the second order of magnitude with respect to the material coefficients, which are supposed to be of the first order of magnitude.

Under the hypotheses above, the system of Equations (22)–(24) is written as

$$\varrho \ddot{u}-(\lambda +2\mu )E_{,x}+\overline{b}{\epsilon }_{,x}-2{\displaystyle \frac{s_1{\tau }_E}{{\tau }_q{A}_q}}q{q}_{,x}=0,$$

(47)

$$\varrho \dot{\epsilon }+q_{,x}-\left[(\lambda +2\mu )E-\overline{b}\epsilon +{\displaystyle \frac{s_1{\tau }_E}{{\tau }_q{A}_q}}{q}^2\right]{\dot{E}}_{,x}=0,$$

(48)

$${\tau }_q{A}_q\dot{q}+q+\overline{\kappa }{A}_K{\epsilon }_{,x}-{\displaystyle \frac{2{\tau }_Q}{\varrho \epsilon }}q{q}_{,x}-{\tau }_{E}q\dot{E}-q_0{L}_0{E}_{,x}=0.$$

(49)

Some considerations on Equation (49) are in order. Although, for the sake of simplicity, we are considering the spatial derivatives of the material coefficients to be negligible, we note the following:

• $\kappa A_K$, with $A_K$ depending on the deformation, is the effective thermal conductivity;
• ${\tau }_q$ is a material parameter, depending on $\epsilon$ and c, and $A_q$ is a material parameter depending on the deformation, such that ${\tau }_q{A}_q$ represents the total relaxation time of the heat flux (i.e., the time elapsed between the application of a temperature difference and the appearance of a heat flux);
• ${\tau }_E$ is the relaxation time of the strain (i.e., the time elapsed between the application of a stress and the appearance of a deformation);
• ${\tau }_Q$ is the relaxation time of the heat carriers, i.e., the quantity ${\displaystyle \frac{\ell }{\overline{v}}}$, with ℓ as the mean free path and $\overline{v}$ as the mean speed of the heat carriers (phonons, electrons, holes).

Equation (49) reduces to the classical Maxwell–Cattaneo equation [10], if $A_q=A_K=1$ and ${\tau }_Q={\tau }_E=q_0=0$.

The nonlinear system of equations above allows the existence of nonregular solutions.

Definition 1. 

An acceleration wave is a traveling surface $\mathcal{S}$ across which a solution $\left\{E,\epsilon ,q\right\}$ of Equations (47)–(49) is continuous, but its first and higher-order derivatives suffer jump discontinuities [28,29].

Remark 1. 

It is worth observing that Equations (47)–(49) hold in the points of the two regions behind and ahead of $\mathcal{S}$, while on $\mathcal{S}$, those equations must be written in terms of the jumps of the discontinuous fields across the wavefront. Such jumps are given by the differences

$$\begin{array}{cccc}\Delta \ddot{u}={\ddot{u}}^{-}-{\ddot{u}}^{+},\hfill & \Delta u_{,x}=u_{,x}^{-}-u_{,x}^{+},\hfill & \Delta u_{,xx}=u_{,xx}^{-}-u_{,xx}^{+},\hfill & \\ \Delta {\epsilon }_{,x}={\epsilon }_{,x}^{-}-{\epsilon }_{,x}^{+},\hfill & \Delta q_{,x}=q_{,x}^{-}-q_{,x}^{+},\hfill \end{array}$$

(50)

where the superscript + denotes the value of the corresponding fields in the region which $\mathcal{S}$ is about to enter, and the superscript − denotes the same value in the region which $\mathcal{S}$ is about to leave.

In order to determine the jumps of the discontinuous fields across $\mathcal{S}$, we first observe that in the one-dimensional case, the sole component of the strain tensor is $E=u_{,x}$. Here and in the following, we suppose that the fields $u_{,x},\epsilon$ and q are continuous across $\mathcal{S}$ but their space and time derivatives suffer jump discontinuities. Meanwhile, we suppose that across $\mathcal{S}$, the time derivatives of the displacement u are discontinuous too. Hence, we introduce the following notation:

$$\delta {\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=\delta {\displaystyle \frac{\partial \left(\partial u/\partial x\right)}{\partial x}}=\delta {\displaystyle \frac{\partial E}{\partial x}}\equiv \delta E,\delta {\displaystyle \frac{\partial \epsilon }{\partial x}}\equiv \delta \epsilon ,\delta {\displaystyle \frac{\partial q}{\partial x}}\equiv \delta q,$$

(51)

with $\delta$ as the jump of the indicated quantities in the one-dimensional case. Moreover, as $u_{,x}$ is continuous across $\mathcal{S}$, by the classical Hadamard identities (see Ref. [29] and Equations (54)–(55) therein), we can write

$$\delta {\displaystyle \frac{\partial u}{\partial t}}=-U\delta {\displaystyle \frac{\partial u}{\partial x}}=-U\delta E,$$

wherein U is the speed of propagation of thermoelastic disturbances along the x direction. Then, the system (47)–(49) yields

$$[\varrho U^2-(\lambda +2\mu )]\delta E-2{\displaystyle \frac{s_1{\tau }_E}{{\tau }_q{A}_q}}q\delta q+\overline{b}\delta \epsilon =0,$$

(52)

$$U[{\displaystyle \frac{s_1{\tau }_E{q}^2}{{\tau }_q{A}_q}}+(\lambda +2\mu )E-\overline{b}\epsilon ]\delta E+\delta q-\varrho U\delta \epsilon =0,$$

(53)

$$(U{\tau }_{E}q-q_0{L}_0)\delta E-[U{\tau }_q{A}_q+{\displaystyle \frac{2{\tau }_Q}{\varrho \epsilon }}q]\delta q+\overline{\kappa }{A}_K\delta \epsilon =0,$$

(54)

wherein it must be understood that the coefficients of the unknown jumps are evaluated in the region which $\mathcal{S}$ is about to enter. Equations (52)–(54) represent a linear and homogeneous algebraic system in the unknown quantities $\delta E$, $\delta \epsilon$ and $\delta q$, which provides, in principle, the values of the jumps we are looking for. The following statement is straightforward.

Theorem 3. 

The system (52)–(54) admits nontrivial solutions if, and only if, the following condition is fulfilled:

$$det\left[\begin{array}{ccc}[\varrho U^2-(\lambda +2\mu )]& {\displaystyle -2\frac{s_1{\tau }_E}{{\tau }_q{A}_q}q}& \overline{b}\\ {\displaystyle U[\frac{s_1{\tau }_E{q}^2}{{\tau }_q{A}_q}+(\lambda +2\mu )E-\overline{b}\epsilon ]}& 1& -\varrho U\\ (U{\tau }_{E}q-q_0{L}_0)& {\displaystyle -(U{\tau }_q{A}_q+\frac{2{\tau }_Q}{\varrho \epsilon }q)}& \overline{\kappa }{A}_K\end{array}\right]=0.$$

(55)

Proof. 

The proof is immediately followed by the observation that the system (52)–(54) admits nontrivial solutions if, and only if, the matrix of the coefficients is singular, i.e., if and only if Equation (55) holds. □

Suppose now that the following additional conditions hold:

$${\tau }_E={\tau }_Q=0.$$

(56)

Remark 2. 

The conditions in (56) mean that the Cauchy stress coincides with that of classical linear thermoelasticity. Moreover, nonlocal effects for the heat flux and viscous effects do not influence the evolution equations for the heat flux (Equation (49)).

Under the hypotheses (56), the system (52)–(54) becomes

$$[\varrho U^2-(\lambda +2\mu )]\delta E+\overline{b}\delta \epsilon =0,$$

(57)

$$U[(\lambda +2\mu )E-\overline{b}\epsilon ]\delta E+\delta q-\varrho U\delta \epsilon =0,$$

(58)

$$-q_0{L}_0\delta E-U{\tau }_q{A}_q\delta q+\overline{\kappa }{A}_K\delta \epsilon =0.$$

(59)

Theorem 4. 

The system (57)–(59) admits nontrivial solutions if, and only if, the following equation is fulfilled:

$$\left[\varrho U^2-(\lambda +2\mu )\right]\left(\overline{\kappa }{A}_K-\varrho U^2{\tau }_q{A}_q\right)+\left[{\overline{b}}^2{U}^2\epsilon {\tau }_q{A}_q-\overline{b}{U}^2{\tau }_q{A}_q(\lambda +2\mu )E\right]+\overline{b}{q}_0{L}_0=0.$$

(60)

Proof. 

To prove the theorem, we observe that the system (57)–(59) admits nontrivial solutions if, and only if,

$$det\left[\begin{array}{ccc}\left[\varrho U^2-(\lambda +2\mu )\right]& 0& \overline{b}\\ U(\lambda +2\mu )E-\overline{b}\epsilon & 1& -\varrho U\\ -q_0{L}_0& -U{\tau }_q{A}_q& \overline{\kappa }{A}_K\end{array}\right]=0,$$

(61)

i.e, if, and only if, Equation (60) holds. □

Remark 3. 

The compatibility condition (60) is satisfied if either two elastic waves with speed

$$U=\pm \sqrt{{\displaystyle \frac{\lambda +2\mu }{\varrho }}},$$

(62)

or two thermoelastic waves with speed

$$U=\pm \sqrt{{\displaystyle \frac{\overline{\kappa }{A}_K}{\varrho {\tau }_q{A}_q}}},$$

(63)

propagate together with two thermoelastic waves with speed

$$U=\pm \sqrt{{\displaystyle \frac{-q_0{L}_0}{\overline{b}\epsilon {\tau }_q{A}_q-{\tau }_q{A}_q(\lambda +2\mu )E}}},$$

(64)

provided the argument of the square root is positive. Note that, since $A_q$ and $A_K$ depend on the deformation, Equation (63) yields the speed of two thermoelastic waves. As a particular case, if $A_q=A_K=1$, Equation (63) yields the speeds of propagation of classical thermal waves (second sound [10]).

Remark 4. 

If all the relaxation times in Equation (49) vanish, then the generalized Fourier law

$$q=-\overline{\kappa }{A}_K{\epsilon }_{,x}-q_0{L}_0{E}_{,x},$$

(65)

holds. Thus, as $q=q(\epsilon ,E,{\epsilon }_{,x},E_{,x})$, we are in the realm of Rational Thermodynamics, i.e., the thermodynamic theory in which heat flux and stress tensor are assigned through suitable constitutive equations, [17].

Corollary 1. 

Under the validity of the generalized Fourier law (65), if ${\displaystyle \lambda +2\mu >\frac{\overline{b}{q}_0{L}_0}{\overline{\kappa }{A}_K}}$ (elastic effects predominant with respect to the thermal ones), the following speeds of propagation are possible:

$$U=\pm \sqrt{{\displaystyle \frac{\lambda +2\mu }{\varrho }}-{\displaystyle \frac{\overline{b}{q}_0{L}_0}{\varrho \overline{\kappa }{A}_K}}}.$$

(66)

Proof. 

The proof is followed by the observation that, if ${\tau }_q=0$, Equation (61) reduces to

$$det\left[\begin{array}{ccc}\left[\varrho U^2-(\lambda +2\mu )\right]& 0& \overline{b}\\ U(\lambda +2\mu )E-\overline{b}\epsilon & 1& -\varrho U\\ -q_0{L}_0& 0& \overline{\kappa }{A}_K\end{array}\right]=0,$$

(67)

which yields

$$U^2={\displaystyle \frac{\lambda +2\mu }{\varrho }}-{\displaystyle \frac{\overline{b}{q}_0{L}_0}{\varrho \overline{\kappa }{A}_K}}$$

(68)

□

Remark 5. 

Equation (66) yields the speed of a thermoelastic wave which, if $q_0{L}_0=0$, reduces to a classical elastic wave. If, instead, ${\displaystyle \lambda +2\mu \le \frac{\overline{b}{q}_0{L}_0}{\overline{\kappa }{A}_K}}$ (thermal effects predominant or of the same intensity with respect to the elastic ones), the solutions of Equation (66) are imaginary or null, and the parabolic regime is restored.

Corollary 2. 

If

$$\overline{b}=q_0{L}_0={\tau }_E={\tau }_Q=0,A_q=A_K=1,$$

(69)

then two elastic waves and two thermal waves propagate with speeds

$$U=\pm \sqrt{{\displaystyle \frac{\lambda +2\mu }{\varrho }}},$$

(70)

and

$$U=\pm \sqrt{{\displaystyle \frac{\overline{\kappa }}{\varrho {\tau }_q}}},$$

(71)

respectively.

Proof. 

To prove the corollary, we observe that, under the hypotheses in (69), Equation (61) yields,

$$det\left[\begin{array}{ccc}\left[\varrho U^2-(\lambda +2\mu )\right]& 0& 0\\ U(\lambda +2\mu )E& 1& -\varrho U\\ 0& -U{\tau }_q& \overline{\kappa }\end{array}\right]=0,$$

(72)

which is equivalent to

$$\left[\varrho U^2-(\lambda +2\mu )\right]\left[U^2\varrho {\tau }_q-\overline{\kappa }\right]=0.$$

(73)

□

Remark 6. 

Under the hypotheses of Corollary 2, the system behaves as a purely elastic body (no thermoelastic coupling), with a Cattaneo-type evolution equation for the heat flux. Thus, the presence of purely elastic and purely thermal waves (second sound) is expected.

Definition 2. 

An acceleration wave is said to propagate in a state of thermal equilibrium with negligible thermal expansion if in the region in which the wave is about to enter

$${\epsilon }_{,x}=q=q_0=q_{,x}=\dot{q}=0,\alpha \simeq 0.$$

(74)

In such a case, the system (52)–(54) yields

$$[\varrho U^2-(\lambda +2\mu )]\delta E=0,$$

(75)

$$U(\lambda +2\mu )E\delta E+\delta q-\varrho U\delta \epsilon =0,$$

(76)

$$-U{\tau }_q{A}_q\delta q+\overline{\kappa }{A}_K\delta \epsilon =0.$$

(77)

Theorem 5. 

The system (75)–(77) admits nontrivial solutions if, and only if, two elastic waves with speed

$$U=\pm \sqrt{{\displaystyle \frac{\lambda +2\mu }{\varrho }}},$$

(78)

propagate together with two thermoelastic waves with speed

$$U=\pm \sqrt{{\displaystyle \frac{\overline{\kappa }{A}_K}{\varrho {\tau }_q{A}_q}}}.$$

(79)

Proof. 

The proof immediately follows by the observation that the system (75)–(77) admits nontrivial solutions if, and only if,

$$det\left[\begin{array}{ccc}[\varrho U^2-(\lambda +2\mu )]& 0& 0\\ U(\lambda +2\mu )E& 1& -\varrho U\\ 0& -U{\tau }_q{A}_q& \overline{\kappa }{A}_K\end{array}\right]=0,$$

(80)

which is equivalent to

$$\left[\varrho U^2-(\lambda +2\mu )\right](\overline{\kappa }{A}_K-U^2{\tau }_q{A}_q)=0.$$

(81)

□

5. Conclusions

In recent years, we have developed a new mathematical tool (generalized Coleman–Noll procedure) to investigate the thermodynamic compatibility of weakly nonlocal materials [20,21,22]. In the present paper, we applied a generalized Coleman–Noll procedure to analyze thermoviscoelastic solids of grade 3 with heat-flux relaxation, undergoing small deformations. A wider class of parabolic models, with respect to the ones allowed by the classical Coleman–Noll analysis, have been determined and studied. We can summarize the present results by saying that we enlarged the set of parabolic models compatible with the second law of thermodynamics, while hyperbolic models occur only under particular conditions. For instance, the dependency of the stress or of the flux of heat flux on $E_{,x}$, $E_{,xx}$$\dot{E}$, $q_{,x}$ and ${\epsilon }_{,x}$ implies the presence of the terms $u_{,xxx}$, $u_{,xxxx}$, $u_{,xxt}$, $q_{,xx}$ and ${\epsilon }_{,xx}$ in the system of balance laws (8)–(10), rendering it parabolic. Hyperbolic theories arise only if T and $\Phi$ are local, while the production of the heat flux depends on $E_{,x}$, $q_{,x}$ and ${\epsilon }_{,x}$, but is independent of $E_{,xx}$. In such a case, propagation of thermomechanical disturbances is possible. For the system at hand, we have studied such a propagation in Section 4, proving that mechanical, thermal, and thermomechanical waves are possible.

The numerical modeling of the materials analyzed above is out of the scope of the present paper. However, in the following, we cite several experimental situations in which the theory developed above could be validated by numerical experiments. In fact, the use of higher-gradient continuum models can play a relevant role in modeling the physical behavior of many complex mechanical systems and structures, which may require the introduction of micro-structured continua endowed with additional kinematical fields (see [30] for an extensive discussion).

• Bone tissues are intrinsically endowed with a multiscale structure which is naturally described by higher-gradient constitutive equations.
• Metamaterials for biological applications have mechanical and biological behavior which must be optimized in order to allow remodeling processes. Generalized continua may supply an effective tool in modeling the aforementioned properties.
• In capillarities, internal boundary layers are formed in bodies which experience high gradients of material properties and deformation fields in very narrow material regions. Microscopically, such a dependency is associated with long-range interactions among the material particles.
• In phase transitions, higher-gradient continuum models are necessary if one wants to describe the onset of the strong space variation of pressure, observed also in equilibrium conditions.
• Long-range interactions are also evident in liquid helium flowing in a narrow capillary at temperature close to the absolute zero.

In future research, we will aim to explore the physical nature of some of such situations by the mathematical procedure presented here and numerical experiments.