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, volume, stress, strain, electric field strength, electric displacement, magnetic induction density, magnetic field strength, temperature, entropy per volume, chemical potential per volume and concentration respectively. Conjugated variable pairs are. There is a convenient and useful combination system in continuum mechanics: variables are used as independent variables and variables are used as dependent variables. In this chapter we only use these conjugated variable pairs, and it is easy to extend to other conjugated variable pairs. In the later discussion we only use the following thermodynamic state functions: the internal energy and the electro-magneto-chemical Gibbs free energy per volume in an electro-magneto-elastic material. They are taken as
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 is the electromagnetic Gibbs free energy. For the case without electromagnetic field is the Gibbs free energy with chemical potential. For the case without chemical potential and electromagnetic field is the Helmholtz free energy.
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 is varied, the extra heat or entropy should be input from the environment. When is varied, the extra chemical potential is also needed. So the general inertial entropy theory (Kuang, 2009b, 2010) is introduced into the continuum thermodynamics. The temperature and diffusion waves etc. with finite phase velocity can easily be obtained from this theory. The second is that usually we consider the first law only as a conservation law of different kinds of energies, but we found that it is also containing a physical variational principle, which gives a true process for all possible process satisfying the natural constrained conditions (Kuang, 2007, 2008a, 2008b, 2009a 2011a, 2011b). Introducing the physical variational principle the governing equations in continuum mechanics and the general Maxwell stress and other theories can naturally be obtained. When write down the energy expression, we get the physical variational principle immediately and do not need to seek the variational functional as that in the usual mathematical methods. The successes of applications of these theories in continuum mechanics are indirectly prove their rationality, but the experimental proof is needed in the further.
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 is omitted. It is also noted that in this section the general Maxwell stress is not considered. The classical thermodynamics discusses the equilibrium system, but when extend it to continuum mechanics we need discuss a dynamic system which is slightly deviated from the equilibrium state. In previous literatures one point is not attentive that the variation of temperature should be supplied extra heat from the environment. Similar to the inertial force in continuum mechanics we modify the thermodynamic entropy equation by adding a term containing an inertial heat or the inertial entropy (Kuang, 2009b), i.e.
where is called the reversible inertial entropy corresponding to the inertial heat; is called the inertial entropy coefficient, is also a constant having the dimension of the time; is the entropy saved in the system, and are the reversible and irreversible parts of the, is the absorbed heat rate of the system from the environment, is the inertial heat rate and is proportional to the acceleration of the temperature; is the external heat source strength, is the heat flow vector per interface area supplied by the environment, is the entropy displacement vector, is the entropy flow vector. Comparing Eq. (2) with the classical entropy equation it is found that in Eq. (2) we use to instead of in the classical theory. In Eq. (2) is still a state function because is reversible. As in classical theory the dissipative energy and its Legendre transformation or “the complement dissipative energy” are respectively
where is the usual heat conductive coefficient. Eq. (4) is just the Fourier’s law.
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 is a material parameter with the dimension of time. After linearization and neglecting many small terms they got the following temperature wave and motion equations for an isotropic material:
where C is the specific heat, is the thermal expansion coefficient, G and are the shear modulus and Poisson’s ratio respectively. From Eq.（5） we can getFrom above equation it is difficult to consider that is a state function.
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 to replace the usual temperature. They used
After linearization and neglecting small terms, finally they get (here we take the form in small deformation for an isotropic material)
where, and γ are material constants.
Now we discuss the inertial entropy theory (Kuang, 2009b). The Helmholtz free energy and the complement dissipative energy assumed in the form
where is the reference (or the environment) temperature, are material constants. In Eq. (9a) it is assumed that when or. It is obvious that.
The constitutive (or state) and evolution equations are
When material coefficients are all constants from（11）we get
Comparing the temperature wave equation Eq. (12b) with the Lord－Shulman theory (Eq. (6)) it is found that in Eq. (12b) a term is lacked (in different notations),but with that in the Green－Lindsay theory (Eq. (8)) is similar (in different notations). For the purely thermal conductive problem three theories are fully the same in mathematical form.
The momentum equation is
Comparing the elastic wave equation Eq. (14) with the Green－Lindsay theory (Eq. (8)) it is found that in Eq. (14) a term is lacked (in different notations), but with the Lord－Shulman theory (Eq. (6)) is similar (in different notations).
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, when the amplitude of the elastic wave will be increased with time. For is about. In the Lord－Shulman theory critical value is about. In order to substantially eliminate the increasing effect of the amplitude of the elastic wave the viscoelastic effect is considered as shown in this section.
where and are the reversible and irreversible parts of the stress,. Comparing Eqs. (9) and (10) with (15) it is found that only a term is added to the rate of the complement dissipative energy in Eq. (15). Substituting the entropy and in Eq. (15) and in (2) into in Eq. (2) we still get the same equation (12).
In one dimensional problem for the isotropic material from Eq. (15) we have
where is the elastic modulus, is a viscose coefficient, is the temperature coefficient.
where for any function. For a plane wave propagating along direction x we assume
In order to have nontrivial solutions for, the coefficient determinant of Eq. (20) should be vanished:
From Eq. (21) we get
where the symbol “+” is applied to the wave number of the temperature wave and the symbol “” is applied to the wave number of the viscoelastic wave. If the temperature wave does not couple with the elastic wave, then is equal to zero. In this case we have
Because due to and due to, a pure viscoelastic wave or a pure temperature waves is attenuated. The pure elastic wave does not attenuate due to.
For the general case in Eq. (22) a coupling term is appeared. It is known that It means that or the temperature wave is always an attenuated wave. If
we get or in this case the elastic wave is an attenuated wave, otherwise is enhanced.
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 in Eq. (1) should keep the temperature variable. The electromagnetic Gibbs free energy and the complement dissipative energy in this case are assumed respectively in the following form
where are material constants. The constitutive equations are
Similar to derivations in sections 2.2 and 2.3 it is easy to get the governing equations:
where is the density of the electric charge. The boundary conditions are omitted here.
2.5. Thermal diffusion wave in linear thermoelastic material
The Gibbs equation of the classical thermodynamics with the thermal diffusion is:
where is the chemical potential, is the flow vector of the diffusing mass, is the concentration. In discussion of the thermal diffusion problem we can also use the free energy (Kuang, 2010), but here it is omitted. Using relationsFrom Eq. (29) (Kuang, 2010) we get:
where is the irreversible heat rate. According to the linear irreversible thermodynamics the irreversible forces are proportional to the irreversible flow (Kuang, 2010; Gyarmati, 1970; De Groet, 1952), we can write the evolution equations in the following form
where is the diffusing coefficients and is the coupling coefficients. The linear irreversible thermodynamics can only give the general form of the evolution equation, the concrete exact formula should be given by experimental results. Considering the experimental facts and the simplicity of the requirement for the variational formula, when the variation of T is not too large, Eq. (31a) can also be approximated by (31b)
Eq. (29) shows that in the equation of the heat flow the role of is somewhat equivalent to. So analogous to the inertial entropy we can also introduce the inertial concentration and introduce a general inertial entropy theory of the thermal diffusion problem. Eq. (29) in the general inertial entropy theory is changed to (Kuang, 2010)
where is the inertial concentration coefficient. Applying the irreversible thermodynamics we can get the Gibbs free energy and the complement dissipative energy as
where are also material constants. The constitutive and evolution equations are:
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:
where is the internal energy per volume, W is the work applied on the body by the environment, is the heat supplied by the environment. According to Gibbs theory when the process is only slightly deviated from the equilibrium state can be substituted by. In practice we prefer to use the free energy:
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 different with displacement, should be divided into local variation and migratory variation, i.e. the variation, where the local variation of is the variation duo to the change of itself and the migratory variation of is the variation of change of due to virtual displacements. In Eqs. (38) and (39) the new force produced by the migratory variation will enter the virtual work or as the same as the external mechanical force. But in the following sections we shall modify Eq. (39) or (38) to deal with this problem. The physical variational principle is inseparable with energy conservation law, so when the expressions of energies are given we get physical variational principle immediately. We need not to seek the variational functional as that in usual mathematical methods. In the following sections we show how to derive the governing equations with the general Maxwell stress of some kind of materials by using the physical variational principle. From this physical variational principle all of the governing equations in the continuum mechanics and physics can be carried out and this fact can be considered as the indirect evidence of the physical variational principle.
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 and the complement dissipative energy can be assumed as that in Eq. (9). The corresponding constitutive and evolution equations are expressed in Eq. (10). As shown in section 3.1, the variation of the virtual temperature is divided into local variation due to the variation of itself and the migratory variation due to:
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 will also produce the general Maxwell stress which is an external temperature stress. The effective general Maxwell stress can be obtained by the energy principle as that in electromagnetic media.
Under assumptions that the virtual mechanical displacement and the virtual temperature satisfy their own boundary conditions on and respectively. The physical variational principle using the free energy in the inertial entropy theory for the thermo-elasticity can be expressed as:
where and are the given mechanical body force, surface traction and surface entropy flow respectively. Eq. (41) is an alternative form of Eq. (39). In Eq. (41) the term is the complement dissipative heat rate per volume corresponding to the inner complement dissipation energy rate. The entropy includes the contribution of. The fact that the complement dissipation energy rate in and the internal irreversible complement heat rate in are simultaneously included in Eq. (41) allows us to get the temperature wave equation and the boundary condition of the heat flow from the variational principle. In Eq. (41) there are two kinds of variational formulas. The first is, in which the integrands contain variables themselves. The second is, in which the integrands contain the time derivatives of variables, so it needs integrate with time t. This is the common feature of the irreversible process because in the irreversible process the integral is dependent to the integral path.
It is noted that
Finishing the variational calculation, we have
where is the effective or equivalent general Maxwell stress which is the external equal axial normal temperature stress. This general Maxwell stress is first introduced and its rationality should be proved by experiments. Obviously can be neglected for the case of the small strain and small change of temperature. In Eq. (43) it is seen that is appeared in a whole. Using and the arbitrariness of and, from Eq. (43) we get
Here is the external temperature body force and is the surface traction.
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
3.3. Physical variational principle in thermo-diffusion theory
Under assumptions that the mechanical displacement, the temperature and the chemical potential satisfy their own boundary conditions, and on,and respectively. When the variation of temperature is not large the physical variational principle for the thermo-elasto-diffusive problem is
In Eqs. (46) and are given values. In Eq. (46)is related to heat (including the heat produced by the irreversible process in the material), is related to the diffusion energy. Eq. (46) shows that there is no term in corresponding to the term, so it should not be included in and.
It is noted that we have the following relations
Due to the arbitrariness of,and, from Eq. (48) we get
where has been used.
The first two formulas in Eq. (51) can be rewritten as
The above variational principle requests prior that the and satisfy their own boundary conditions, so in governing equations the following equations should also be added
If we neglect the term in Eq. (32), or is adopted, then we easily get
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 is taken as
where and are the electrostrictive and megnetostrictive constants respectively; and may be asymmetric. The corresponding constitutive equations are
where have been used, and are the polarization density and magnetization density, and are the dielectric constant and magnetic permeability in vacuum respectively. The terms containing in and in Eq. (58) have been neglected. In the usual electromagnetic theory the electromagnetic body couple is. From Eq. (58) it is seen that or the electromagnetic body couple is balanced by the moment produced by the asymmetric stresses.
Because the value of the term is much less than that of other terms, it can be neglected.
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 satisfy their
boundary conditions on their own boundaries and the continuity conditions on the interface. The Physical variational principle in the nonlinear electro-magneto-elastic analysis is
where the superscript “env” means the variable in environment, “int” means the variable on the interface, are the given values on the corresponding surfaces. Eq. (59) is an alternative form of Eq. (39) and the electromagnetic force is directly enclosed in the formula (Kuang, 2008a, 2009a).
where is the part of due to the local variations of; is the part of due to the migratory variations of. Substituting the following identity
into in Eq. (61) we get
where is the Maxwell stress:
where is the pseudo total stress (Jiang and Kuang, 2004), which is not the true stress in electromagnetic media. From the expression of it is known that is symmetric though and are asymmetric. Due to the arbitrariness of and, from Eq. (65) we get
For the environment we have the similar formula:
Using and on the boundary surface we get
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
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
where is the static magnetoelectric coupling coefficient. The electromagnetic body couple is still balanced by the asymmetric stress, i.e. In this case though the constitutive equations are changed, but the electromagnetic Gibbs free energy in Eq. (56b), governing equations (66)－(69) and the Maxwell stress (64) are still tenable.
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.