## 1. Introduction

Classical thermodynamics discusses the thermodynamic system, its surroundings and their common boundary. It is concerned with the state of thermodynamic systems at equilibrium, using macroscopic, empirical properties directly measurable in the laboratory (Wang, 1955; Yunus, Michael and Boles, 2011). Classical thermodynamics model exchanges of energy, work and heat based on the laws of thermodynamics. The first law of thermodynamics is a principle of conservation of energy and defines a specific internal energy which is a state function of the system. The second law of thermodynamics is a principle to explain the irreversibile phenomenon in nature. The entropy of an isolated non-equilibrium system will tend to increase over time, approaching a maximum value at equilibrium. Thermodynamic laws are generally valid and can be applied to systems about which only knows the balance of energy and matter transfer. The thermodynamic state of the system can be described by a number of state variables. In continuum mechanics state variables usually are pressure

(1) |

Other thermodynamic state functions and their applications can be seen in many literatures (Kuang, 2007, 2008a, 2008b, 2009a, 2009b, 2010, 2011a, 2011b). For the case without chemical potential

In this chapter two new problems in the continuum thermodynamics will be discussed. The first is that in traditional continuum thermodynamics including the non-equilibrium theory the dynamic effect of the temperature is not fully considered. When the temperature

## 2. Inertial entropy theory

### 2.1. Basic theory in linear thermoelastic material

In this section we discuss the linear thermoelastic material without chemical reaction, so in Eq. (1) the term

(2) |

where

Using the theory of the usual irreversible thermodynamics (Groet, 1952; Gyarmati, 1970; Jou, Casas-Vzquez, Lebon, 2001; Kuang, 2002) from Eq. (3) we get

where

### 2.2. Temperature wave in linear thermoelastic material

The temperature wave from heat pulses at low temperature propagates with a finite velocity. So many generalized thermoelastic and thermopiezoelectric theories were proposed to allow a finite velocity for the propagation of a thermal wave. The main generalized theories are: Lord－Shulman theory (1967), Green－Lindsay theory (1972) and the inertial entropy theory (Kuang, 2009b).

In the Lord－Shulman theory the following Maxwell-Cattaneo heat conductive formula for an isotropic material was used to replace the Fourier’s law, but the classical entropy equation is kept, i.e. they used

where

(6) |

where *C* is the specific heat,

The Green－Lindsay theory with two relaxation times was based on modifying the Clausius-Duhemin inequality and the energy equation; In their theory they used a new temperature function

After linearization and neglecting small terms, finally they get (here we take the form in small deformation for an isotropic material)

where*γ* are material constants.

Now we discuss the inertial entropy theory (Kuang, 2009b). The Helmholtz free energy

(9) |

where

The constitutive (or state) and evolution equations are

Using Eq. (10), Eq. (9a) can be rewritten as (9b) where

Substituting the entropy

When material coefficients are all constants from（11）we get

(12) |

Eq. (12a) is a temperature wave equation with finite phase velocity. For an isotropic elastic material and the variation of the temperature is not large, from Eq. (12a) we get (12b)

Comparing the temperature wave equation Eq. (12b) with the Lord－Shulman theory (Eq. (6)) it is found that in Eq. (12b) a term

The momentum equation is

where

Comparing the elastic wave equation Eq. (14) with the Green－Lindsay theory (Eq. (8)) it is found that in Eq. (14) a term

### 2.3. Temperature wave in linear thermo - viscoelastic material

In the pyroelectric problem (without viscous effect) through numerical calculations Yuan and Kuang（2008, 2010）pointed out that the term containing the inertial entropy attenuates the temperature wave, but enhances the elastic wave. For a given material there is a definite value of

Using the irreversible thermodynamics (Groet, 1952; Kuang, 1999, 2002) we can assume

(15) |

where

Substituting the stress

In one dimensional problem for the isotropic material from Eq. (15) we have

where

When there is no body force and body heat source, Eqs. (12) and (16) are reduced to

where *x* we assume

where

In order to have nontrivial solutions for

(21) |

From Eq. (21) we get

(22) |

where the symbol “+” is applied to the wave number

Because

For the general case in Eq. (22) a coupling term

(24) |

we get

Introducing the viscoelastic effect in the elastic wave as shown in this section can substantially eliminate the increasing effect of the amplitude of the elastic wave with time.

### 2.4. Temperature wave in thermo-electromagneto-elastic material

In this section we discuss the linear thermo-electromagneto-elastic material without chemical reaction and viscous effect, so the electromagnetic Gibbs free energy

(25) |

where

(26) |

Similar to derivations in sections 2.2 and 2.3 it is easy to get the governing equations:

where

### 2.5. Thermal diffusion wave in linear thermoelastic material

The Gibbs equation of the classical thermodynamics with the thermal diffusion is:

where

(30) |

where

(31) |

where *T* is not too large, Eq. (31a) can also be approximated by (31b)

Especially the coefficients

Eq. (29) shows that in the equation of the heat flow the role of

(32) |

where

(33) |

where

(34) |

Using Eq. (34)

where

(35) |

If we neglect the term in second order

The formulas in literatures analogous to Eq. (34) can be found, such as in Sherief, Hamza, and Saleh’s paper (2004), where they used the Maxwell-Cattaneo formula.

The momentum equation is

The above theory is easy extended to more complex materials.

## 3. Physical variational principle

### 3.1. General theory

Usually it is considered that the first law of thermodynamics is only a principle of the energy conservation. But we found that the first law of thermodynamics is also a physical variational principle (Kuang, 2007, 2008a, 2008b, 2009a 2011a, 2011b). Therefore the first law of the classical thermodynamics includes two aspects: energy conservation law and physical variational principle:

(38) |

where *W* is the work applied on the body by the environment,

(39) |

Here the physical variational principle is considered to be one of the fundamental physical law, which can be used to derive governing equations in continuum mechanics and other fields. We can also give it a simple explanation that the true displacement is one kind of the virtual displacement and obviously it satisfies the variational principle. Other virtual displacements cannot satisfy this variational principle, otherwise the first law is not objective. The physical variational principle is different to the usual mathematical variational method which is based on the known physical facts. In many problems the variation of a variable

### 3.2. Physical variational principle in thermo-elasticity

In the thermo-elasticity it is usually considered that only the thermal process is irreversible, but the elastic process is reversible. So the free energy

In previous paper (Kuang, 2011a) we showed that the migratory variation of virtual electric and magnetic potentials will produce the Maxwell stress in electromagnetic media, which is also shown in section 3.4 of this paper. Similarly the migratory variation

Under assumptions that the virtual mechanical displacement

(41) |

where

It is noted that

(42) |

Finishing the variational calculation, we have

(43) |

where

(44) |

Here

The above variational principle requests prior that displacements and the temperature satisfy the boundary conditions, so in governing equations the following equations should also be added

Eqs. (44) and (45) are the governing equations of the thermo-elasticity derived from the physical variational principle.

### 3.3. Physical variational principle in thermo-diffusion theory

The electro-chemical Gibbs free energy

Under assumptions that the mechanical displacement

(46) |

In Eqs. (46)

It is noted that we have the following relations

(47) |

The further derivation is fully similar to that in the thermo-elasticity. Combining Eqs. (46) and (47) we get

(48) |

where

Due to the arbitrariness of

and

(51) |

where

The first two formulas in Eq. (51) can be rewritten as

The last equation in Eq. (52) is just the same as that in Eq. (32).

The above variational principle requests prior that the

Eqs. (49)-(53) are the governing equations of the generalized thermodiffusion theory.

If we neglect the term

If we also assume that

## 3.4. Physical variational principle in electro-magneto-elastic analysis

In this section we discuss the nonlinear electro-magneto-elastic media. Here we extend the theory in previous paper (Kuang, 2011) to the material with the electromagnetic body couple. Because the asymmetric part of the stress is introduced by the electromagnetic body couple, the specific electromagnetic Gibbs free energy

(56) |

where

(57) |

(58) |

where

Using Eq. (57), Eq. (56a) can be reduced to (56b)

Because the value of the term

In the nonlinear electro-magneto-elastic analysis the medium and its environment should be considered together as shown in Fig. 1 (Kuang, 2011a, 2011b), because the electromagnetic field exists in all space. Under the assumption that

boundary conditions on their own boundaries

(59) |

where the superscript “env” means the variable in environment, “int” means the variable on the interface,

As shown in previous paper (Kuang, 2011a, 2011b) and in section 3.1. the variations of

(60) |

Noting that in Eq. (59) we have

(61) |

where

(62) |

into

(63) |

where

Substituting Eq. (63) into Eq. (61) we get

(65) |

where

(66) |

For the environment we have the similar formula:

(67) |

Using

The above variational principle requests prior that the displacements, the electric potential and the magnetic potential satisfy their own boundary conditions and the continuity conditions on the interface, so the following equations should also be added to governing equations

(69) |

Eqs. (66)－(69) are the governing equations. It is obvious that the above physical variational principle is easy to extend to other materials.

## 3.5. Materials with static magnetoelectric coupling effect

In this section we discuss the electro-magneto-elastic media with static magnetoelectric coupling effect shortly. For these materials the constitutive equations are

(70) |

where

## 4. Conclusions

In this chapter some advances of thermodynamics in continuum mechanics are introduced. We advocate that the first law of the thermodynamics includes two contents: one is the energy conservation and the other is the physical variational principle which is substantially the momentum equation. For the conservative system the complete governing equations can be obtained by using this theory and the classical thermodynamics. For the non-conservative system the complete governing equations can also be obtained by using this theory and the irreversible thermodynamics when the system is only slightly deviated from the equilibrium state. Because the physical variational principle is tensely connected with the energy conservation law, so we write down the energy expressions, we get the physical variational principle immediately and do not need to seek the variational functional as that in usual mathematical methods.

In this chapter we also advocate that the accelerative variation of temperature needs extra heat and propose the general inertial entropy theory. From this theory the temperature wave and the diffusion wave with finite propagation velocities are easily obtained. It is found that the coupling effect in elastic and temperature waves attenuates the temperature wave, but enhances the elastic wave. So the theory with two parameters by introducing the viscous effect in this problem may be more appropriate.

Some explanation examples for the physical variational principle and the inertial entropy theory are also introduced in this chapter, which may indirectly prove the rationality of these theories. These theories should still be proved by experiments.