Nonlinear Propagation of Coupled First- and Second-Sound Waves in Thermoelastic Solids

Abstract

We study coupled nonlinear first- and second-sound propagation along equilibrium and nonequilibrium states of a thermoelastic system undergoing small perturbations. We apply a nonlinear constitutive equation for the Cauchy stress and a nonlinear heat-transport equation ruling the evolution of the heat flux. Both of them account for relaxational and nonlinear effects, as well as for the coupling between strain tensor and heat flux. The speeds of thermomechanical waves are obtained, and we show that they depend on whether the waves are travelling along, or against, a superimposed constant heat flux.

Appendix: Thermodynamic Compatibility

According to the entropy principle [32], the constitutive equations for the specific internal energy and for the Cauchy stress tensor have to be postulated in such a way that any solution of the system of differential equations in Eqs. (3a)–(4) represents an admissible thermodynamic process, namely, a process which is in accordance with second law of thermodynamics [5, 16, 17, 44, 45].

In continuum thermodynamics such a law is analyzed by considering the local balance of entropy

$$ \varrho \dot{s}+J_{i,i}^{ (s )}=\varrho \sigma ^{ (s )}, $$

(47)

wherein $s$ is the specific entropy, and $J_{i}^{ (s )}$ and $\sigma ^{ (s )}$ are, respectively, the specific-entropy flux and production. In fact, second law of thermodynamics imposes that $\sigma ^{ (s )}$ is non-negative along any admissible thermodynamic process [32, 44]. This implies nontrivial consequences on the constitutive equation for $T_{ij}$, leading to the form given in Eq. (6). Here we prove that Eq. (6) is true by applying the classical Coleman-Noll procedure for the exploitation of the entropy inequality [44]. To this end we observe that, under the constitutive assumption

$$ J_{i}^{ (s )}=\frac{q_{i}}{\theta }, $$

(48)

from Eq. (47) we have

$$ 0\leq \varrho \dot{s}-\frac{q_{i}\theta _{,i}}{\theta ^{2}}+\frac{q_{i, {i}}}{\theta }. $$

(49)

If the Helmholtz free energy $\psi =e-s\theta$ is introduced into the inequality above, and use of Eqs. (3b) and (4) is made, we get

$$\begin{aligned} 0 \leq& \varrho (\dot{e}-\dot{\psi }-s\dot{\theta } )+q _{i,{i}}- \frac{q_{i}\theta _{,i}}{\theta } \\ =&-\varrho \dot{\psi }-\varrho s\dot{\theta }+T_{ij} \dot{E}_{ij}+\frac{ \tau _{q}}{\kappa \theta } (1+a_{\tau }E_{hk}E_{hk} )q_{i} \dot{q}_{i}+ \frac{q_{i}q_{i}}{\kappa \theta }- \frac{2\tau _{q}}{ \varrho c_{v}\kappa \theta ^{2}}q_{j}q_{j,{i}}q_{i} \\ &{}- \frac{\tau _{E}}{ \kappa \theta }q_{j}\dot{E}_{ij}q_{i}. \end{aligned}$$

(50)

Theorem 6

The inequality (50) is satisfied whatever the thermodynamic process is if, and only if, the following thermodynamic restrictions hold

$$\begin{aligned} &\frac{\partial \psi }{\partial \theta }=-s, \end{aligned}$$

(51a)

$$\begin{aligned} &\frac{\partial \psi }{\partial \theta _{,_{i}}}=0, \end{aligned}$$

(51b)

$$\begin{aligned} &\frac{\partial \psi }{\partial q_{i}}=\frac{\tau _{q}}{\varrho \kappa \theta } (1+a_{\tau }E_{hk}E_{hk} )q_{i}, \end{aligned}$$

(51c)

$$\begin{aligned} &\frac{\partial \psi }{\partial q_{i,_{j}}}=0, \end{aligned}$$

(51d)

$$\begin{aligned} &\frac{\partial \psi }{\partial E_{ij}}=\frac{T_{ij}}{\varrho }-\frac{ \tau _{E}}{\varrho \kappa \theta }q_{i}q_{j}, \end{aligned}$$

(51e)

$$\begin{aligned} &\frac{q_{i}q_{j}}{\kappa \theta } \biggl(\delta _{ij}-\frac{2\tau _{q}}{ \varrho c_{v}\theta }q_{i,j} \biggr)\geq 0. \end{aligned}$$

(51f)

Proof

Indeed, once account is taken of the thermodynamic variables spanning the state space $\Sigma$, the inequality (50) can be put in the following form

$$\begin{aligned} 0\leq\; &-\varrho \biggl(\frac{\partial \psi }{\partial \theta }+s \biggr) \dot{\theta }+ \biggl[ \frac{\tau _{q}}{\kappa \theta } (1++a_{ \tau }E_{hk}E_{hk} )q_{i}-\varrho \frac{\partial \psi }{\partial q_{i}} \biggr]\dot{q}_{i}+ \frac{q_{i}q_{i}}{\kappa \theta } \\ &{}-\frac{2\tau _{q}}{\varrho c_{v}\kappa \theta ^{2}}q_{j}q_{j,{i}}q _{i}+ \biggl(T_{ij}-\varrho \frac{\partial \psi }{\partial E_{ij}}-\frac{ \tau _{E}}{\kappa \theta }q_{i}q_{j} \biggr)\dot{E}_{ij}- \varrho \frac{ \partial \psi }{\partial \theta _{,i}}\dot{\theta }_{,i}-\varrho \frac{ \partial \psi }{\partial q_{i,_{j}}}\dot{q}_{i,{j}}. \end{aligned}$$

(52)

This inequality is linear in the time derivatives of the elements of the state space $\dot{\theta }$, $\dot{q}_{i}$, $\dot{E}_{ij}$$\dot{\theta }_{,i}$ and $\dot{q}_{i,{j}}$. Moreover, it is evident that in any point of the body and at every instant of time, those derivatives are independent of the elements of the state space and, due to the arbitrariness of the thermodynamic process, they can assume completely arbitrary values. Hence, since their coefficients in the entropy inequality (52) are defined on the state space, the time derivatives $\dot{\theta }$, $\dot{q}_{i}$, $\dot{E}_{ij}$$\dot{\theta }_{,i}$ and $\dot{q}_{i,{j}}$ and their coefficients are completely independent, and their product can assume an arbitrary sign. Thus, the entropy inequality is always satisfied if, and only if, all the coefficients of those time derivatives vanish [44]. As a consequence, the conditions (51a)–(51f) are necessary and sufficient for the fulfillment of inequality (52) along arbitrary thermodynamic processes. □

Remark 5

Once Eqs. (51a)–(51e) are fulfilled, inequality (52) reduces to (51f), namely,

$$ \biggl( \frac{q_{i}}{\kappa \theta }- \frac{2\tau _{q}}{\varrho c_{v}\kappa \theta ^{2}}q_{j}q_{j,{i}} \biggr)q _{i}\ge 0, $$

which represents an unilateral differential constraint defined on $\Sigma$. Observing that the functions ${q}_{i}$ and $q_{j,{i}}$ belong to the state space, the reduced entropy inequality above requires that the following matrix

$$ q_{i,j}- \biggl(\frac{\varrho c_{v}\theta }{2\tau _{q}} \biggr)\delta _{ij}, $$

(53)

has to be negative semidefinite on $\Sigma$.

At this point, we firstly observe that the restrictions above do not impose any constraint on the internal energy. Hence, nothing prevents to assume that the linear relation $e=c_{v}\theta$ holds. Under such hypothesis our last task is to derive by the restrictions above a suitable constitutive equation for the Cauchy stress tensor and, in such a way, to close the system (3a)–(4). From Eqs. (51b) and (51d) directly follows the relation

$$ \psi =\psi (\theta ;q_{i};E_{ij} ). $$

(54)

The coupling of Eqs. (51d) and (54), instead, yields

$$ \psi =\psi _{0} (\theta ;E_{ij} )+\frac{\tau _{q}}{2\varrho \kappa \theta } (1+a_{\tau }E_{hk}E_{hk} )q_{i}q_{i}. $$

(55)

Finally, if the classical assumption of thermoelasticity

$$ \varrho \frac{\partial \psi _{0}}{\partial E_{ij}}= (\lambda E _{kk}-b\theta )\delta _{ij}+2\mu E_{ij} $$

(56)

is used, then from Eq. (51e) we get the constitutive equation (6).