Energy Conversion and Entropy Change During the Thermoelastic Effect

1. Introduction

Theoretical studies of themoelasticity have profound historical origins and abundant research backgrounds, with many important contributions documented in existing literature. For brevity, only an extremely concise and incomprehensive introduction to the advances in the theoretical analyses in this area is provided here. The initial theoretical framework was established independently by Duhamel (1837) [1] and Neumann (1841) [2]. By introducing thermal strains into the elastic constitutive relations, they derived the first uncoupled thermoelastic equations for thermal stress analysis. Then, an important advance was made in 1956 by Biot [3], who developed the coupled thermoelastic theory by considering elasticity, heat conduction, and irreversible thermodynamics simutaneously. It is worth noting that both entropy production and energy dissipation during the thermoelastic process were considered in Biot’s approach. Later, the focus of researchers gradually shifted to thermoelasticity under thermal shock, i.e., the so-called second sound effect. During thermal shock, the traditional Fourier’s law is no longer valid since it indicates an infinitely large thermal conduction rate. In addition, the free energy and the thermal stress can also depends on the rate of change of temperature. Revisions to the heat conduction law and energy equations help converting the heat conduction equation into a wave equation. Models developed for this analysis include the Lord-Shulman model [4], Green-Lindsay model [5], Green-Naghdi model [6] and Tzou’s dual-phase-lag model [7]. Detailed elaborations on the development of the fundamental theory on thermoelasitcity are available in these references [8,9].

However, since thermoelastiticy is a coupling effect between the thermal field and the elastic field, it essentially refects that energy is converted between these two coupled fields. Then how is energy converted between these two fields and how can this process be formulated? Yet, in all traditional apporaches mentioned above, this issue has never been addressed. Furthermore, thermoelastic process is a typical irreversible process which leads to the production of entropy. Then, what is the total entropy change during this process and what contributed to this change? These questions are also left unanswered in traditional approaches. Hence, in this paper, the majority of the discussion is devoted to the analysis of the energy conversion as well as the entropy change during the thermoelastic effect. Compared with traditinal approaches that expresses free energy via Taylor expansions, our appoach adopts the exact thermal and elastic energy and entropy functionals to conduct the analysis. Furthermore, rather than resorting to the free energy and the entropy inequality for analysis by traditional approaches, the first and second law of thermodynamics are directly applied in our approach. As a result, not only the governing equations derived herein strictly abide by with the first and second laws of thermodynamics, but also the contributions leading to the energy conversion and entropy change during the process can be discussed in greater details. The rest of this paper is organized as follows. In the second and third sections, the energy conversion and entropy change of thermoelasticity are analysed, respectively. In the fourth section, two additional remarks are presented. Then, in the end conclusions are given.

2. Energy Conversion During the Thermoelastic Effect

Pior to analyzing energy conservation for thermoelasticity, energy conservation is first reviewed for the simple and uncoupled fields, respectively. Consider an arbitray volume V within a medium, then the total thermal energy $E_T$ stored within is expressed as

$$\begin{array}{ccc}\hfill E_T& =& {\int }_V{e}_{T}dv={\int }_V{\int }_0^T\rho C_p\left(\theta \right)d\theta dv,\hfill \end{array}$$

(1)

where $e_T$ is the thermal energy density, $C_p\left(\theta \right)$ is the specific heat capacity and $\rho$ is the mass density. Then the law of conservation of energy in the thermal field is

$$\begin{array}{ccc}\hfill {\oint }_S{\overrightarrow{J}}_q·\overrightarrow{n}ds& =& {\displaystyle \frac{d{E}_T}{dt}}\hfill \\ \hfill {\int }_V\nabla ·{\overrightarrow{J}}_{q}dv& =& {\displaystyle \frac{d}{dt}}{\int }_V{\int }_0^T\rho C_{p}d\theta dv={\int }_V{\displaystyle \frac{\partial }{\partial t}}\left({\int }_0^T\rho C_{p}d\theta \right)dv\hfill \\ & =& {\int }_V\rho {\displaystyle \frac{\partial }{\partial T}}\left({\int }_0^T{C}_{p}d\theta \right){\displaystyle \frac{\partial T}{\partial t}}dv={\int }_V\rho C_p{\displaystyle \frac{\partial T}{\partial t}}dv.\hfill \end{array}$$

(2)

where ${\overrightarrow{J}}_q$ is the heat flux and S is the total surface area of the volume V. Thus,

$$\begin{array}{c}\hfill {\int }_V(\rho C_p{\displaystyle \frac{\partial T}{\partial t}}-\nabla ·{\overrightarrow{J}}_q)dv=0,\end{array}$$

Since the volume V is arbitrary, then

$$\begin{array}{c}\hfill \rho C_p{\displaystyle \frac{\partial T}{\partial t}}-\nabla ·{\overrightarrow{J}}_q=0.\end{array}$$

(3)

Fourier’s law of thermal conduction is

$$\begin{array}{c}\hfill {\overrightarrow{J}}_q=-k_T\nabla T\end{array}$$

(4)

where $k_T$ is thermal conductivity. Substitute it into Eqn (3), then the following governing equation of thermal conduction is obtained,

$$\begin{array}{ccc}\hfill \rho C_p{\displaystyle \frac{\partial T}{\partial t}}& =& k_T{\nabla }^{2}T\hfill \\ \hfill {\displaystyle \frac{\partial T}{\partial t}}& =& {\displaystyle \frac{k_T}{\rho C_p}}{\nabla }^{2}T=k{\nabla }^{2}T\hfill \end{array}$$

(5)

where k is the thermal diffusivity. It is evident that the above governing equation satisfys the law of conservation of energy in the thermal field.

For the elastic field, the total elastic energy $E_E$ and kinetic energy K inside the volume V is expressed as

$$\begin{array}{c}\hfill E_E={\int }_V{e}_{E}dv={\int }_V{\displaystyle \frac{1}{2}}{C}_{ijkl}{\epsilon }_{ij}^e{\epsilon }_{kl}^{e}dv,K={\int }_V{\displaystyle \frac{1}{2}}\rho v_i{v}_{i}dv,\end{array}$$

(6)

where $e_E$ is the elastic energy density, $C_{ijkl}$ is the elastic modulus and assumed constant for simplicity, ${\epsilon }_{ij}^e$ is the infinitesimal strain and $v_i$ is the velocity. Then the law of conservation of energy is

$$\begin{array}{ccc}\hfill {\int }_V\rho f_i{v}_{i}dv+{\oint }_S{v}_i{t}_{i}ds& =& {\displaystyle \frac{d{E}_E}{dt}}+{\displaystyle \frac{dK}{dt}}\hfill \\ & =& {\displaystyle \frac{d}{dt}}{\int }_V{\displaystyle \frac{1}{2}}{C}_{ijkl}{\epsilon }_{ij}^e{\epsilon }_{kl}^{e}dv+{\displaystyle \frac{d}{dt}}{\int }_V{\displaystyle \frac{1}{2}}\rho v_i{v}_{i}dv\hfill \\ & =& {\int }_V{c}_{ijkl}{\epsilon }_{kl}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^e}{\partial t}}dv+{\int }_V\rho v_i{a}_{i}dv\hfill \\ & =& {\int }_V{\sigma }_{ij}{v}_{i,j}dv+{\int }_V\rho v_i{a}_{i}dv\hfill \\ & =& {\oint }_S{\sigma }_{ij}{v}_i{n}_{j}ds-{\int }_V({\sigma }_{ij,j}-\rho a_i)v_{i}dv\hfill \\ & =& {\oint }_S{v}_i{t}_{i}ds-{\int }_V({\sigma }_{ij,j}-\rho a_i)v_{i}dv\hfill \end{array}$$

(7)

where $f_i$ is the body force density, $t_i$ is the surface traction, ${\sigma }_{ij}=c_{ijkl}{\epsilon }_{kl}^e$ is the elastic stress and $a_i$ is the acceleration; at the third step, ${\displaystyle \frac{\partial {\epsilon }_{ij}^e}{\partial t}}={\displaystyle \frac{\partial }{\partial t}}\left[{\displaystyle \frac{1}{2}}(u_{i,j}+u_{j,i})\right]={\displaystyle \frac{1}{2}}(v_{i,j}+v_{j,i})$ and the symmetry of indices i and j is used. Rearranging the above equation directly leads to

$$\begin{array}{c}\hfill {\int }_V({\sigma }_{ij,j}+\rho f_i-\rho a_i)v_{i}dv=0\end{array}$$

Since the volume V is arbitrary, then Cauchy’s equation of motion is obtained,

$$\begin{array}{c}\hfill {\sigma }_{ij,j}+\rho f_i=\rho a_i.\end{array}$$

(8)

It is also evident that Cauchy’s equation of motion satisfys the law of conservation of energy in the elastic field.

Next, the law of conservation of energy is examined during the coupled thermoelastic effect. Given the inherent coupling between the thermal and elastic fields, the two energies should depends on both field variables. Accordingly, for the thermal field, the total thermal energy should be written as

$$\begin{array}{ccc}\hfill E_T& =& {\int }_V{e}_{T}dv={\int }_V{\int }_0^T\rho C_p(\theta ,{\epsilon }_{ij})d\theta dv,\hfill \end{array}$$

(9)

where ${\epsilon }_{ij}$ is the total strain which consists of the elastic strain ${\epsilon }_{ij}^e$ and the thermal strain ${\epsilon }_{ij}^T$, i.e., ${\epsilon }_{ij}={\epsilon }_{ij}^e+{\epsilon }_{ij}^T$. The thermal strain is expresses as

$$\begin{array}{c}\hfill {\epsilon }_{ij}^T=\alpha {\delta }_{ij}(T-T_0),\end{array}$$

(10)

where $\alpha$ is the coefficient of linear thermal expansion, $T_0$ is the reference temperature and $T-T_0$ is a reasonable temperature variation so that the linear expansion of the thermal strain is valid. As a result, in Eqn (9), the explicit dependence of $C_p$ on temperature is involved in the integration with respect to $\theta$ from 0 to T, while the implicit dependence of $C_p$ on temperature via the total strain ${\epsilon }_{ij}$ is not involved in this integration since $T-T_0$ is a reasonable or small temperature variation.

Note that the specific heat capacity in Eqn (9) is expressed a function of ${\epsilon }_{ij}$. As a matter of fact, the heat capacity has already been shown to depend on elastic strains by experiments, simulations and theoretical analyses [10,11,12]. For instance, first-principle calculations have already revealed that the heat capacity of graphene increases with increasing uniaxial strains [11]; experimental measurements also have shown that the heat capacity increases with elongation in lightly cross-linked cis-1,4-polyisoprene [10]. Thus, it is reasonable to set the heat capacity as a function of elastic strains. In addition, consider the case of free thermal expansion, within the continuum body only thermal strains exist; that is, in this case there are no elastic stresses and strains and thus no elastic potential energy present. This in fact indicates the elastic potential energy can not depend on thermal strains. Since thermal strains are not related to the elastic kinetic energy either, leaving the thermal energy as the only energy thermal strains can be related to, then the heat capacity also has to be a function of thermal strains. Moreover, since both elastic strains and thermal strains correponds to the variations in interatomic or intermolecular distances, it is physically reasonable to treat the specific heat capacity as a function of both, or the total strain. Finally, in fact in tradional approaches, the entropy is usually treated as a function of the total strain [8,9]. Given the actual expression of entropy in Eqn (28) to be discussed in the next section, this also indicates the heat capacity has to be a function of the total strain.

For the elastic field, the total elastic energy is

$$\begin{array}{c}\hfill E_E={\int }_V{e}_{E}dv={\int }_V{\displaystyle \frac{1}{2}}{C}_{ijkl}({\epsilon }_{ij}-{\epsilon }_{ij}^T)({\epsilon }_{kl}-{\epsilon }_{kl}^T)dv,\end{array}$$

(11)

where the elastic strain, ${\epsilon }_{ij}^e={\epsilon }_{ij}-{\epsilon }_{ij}^T$, is substituted into $e_E$. As a result, the total thermal and elastic energy E for the coupled fields is

$$\begin{array}{c}\hfill E=E_T+E_E={\int }_V(e_T+e_E)dv={\int }_V\left[\rho {\int }_0^T{C}_p({\epsilon }_{ij},\theta )d\theta +{\displaystyle \frac{1}{2}}{C}_{ijkl}({\epsilon }_{ij}-{\epsilon }_{ij}^T)({\epsilon }_{kl}-{\epsilon }_{kl}^T)\right]dV.\end{array}$$

(12)

Hence, the law of conservation of energy in the coupled fields is

$$\begin{array}{ccc}& & {\int }_V\rho f_i{v}_{i}dV+{\oint }_S{v}_i{t}_{i}dS-{\oint }_S{\overrightarrow{J}}_q·\overrightarrow{n}dS\hfill \end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{dE}{dt}}+{\displaystyle \frac{dK}{dt}}\hfill \\ & =& {\int }_V\left[{\displaystyle \frac{\partial (e_T+e_E)}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{\partial (e_T+e_E)}{\partial {\epsilon }_{ij}}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right]dV+{\displaystyle \frac{d}{dt}}{\int }_V{\displaystyle \frac{1}{2}}\rho v_i{v}_{i}dV\hfill \\ & =& {\int }_V\left\{\left[\rho C_p({\epsilon }_{ij},T)-C_{ijkl}{\epsilon }_{kl}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right]{\displaystyle \frac{\partial T}{\partial t}}+\left[\rho {\int }_0^T{\displaystyle \frac{\partial C_p({\epsilon }_{ij},\theta )}{\partial {\epsilon }_{ij}}}d\theta +C_{ijkl}({\epsilon }_{kl}-{\epsilon }_{kl}^T)\right]{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right\}dV\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & +{\int }_V\rho v_i{a}_{i}dV\hfill \end{array}$$

(14)

where ${\displaystyle \frac{\partial (e_T+e_E)}{\partial {\epsilon }_{ij}}}$ can be defined as the generalized stress ${\sigma }_{ij}^g$,

$$\begin{array}{c}\hfill {\sigma }_{ij}^g={\displaystyle \frac{\partial (e_T+e_E)}{\partial {\epsilon }_{ij}}}=\rho {\int }_0^T{\displaystyle \frac{\partial C_p({\epsilon }_{ij},\theta )}{\partial {\epsilon }_{ij}}}d\theta +C_{ijkl}({\epsilon }_{kl}-{\epsilon }_{kl}^T)\end{array}$$

(15)

Then, the surface traction in the coupled fields is

$$\begin{array}{c}\hfill t_i=({\sigma }_{ij}^g+P{\delta }_{ij})n_j\end{array}$$

(16)

where P is the atmospheric pressure. Hence in Eqn (13), the work done by the surface traction becomes

$$\begin{array}{ccc}\hfill {\oint }_S{v}_i{t}_{i}dS& =& {\int }_S{v}_i({\sigma }_{ij}^g+P{\delta }_{ij})n_{j}dS\hfill \\ & =& {\int }_V{(v_i{\sigma }_{ij}^g+v_{i}P{\delta }_{ij})}_{,j}dV={\int }_V(v_{i,j}{\sigma }_{ij}^g+v_i{\sigma }_{ij,j}^g+P{v}_{i,i})dV\hfill \\ & =& {\int }_V(v_{i,j}{\sigma }_{ij}^g+v_i{\sigma }_{ij,j}^g+P{\displaystyle \frac{\partial {\epsilon }_{ii}}{\partial t}})dV.\hfill \end{array}$$

(17)

Substituting Eqn (17) into Eqn (14) leads to

$$\begin{array}{ccc}& & {\int }_V\rho f_i{v}_{i}dV+{\int }_V(v_{i,j}{\sigma }_{ij}^g+v_i{\sigma }_{ij,j}^g)dV+{\int }_{V}P{\displaystyle \frac{\partial }{\partial t}}\left({\epsilon }_{ii}\right)dV-{\int }_V\overrightarrow{\nabla }·{\overrightarrow{J}}_{q}dV\hfill \\ & =& {\int }_V\left\{\left[\rho C_p({\epsilon }_{ij},T)-C_{ijkl}{\epsilon }_{kl}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right]{\displaystyle \frac{\partial T}{\partial t}}+{\sigma }_{ij}^g{v}_{i,j}\right\}dV+{\int }_V\rho v_i{a}_{i}dV\hfill \end{array}$$

(18)

Rearrange the above equation, then we have

$$\begin{array}{ccc}& & {\int }_V(\rho f_i+{\sigma }_{ij,j}^g-\rho a_i)v_{i}dV-{\int }_V\overrightarrow{\nabla }·{\overrightarrow{J}}_{q}dV\hfill \\ & =& {\int }_V\left[\rho C_p({\epsilon }_{ij},T)-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right]{\displaystyle \frac{\partial T}{\partial t}}dV-{\int }_{V}P{\displaystyle \frac{\partial {\epsilon }_{ii}}{\partial t}}dV.\hfill \end{array}$$

(19)

Clearly, to ensure that energy is conserved in the elastic field, the integrand in first volume integral on the left-hand side of the above equation should be zero. Thus,

$$\begin{array}{c}\hfill \rho f_i+{\sigma }_{ij,j}^g-\rho a_i=0,\end{array}$$

(20)

then the equation of motion in the coupled fields is determined. Consequently, the rest of Eqn (19) gives the law of conservation of energy in the thermal field, which leads to

$$\begin{array}{ccc}\hfill -{\int }_V\overrightarrow{\nabla }·{\overrightarrow{J}}_{q}dV& =& {\int }_V\rho C_p({\epsilon }_{ij},T){\displaystyle \frac{\partial T}{\partial t}}dv-{\int }_V{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}dV-{\int }_{V}P{\displaystyle \frac{\partial {\epsilon }_{ii}}{\partial t}}dV,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill -\overrightarrow{\nabla }·{\overrightarrow{J}}_q& =& \left[\rho C_p({\epsilon }_{ij},T)-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right]{\displaystyle \frac{\partial T}{\partial t}}-P{\displaystyle \frac{\partial {\epsilon }_{ii}}{\partial t}}\hfill \end{array}$$

(22)

Substituting Eqn (4) into the above equation yields the governing equation for temperature variations in the coupled fields,

$$\begin{array}{c}\hfill \left[\rho C_p({\epsilon }_{ij},T)-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right]{\displaystyle \frac{\partial T}{\partial t}}-P{\displaystyle \frac{\partial {\epsilon }_{ii}}{\partial t}}=k_T{\nabla }^{2}T\end{array}$$

(23)

Note that the physical content of the above three equations, especially that of Eqn (21) needs careful examination. The term on the left-hand side of Eqn (21) is the total heat flowing into the considered volume V. While the first term on the right-hand side of Eqn (21) is the rate of increase of the thermal energy stored within the volume V; the third term is no doubt the rate of work done on the surrounding environment by the medium due to its volume expansion, which consequently leads to a decrease in the thermal energy and thus is negative; the integrand of the sencond volume integral is in fact ${\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}$, that is, the rate of work done by the elastic stresses on thermal strains. This rate of work has a negative sign and thus also leads to a decrease in the thermal energy, similar to the case in the third term. It is argued here that ${\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}$ is exactly the term responsible for the energy conversion between the thermal and elastic fields. To address this problem, it is quite benerficial to consider the rate of change of the total thermal and elastic energy in the coupled fields. The derviative of this total energy with time yields

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dE}{dt}}& =& {\displaystyle \frac{d}{dt}}\left\{{\int }_V\left[\rho {\int }_0^T{C}_p({\epsilon }_{ij},\theta )d\theta +{\displaystyle \frac{1}{2}}{C}_{ijkl}({\epsilon }_{ij}-{\epsilon }_{ij}^T)({\epsilon }_{kl}-{\epsilon }_{kl}^T)\right]dV\right\}\hfill \\ & =& {\int }_V\left\{\left[\rho C_p({\epsilon }_{ij},T)-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right]{\displaystyle \frac{\partial T}{\partial t}}+\left[\rho {\int }_0^T{\displaystyle \frac{\partial C_p({\epsilon }_{ij},\theta )}{\partial {\epsilon }_{ij}}}d\theta +{\sigma }_{ij}^e\right]{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right\}dV\hfill \\ & =& {\int }_V\left[\rho C_p({\epsilon }_{ij},T){\displaystyle \frac{\partial T}{\partial t}}-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}+\rho {\int }_0^T{\displaystyle \frac{\partial C_p({\epsilon }_{ij},\theta )}{\partial {\epsilon }_{ij}}}d\theta {\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}+{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^e}{\partial t}}+{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}\right]dV,\hfill \end{array}$$

At the second step, the first portion in the integrand is evidently the variation of this total energy due to the temperature variation in the thermal field; while the second portion is the variation of this total energy due to the variation of the total strain in the elastic field. Then, substitution of the expression for the total strain leads to the terms in the third step. It is straightforward to see that the second term and the fifth term within the square bracket at the third step cancel each other. They are in fact both the rate of work done by the elastic stress on the thermal strain. The second term, which is negative, leads to a decrease of the energy in the thermal field; while the fifth term, which is positive, results in an increase in the energy of the elastic field. Consequently, the total energy is conserved in the coupled fields and the term, ${\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}$, helps achieving energy conversion between these two fields.

Finally, in this section, Eqn (20) is revisited and its physical content is examined. During the thermoelastic process, there are thermal strains and elastic strains. Thermal strains arise from the thermal displacement due to thermal expansion of the medium, while elastic strains arise from the elastic displacement owing to externally applied loads. As a result, the total displacement $u_i$ consists of both the thermal displacement $u_i^T$ and the elastic displacement $u_i^e$, i.e.,

$$\begin{array}{c}\hfill u_i=u_i^T+u_i^e.\end{array}$$

Accordingly, the acceleration $a_i$ in Eqn (20) is also the total acceleration consisting of the following two contributions, i.e.,

$$\begin{array}{c}\hfill a_i=a_i^T+a_i^e={\displaystyle \frac{{\partial }^2{u}_i^T}{\partial t^2}}+{\displaystyle \frac{{\partial }^2{u}_i^e}{\partial t^2}}.\end{array}$$

Hence Eqn (15) and (20) can be rewritten as

$$\begin{array}{ccc}\hfill \rho f_i+{\sigma }_{ij,j}^g& =& \rho a_i^T+\rho a_i^e\hfill \\ \hfill {\sigma }_{ij}^g& =& \rho {\int }_0^T{\displaystyle \frac{\partial C_p({\epsilon }_{ij},\theta )}{\partial {\epsilon }_{ij}}}d\theta +C_{ijkl}{\epsilon }_{kl}^e\hfill \end{array}$$

(24)

Compare Eqn (24) with Eqn (8), it is straightforward to see that in the absence of elastic fields,

$$\begin{array}{ccc}\hfill {\sigma }_{ij,j}^{\prime g}& =& \rho a_i^T=\rho {\displaystyle \frac{{\partial }^2{u}_e^T}{\partial t^2}},\hfill \\ \hfill {\sigma }_{ij}^{\prime g}& =& \rho {\int }_0^T{\displaystyle \frac{\partial C_p({\epsilon }_{ij}^T,\theta )}{\partial {\epsilon }_{ij}^T}}d\theta .\hfill \end{array}$$

(25)

Note that, the body force generally induces elastic stresses and thus is mechanically balanced. As a result, in the case of free thermal expansion, the driving force for thermal expansion essentially comes from the thermal energy and it directly arises from the dependence of thermal energy on thermal strains.

3. Entropy Change During the Thermoelastic Effect

In this section, the entropy change during the thermoelastic effect is discussed. To begin with, the entropy change in each simple and uncoupled field is reviewed similarly for clarity. In the simple elastic field, the elastic potential energy shown in Eqn (6) is totally independent of thermperture since both the elastic strain and modulus are temperature-independent. As a result the entropy associated with the elastic process is $S_e=-{\displaystyle \frac{\partial E_e}{\partial T}}=0$. It is reasonable since the ideal elastic process is reversible and thus no energy is dissipated. On constrast, in the simple thermal field, its entropy is given by

$$\begin{array}{ccc}\hfill S_T& =& {\int }_V\rho {\int }_0^T{\displaystyle \frac{C_p\left(\theta \right)}{\theta }}d\theta dV.\hfill \end{array}$$

(26)

Thus, the rate of change of the thermal entropy [13] is

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{S}_T}{dt}}& =& {\displaystyle \frac{d}{dt}}{\int }_V\rho {\int }_0^{\tau }{\displaystyle \frac{C_P\left(\theta \right)}{\theta }}d\theta dV\hfill \\ & =& {\int }_V\rho {\displaystyle \frac{\partial }{\partial T}}\left({\int }_0^T{\displaystyle \frac{C_P}{\theta }}d\theta \right){\displaystyle \frac{\partial T}{\partial t}}dV\hfill \\ & =& {\int }_V\rho {\displaystyle \frac{C_P}{T}}{\displaystyle \frac{\partial T}{\partial t}}dV=-{\int }_V{\displaystyle \frac{\rho }{T}}\nabla ·{\overrightarrow{J}}_{q}dV\hfill \\ & =& -{\int }_V\left[\nabla ·\left(\rho {\displaystyle \frac{{\overrightarrow{J}}_q}{T}}\right)-\rho \nabla \left({\displaystyle \frac{1}{T}}\right)·{\overrightarrow{J}}_q\right]dV\hfill \\ & =& -{\oint }_S\rho {\displaystyle \frac{{\overrightarrow{J}}_q}{T}}·\overrightarrow{n}dS+{\int }_V\rho \left(-{\displaystyle \frac{1}{T^2}}\right)\nabla T·{\overrightarrow{J}}_{q}dV\hfill \\ & =& -{\oint }_S\rho {\displaystyle \frac{{\overrightarrow{J}}_q}{T}}·\overrightarrow{n}dS-{\int }_V{\displaystyle \frac{\rho }{T^2}}\nabla T·{\overrightarrow{J}}_{q}dV\hfill \\ & =& -{\oint }_S\rho {\displaystyle \frac{{\overrightarrow{J}}_q}{T}}·\overrightarrow{n}dS+{\int }_V{\displaystyle \frac{\rho }{T^2}}{\kappa }_T{\left|\nabla T\right|}^{2}dV\hfill \end{array}$$

(27)

where at the final steps, $\rho {\displaystyle \frac{{\overrightarrow{J}}_q}{T}}$ in the surface integral is the entropy flux; at the last step, after the substitution of Eqn (4), it is straightforward to see that in an isolated system with the entropy flux being zero, the total entropy is non-decreasing as time evolves, and thus, the second law of thermodynamics is satisfied.

Next, the entropy change for the couple thermoelastic effect can be examined. In the coupled fields, similar to the case in Eqn (9), the total entropy S can be rewriten as

$$\begin{array}{ccc}\hfill S& =& {\int }_V\rho {\int }_0^T{\displaystyle \frac{C_p({\epsilon }_{ij},\theta )}{\theta }}d\theta dV,\hfill \end{array}$$

(28)

that is, the total entropy is a function of both the temperature T and the total strain ${ϵ}_{ij}$ since, as argued previously, the heat capacity depends on both variables. Moreover, the elastic process in the coupled fields is still assumed to be an ideal reversible process so that its entropy remains zero. Then, the rate of change of the total entropy is

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& {\displaystyle \frac{d}{dt}}{\int }_V\rho {\int }_0^T{\displaystyle \frac{C_p({\epsilon }_{ij},\theta )}{\theta }}d\theta dV\hfill \\ & =& {\int }_V\left[\rho {\displaystyle \frac{\partial }{\partial T}}\left({\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right){\displaystyle \frac{\partial T}{\partial t}}+\rho {\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left({\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right){\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right]dV\hfill \\ & =& {\int }_V\rho {\displaystyle \frac{C_p}{T}}{\displaystyle \frac{\partial T}{\partial t}}dV+{\int }_V\rho \left({\int }_0^T{\displaystyle \frac{1}{\theta }}{\displaystyle \frac{\partial C_p}{\partial {\epsilon }_{ij}}}d\theta \right)v_{i,j}dV,\hfill \end{array}$$

(29)

where on the right-hand side of the above equation, the first and second terms are the contributions to the entropy change due to temperature variation and elastic process, respectively. Here, the second term can be rewritten as

$$\begin{array}{ccc}\hfill {\int }_V\rho {\int }_0^T{\displaystyle \frac{1}{\theta }}{\displaystyle \frac{\partial C_p}{\partial {\epsilon }_{ij}}}d\theta v_{i,j}dV& =& {\int }_V\left[{\left(\rho {\int }_0^T{\displaystyle \frac{1}{\theta }}{\displaystyle \frac{\partial C_p}{\partial {\epsilon }_{ij}}}d\theta v_i\right)}_{,j}-{\left(\rho {\int }_0^T{\displaystyle \frac{1}{\theta }}{\displaystyle \frac{\partial C_p}{\partial {\epsilon }_{ij}}}d\theta \right)}_{,j}{v}_i\right]dV\hfill \\ & =& {\oint }_S\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right)n_j\right]v_{i}dS-{\int }_V{\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right)\right]}_{,j}{v}_{i}dV.\hfill \end{array}$$

(30)

Note that ${\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{C}_{p}d\theta \right)$ is a portion of the generalized stress ${\sigma }_{ij}^g$ as shown in Eqn (15), thus

$$\begin{array}{c}\hfill {\oint }_S\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{C}_{p}d\theta \right)n_j\right]v_{i}dS-{\int }_V{\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{C}_{p}d\theta \right)\right]}_{,j}{v}_{i}dV\end{array}$$

(31)

is actually a portion of the rate of the work, ${\oint }_S{\sigma }_{ij}^g{n}_j{v}_{i}dS-{\int }_V{\left({\sigma }_{ij}^g\right)}_{,j}{v}_{i}dV$, done by the generalized stress. Comparing Eqn (31) with Eqn (30), it is straightforward to see that terms in Eqn (30) are actually the rates of production of entropy associated with the rates of work on the surface and within the bulk from the generalized stress, respectively. Similar to the case of naming ${\displaystyle \frac{{\overrightarrow{J}}_q}{T}}$ as the entropy flux when compared to the heat flux ${\overrightarrow{J}}_q$, it could also be reasonable to name ${\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right)$ as the entropy stress and ${\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right)n_j$ as the entropy contraction when compared to the elastic stress and elastic contraction. The elastic stress and contraction lead to the conversion of energy, while the entropy stress and contraction contribute to the production of entropy.

Of particulare interest is the first term on the right-hand side of Eqn (29), it can be rewritten as

$$\begin{array}{ccc}& & {\int }_V{\displaystyle \frac{1}{T}}\rho C_p{\displaystyle \frac{\partial T}{\partial t}}dV\hfill \\ & =& {\int }_V{\displaystyle \frac{1}{T}}\left[-\nabla ·{\overrightarrow{J}}_q+{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}+P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right]dV\hfill \\ & =& -{\int }_V\nabla ·\left({\displaystyle \frac{{\overrightarrow{J}}_q}{T}}\right)dV+{\int }_V\nabla \left({\displaystyle \frac{1}{T}}\right)·{\overrightarrow{J}}_{q}dV+{\int }_V\left[{\displaystyle \frac{1}{T}}{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{1}{T}}P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right]dV\hfill \\ & =& -{\oint }_S{\displaystyle \frac{{\overrightarrow{J}}_q}{T}}·\overrightarrow{n}dS+{\int }_V-{\displaystyle \frac{1}{T^2}}\nabla T·{\overrightarrow{J}}_{q}dV+{\int }_V{\displaystyle \frac{1}{T}}{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}dV+{\int }_V{\displaystyle \frac{1}{T}}P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}dV,\hfill \\ & =& -{\oint }_S{\displaystyle \frac{{\overrightarrow{J}}_q}{T}}·\overrightarrow{n}dS+{\int }_V{\displaystyle \frac{{\kappa }_T}{T^2}}{|\nabla T|}^{2}dV+{\int }_V{\displaystyle \frac{1}{T}}{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}dV+{\int }_V{\displaystyle \frac{1}{T}}P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}dV,\hfill \end{array}$$

(32)

where at the first step, Eqn (22) is substituted; and at the last step, Eqn (4) is substituted. It can be readily observed that at the last step, on right-hand side of Eqn (32), the first term is the net entropy flowing out of the surface of the considered volume; the second, third and fourth term are the rates of production of entropy during the processes of heat conduction, energy conversion and the work of volumetric expansion. Moreover, since both the energy conversion and the work of volumetric expansion substantially come from the thermal energy, thus their rate of production of entropy is ${\displaystyle \frac{1}{T}}{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial t}}$ and ${\displaystyle \frac{1}{T}}P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}$ respectively, similar to the relation $dS={\displaystyle \frac{dQ}{T}}$. Evidently, in the above analysis, each contribution to the entropy change can be explicitly demonstrated and its physical content can be clearly shown.

Finally, a prelimary discussion on the thermoelastic effect under thermal shock is presented in this section. Usually, due to thermal shock, the coupled effect is far from equilibrium and its governing equation will be a wave equation. In order to yield such a wave equation, it could be reasonable to include a kinetic contribution in the above total entropy in Eqn (28), i.e.,

$$\begin{array}{ccc}\hfill S& =& {\int }_V(\rho {\int }_0^T{\displaystyle \frac{C_p({\epsilon }_{ij},\theta )}{\theta }}d\theta -{\displaystyle \frac{1}{2}}{k}_J^{\prime }|{\overrightarrow{J}}_q{|}^2)dV,\hfill \end{array}$$

(33)

where $k_J^{\prime }$ is a proportional coefficient which describe the contribution of the heat flux ${\overrightarrow{J}}_q$ to the total entropy. Usually at the intial time of a thermal shock, the distribution of temperature is highly heterogeneous. Thus, at the area around the thermal shock zone, there is usually a very large heat flux present. Thus, it is reasonable to assume that the total entropy, under the influence thermal shock, also dependends on the heat flux. This contribution is assigned a negative sign so that the total entropy is maximized when eventually the thermal equilibrium is achieved, i.e., ${\overrightarrow{J}}_q=0$. Thus the rate of change of the total entropy is

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& {\displaystyle \frac{d}{dt}}{\int }_V(\rho {\int }_0^T{\displaystyle \frac{C_p({\epsilon }_{ij},\theta )}{\theta }}d\theta -{\displaystyle \frac{1}{2}}{k}_J^{\prime }|{\overrightarrow{J}}_q{|}^2)dV\hfill \\ & =& {\int }_V\left[\rho {\displaystyle \frac{\partial }{\partial T}}\left({\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right){\displaystyle \frac{\partial T}{\partial t}}+\rho {\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left({\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right){\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}-k_J^{\prime }{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}}·{\overrightarrow{J}}_q\right]dV.\hfill \end{array}$$

Following the same steps in the above analysis, it is straightforward to show that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dS}{dt}}& =& -{\oint }_S{\displaystyle \frac{{\overrightarrow{J}}_q}{T}}·\overrightarrow{n}dS-{\int }_V\left({\displaystyle \frac{1}{T^2}}\nabla T+k_J^{\prime }{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}}\right)·{\overrightarrow{J}}_{q}dV+{\int }_V{\displaystyle \frac{1}{T}}{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}dV+{\int }_V{\displaystyle \frac{1}{T}}P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}dV\hfill \\ & & +{\oint }_S\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right)n_j\right]v_{i}dS-{\int }_V{\left[{\displaystyle \frac{\partial }{\partial {\epsilon }_{ij}}}\left(\rho {\int }_0^T{\displaystyle \frac{C_p}{\theta }}d\theta \right)\right]}_{,j}{v}_{i}dV\hfill \end{array}$$

Note that in a system without elasticity present, then the third term to the sixth term can all be neglected. Furthermore, if the system is isolated, then the first term is zero. In this case, to guarantee that the second law of thermodynamics is satisfied, i.e., the total entropy is non-decreasing, then the heat flux can be defined as

$$\begin{array}{c}\hfill {\overrightarrow{J}}_q=-k({\displaystyle \frac{1}{T^2}}\nabla T+k_J^{\prime }{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}})=-k_T\nabla T-k_J{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}},\end{array}$$

where k is a proportional coefficient, $k_T=k{T}^2$ and $k_J=k{k}_J^{\prime }$. The above equation can also be rewritten as

$$\begin{array}{c}\hfill {\overrightarrow{J}}_q+k_J{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}}=-k_T\nabla T,\end{array}$$

(34)

as a result, the heat relaxation correction to Fourier’s law of thermal conduction is obtained. Taking the divergence of both sides of this equation leads to

$$\begin{array}{c}\hfill \nabla ·{\overrightarrow{J}}_q+k_J{\displaystyle \frac{\partial \nabla ·{\overrightarrow{J}}_q}{\partial t}}=-\nabla ·\left(k_T\nabla T\right).\end{array}$$

(35)

Substitution of Eqn(22) into the above equation finally yields

$$\begin{array}{c}\hfill \left(\rho C_p-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right){\displaystyle \frac{\partial T}{\partial t}}-P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}+k_J{\displaystyle \frac{\partial }{\partial t}}\left[\left(\rho C_p-{\sigma }_{ij}^e{\displaystyle \frac{\partial {\epsilon }_{ij}^T}{\partial T}}\right){\displaystyle \frac{\partial T}{\partial t}}-P{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}\right]=k_T{\nabla }^{2}T,\end{array}$$

(36)

which is apparently a wave equation. Note that the heat capacity is no longer a constant but a function of both the total strain and temperature, i.e., $C_p=C_p({\epsilon }_{ij},T)$. Thus, after a full expansion of the above equation, besides terms such as ${\displaystyle \frac{{\partial }^{2}T}{\partial t^2}}$ and ${\displaystyle \frac{{\partial }^2{\epsilon }_{ij}}{\partial t^2}}$, there are also nonlinear terms such as ${\displaystyle \frac{\partial T}{\partial t}}{\displaystyle \frac{\partial {\epsilon }_{ij}}{\partial t}}$ and ${\left({\displaystyle \frac{\partial T}{\partial t}}\right)}^2$ present in the equation. Though this governing equation is much more complex owing to thermal shock, the physical content of Eqn (36) is still straightforward. In Eqn (22), the right-hand side is in fact the rate of the net increase in the internal energy of the system, which is the increase in the thermal energy minus the sum of the energy converted into the elastic energy and the work used for volumetric expansion. Thus, the left-hand side of Eqn (36) consists of the first-order and second-order rates of the net increase in the internal energy, while the right-hand side is the rate of thermal energy conducted via the thermal gradient. In brief, compared with Eqn (23), for thermal conductions under thermal shock or far away from equilibrium, the thermal energy conducted via the thermal gradient not only contributes to the first-order rate of the net increase in the internal energy, but also contributes to its second-order rate. The underlying physical mechanism is very interesting but stil needs further and careful examinations.

4. Additional Remarks

In this section, two additional remarks are presented. The first one is on the differences between traditional approaches and this approach. First, in traditional approaches, the thermal energy and entropy are usually obtained as taylor expansions of temperature increments, thus the physical intepretations of these quantities are obscure and the same also holds for the terms in equations derived subsequently, because the coefficients of taylor expansions usually do not contain clear physical contents. Second, traditional approaches often resort to the free energy function to obtain the constitutive equations; then use the relation between heat and entropy, and treat entropy as a function of the strain and temperature to obtain a governing equation for the thermal field. Nevertheless, conservation of energy is not thoroughly discussed and especially conversion of energies are not addressed, even though both issues play crucial roles in coupling effects. Third, in traditional approaches, the production of entropy is discussed; however, it only serves as a thermodynamic criterion rather than a guideline to construct governing equations. On constrast, in this approach, exact physical expressions for energy and entropy are used so that the equations derived subsequently contain straightforward physical contents and the contributions to the change of energy or entropy can be discussed explicitly. Moreover, the law of conservation of energy is used directly to construct governing equations for both fields so that the conservation of energy, especially the coversion of energy between the coupled two fields can also be discussed explicitly. In addition, in the case of thermoelastic effect under thermal shock, it is shown that the second law of thermodynamics can be used directly to find the heat flux and thus help determine the governing equation for the thermal field. In fact in this approach, both the first and second laws of thermodynamics are used directly to guide the construction of governing equations, which thereby guarantees their consistency with fundamental thermodynamics laws. It is argued here that this approach has the potential of being developed as a general approach to analyse typical coupling effects, especially those involving the thermal field. That is, the second law of thermodynamics can be used to find an expression for the thermal flux or any other physical quantity which determines the evolving direction of the process; while the first law can be use to obtain the governing equations for all fields involved.

The second remark is about systems highly nonlinear or far away from equilibrium. Note that in the total entropy for the thermoelastic effect under thermal shock, as shown in Eqn (28), a kinetic contribution depending on the heat flux ${\overrightarrow{J}}_q$ is included. For the sake of consistency, a kinetic energy depending on ${\overrightarrow{J}}_q$, which is associated with the thermal field, should also be included in Eqn (13) while considering the conservation of energy. However, such an update in Eqn (13) leads to extremely messy terms such as ${\overrightarrow{J}}_q·{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}}$, $|{\displaystyle \frac{\partial {\overrightarrow{J}}_q}{\partial t}}{|}^2$, ${\overrightarrow{J}}_q·{\displaystyle \frac{{\partial }^2{\overrightarrow{J}}_q}{\partial t^2}}$ and no clear physical interpretation can be obtained for these terms. Similar situations occur in the case of the coupling effects between the elastic and electromagnetic fields. For electrostriction and magnetostriction, when the updated Maxwell’s second and fourth equations, which are Eqn (27) and (28) in our previous work [14] and are themselves highly nonlinear equations, are resubstituted into the equation of the law of conservation of momentumn, the Maxwell’s stress will then be updated and hence the energy associated with electrostriction and magnetostriction. Yet the expression for such an update in the energy has an extremely complex form and similarly no clear physical interpretation can be obtained for it. For thermodynamic systems under thermal shock or systems which behave highly nonlinearly, they are all systerms far away from equilibrium. For such systems, the traditional approaches resorting to linear consitutive equations and quadratic energy terms, which usually work for systems near equilibrium, often do not perform so well. Linear consitutive equations and quadratic energy terms are in fact the first or second order taylor expansion of physcial quantities, and the relevant governing equations are usually linear equations. It is hardly convincing that linear discriptions of physical systems based on the first or second order taylor expansion can be considered to be accurate and fully developed. However, for systems far away from equilibrium, their governing equations should also be concise and with each term carrying a clear physical interpretation. Thus, it is quite reasonable to argue that for systems far away from equilibrium, or in the highly nonlinear area of such systems, those concise yet profound physical laws are probably never given a chance to be fully and explicitly revealed.

5. Conclusions

In this paper, the law of conservation of energy in the coupled thermal and elastic fields is applied to analyse the conversion of energy for the thermoelastic effect. It is found that the work done by the elastic stresses on the thermal strains is responsible for the energy conversion, and governing equations which abide by the law of conservation of energy are obtained for both fields. Moreover, it is also shown that for free thermal expansion, the driving force which induces thermal expansion directly comes from the thermal energy owing to its dependence on thermal strains. Furthermore, entropy change during the thermoelastic effect is also analysed. it is found that a portion of the work done by the generalized stress on the surface and within the bulk of the system both leads to productions of entropy for the coupled effect; while for the process of thermal conduction, besides the net entropy flowing out of the surface due to the entropy flux, the processes of heat conduction, energy conversion and the work of volumetric expansion all leads to productions of entropy. All contributions for the above entropy changes are explicitly demonstrated and their physical contents are clearly shown. In addition, a prelimary discussion on the thermoelastic effect under thermal shock is presented. With an extra kinetic contribution included in the total entropy functional, the heat relaxation correction is obtained in the thermal conduction law and everntually a wave type governing equation is found for the conduction of heat under thermal shock. In the end, two additional remarks are presented.