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Some Thermodynamic Problems in Continuum Mechanics

Written By

Zhen-Bang Kuang

Submitted: 21 November 2010 Published: 22 September 2011

DOI: 10.5772/22610

From the Edited Volume

Thermodynamics - Kinetics of Dynamic Systems

Edited by Juan Carlos Moreno Piraján

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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 pressurep, volumeV, stressσ, strainε, electric field strengthE, electric displacementD, magnetic induction densityB, magnetic field strengthH, temperatureT, entropy per volumes, chemical potential per volume μand concentration c respectively. Conjugated variable pairs are(p,V),(σ,ε),(E,D),(H,B),(T,S),(μ,c). There is a convenient and useful combination system in continuum mechanics: variables V,ε,E,H,T,μ are used as independent variables and variables p,σ,D,B,S,c 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 U and the electro-magneto-chemical Gibbs free energy geμ(ε,E,H,T,μ) 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 geμ=ge is the electromagnetic Gibbs free energy. For the case without electromagnetic field geμ=gμ is the Gibbs free energy with chemical potential. For the case without chemical potential and electromagnetic field geμ=g 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 T is varied, the extra heat or entropy should be input from the environment. When c is varied, the extra chemical potential μ 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 DdEBdHcdμ 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 s(a) is called the reversible inertial entropy corresponding to the inertial heat; ρsis called the inertial entropy coefficient, ρs0is also a constant having the dimension of the time; sis the entropy saved in the system, s˙(r)and s˙(i) are the reversible and irreversible parts of thes, Ts˙is the absorbed heat rate of the system from the environment, Ts˙(a)=ρsTT¨is the inertial heat rate and s˙(a) is proportional to the acceleration of the temperature; ris the external heat source strength, qis the heat flow vector per interface area supplied by the environment, ηis the entropy displacement vector, η˙is the entropy flow vector. Comparing Eq. (2) with the classical entropy equation it is found that in Eq. (2) we use Ts˙+Ts˙(a) to instead of Ts˙ in the classical theory. In Eq. (2) sis still a state function because s(a) is reversible. As in classical theory the dissipative energy h˙ and its Legendre transformation or “the complement dissipative energy” hare respectively


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 λ 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 τ0 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, αis the thermal expansion coefficient, G and ν are the shear modulus and Poisson’s ratio respectively. From Eq.(5) we can getTs˙τ0Ts¨=λT,ii+(r˙+τ0r¨)From above equation it is difficult to consider that s 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 ϕ(T,T˙) to replace the usual temperatureT. They used


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


whereτ0, τ1and γ are material constants.

Now we discuss the inertial entropy theory (Kuang, 2009b). The Helmholtz free energy g and the complement dissipative energy h assumed in the form


where T0 is the reference (or the environment) temperature, Cijkl,αijare material constants. In Eq. (9a) it is assumed that s=0 when T=T0 orϑ=0. It is obvious thatT,j=ϑ,j,T˙=ϑ˙.

The constitutive (or state) and evolution equations are


Using Eq. (10), Eq. (9a) can be rewritten as (9b) where g(T) is the energy containing the effect of the to temperature.

Substituting the entropy s and Tη˙i in Eq. (10) and s(a) in (2) into Ts˙+Ts˙(a)=r˙(Tη˙i),i in Eq. (2) we get


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


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 τ0ε¨jj 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


where f is the body force per volume, ρis the density. Substituting the stress σ in Eq. (10) into (13) we get


Comparing the elastic wave equation Eq. (14) with the Green-Lindsay theory (Eq. (8)) it is found that in Eq. (14) a term γτ1ϑ˙,i 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 ofρs0, when ρs0>ρs0 the amplitude of the elastic wave will be increased with time. For BaTio3ρs0 is about1013s. In the Lord-Shulman theory critical value τ0is about108s. 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.

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


where σij(r) and σij(i) are the reversible and irreversible parts of the stressσij,ε˙ij=dεij/dt. Comparing Eqs. (9) and (10) with (15) it is found that only a term βijklε˙jiδε˙lk is added to the rate of the complement dissipative energy in Eq. (15). Substituting the entropy s and Tη˙i in Eq. (15) and s(a) in (2) into Ts˙+Ts˙(a)=r˙(Tη˙i),i in Eq. (2) we still get the same equation (12).

Substituting the stress σ in Eq. (15) into (13) we get


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


where Y is the elastic modulus, βis a viscose coefficient, αis the temperature coefficient.

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


where φ˙=φ/t,φ=φ/x for any functionφ. For a plane wave propagating along direction x we assume


where U,Θ are the amplitudes of u and ϑ respectively, kis the wave number and ω is the circular frequency. Substituting Eq. (19) into (18) we obtain


In order to have nontrivial solutions forU,Θ, the coefficient determinant of Eq. (20) should be vanished:


From Eq. (21) we get


where the symbol “+” is applied to the wave number kT of the temperature wave and the symbol “” is applied to the wave number of the viscoelastic wavekY. If the temperature wave does not couple with the elastic wave, then αis equal to zero. In this case we have


Because θY>0due to β>0 and θT>0 due toρs0>0, a pure viscoelastic wave or a pure temperature waves is attenuated. The pure elastic wave does not attenuate due toβ=0.

For the general case in Eq. (22) a coupling term iα2T0ωk2 is appeared. It is known that |Im(CrYrTeiθT+λρω2eiθY+iα2T0ωeiθY)|>|Im(CrYrTeiθTλρω2eiθY+iα2T0ωeiθY)|It means that ImkT>0 or the temperature wave is always an attenuated wave. If


we get ImkY>0 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 ge in Eq. (1) should keep the temperature variable. The electromagnetic Gibbs free energy ge and the complement dissipative energy he in this case are assumed respectively in the following form


where ekije,kl,τie,ekijm,μkl,τim 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 ρe 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 μis the chemical potential, dis the flow vector of the diffusing mass, cis the concentration. In discussion of the thermal diffusion problem we can also use the free energy g˙c=σ:ε˙sT˙+μc˙ (Kuang, 2010), but here it is omitted. Using relationsT1qi,i=(T1qi),i+T2qiT,i,T1μdi,i=(T1μdi),idi(T1μ),iFrom Eq. (29) (Kuang, 2010) we get:


where Ts˙(i) 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 Dij is the diffusing coefficients and Lij 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)

Especially the coefficients λij,Lij,Dij,λ^ij,L^ij,D^ij in Eq. (31b) can all be considered as symmetric constants which are adopted in following sections. Eq. (31) is the extension of the Fourier’s law and Fick’s law.

Eq. (29) shows that in the equation of the heat flow the role of Ts˙is somewhat equivalent toμc˙. So analogous to the inertial entropy s(a) we can also introduce the inertial concentration c(a) 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 ρc is the inertial concentration coefficient. Applying the irreversible thermodynamics we can get the Gibbs free energy gμ and the complement dissipative energy hμ as


where a,b,bij are also material constants. The constitutive and evolution equations are:


Using Eq. (34) gμin Eq. (33a) can also be rewritten as (33b)

where gμ(T) is the energy containing the effects of temperature and concentration. Substituting Eq. (34) into Eq. (32) we get


If we neglect the term in second order μdi,i in Eq. (29), i.e. we take Ts˙=r˙qi,i and assume that T,i andμ,j are not dependent each other, i.e. in Eq. (31b) we assumeη˙i=λijT1T,j,ξ˙i=Dijμ,j, then forr˙=0, Eq. (35) becomes


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 U is the internal energy per volume, W is the work applied on the body by the environment, Qis the heat supplied by the environment. According to Gibbs theory when the process is only slightly deviated from the equilibrium state dQ can be substituted byVTdsdV. In practice we prefer to use the free energyg:


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 ϕ different with displacementu, should be divided into local variation and migratory variation, i.e. the variationδϕ=δϕϕ+δuϕ, where the local variation δϕϕ of ϕ is the variation duo to the change of ϕ itself and the migratory variation δuϕ of ϕ is the variation of change of ϕ due to virtual displacements. In Eqs. (38) and (39) the new force produced by the migratory variation δuϕ will enter the virtual work δW or δW 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 g 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 ϑ is divided into local variation δϑϑ due to the variation of ϑ itself and the migratory variation δuϑ due toδu:


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 δuϑwill 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 u and the virtual temperature ϑ(orT) satisfy their own boundary conditions ui=ui,ϑ=ϑ on auand aT respectively. The physical variational principle using the free energy in the inertial entropy theory for the thermo-elasticity can be expressed as:


where fk,Tk and η˙=η˙ini 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 0ts˙(i)δϑdτ=0tηiδT,idτ is the complement dissipative heat rate per volume corresponding to the inner complement dissipation energy rateδh. The entropy s includes the contribution of0ts˙(i)δϑdτ. The fact that the complement dissipation energy rate VδhdV in δΠT and the internal irreversible complement heat rate V0ts˙(i)δϑdτdV in δQ 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 isVδgdV+Vg(T)δuk,kdVδW, in which the integrands contain variables themselves. The second isVδhdVδQ, 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 σT 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 σT can be neglected for the case of the small strain and small change of temperature. In Eq. (43) it is seen that δϑ=δϑϑ+δuϑ is appeared in a whole. Using T(λijT1ϑ,i),jTs˙(i)=Tη˙j,jTs˙(i)=Tη˙j,j+η˙iT,i=(Tη˙i),i=qi,iand the arbitrariness of δui andδϑ, from Eq. (43) we get


Here σT is the external temperature body force and nσTis 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


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 gμ and the complement dissipative energy hμ are expressed in Eq. (33) and the constitutive and evolution equations are expressed in Eq. (34).

Under assumptions that the mechanical displacementu, the temperature ϑ and the chemical potential μ satisfy their own boundary conditionsu=u, ϑ=ϑand μ=μ onau,aT and aμrespectively. When the variation of temperature is not large the physical variational principle for the thermo-elasto-diffusive problem is


In Eqs. (46) fk,Tkη˙=η˙ini and ξ˙=ξ˙ini are given values. In Eq. (46)δQ is related to heat (including the heat produced by the irreversible process in the material), δΦis related to the diffusion energy. Eq. (46) shows that there is no term in VδhμdVcorresponding to the terma0tT1μξ˙nδϑdτda, so it should not be included in δQ andV0tT1μ,iξ˙iδϑdτdVa0tT1μξ˙iniδϑdτda=V0tT1μξ˙i,iδϑdτdV.

It is noted that we have the following relations


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




Due to the arbitrariness ofδu,δϑ andδμ, from Eq. (48) we get




where T1(λijT,i+Lijμ,i),jT1(T,iη˙iμξ˙i,i)=T1(qj,j+μξ˙i,i) has been used.

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 u,ϑ and μ satisfy their own boundary conditions, so in governing equations the following equations should also be added


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

If we neglect the term μ(c˙+c˙(a)) in Eq. (32), or T(s˙+s˙(a))=r˙qi,i is adopted, then we easily get


If we also assume that T,i andμ,j are not dependent each other, then forr˙=0, the Eq. (54) becomes Eq. (36), i.e.


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 gem is taken as


where lijkleand lijklm are the electrostrictive and megnetostrictive constants respectively; and μ may be asymmetric. The corresponding constitutive equations are

Let σs and σa be the symmetric and asymmetric parts of σ respectively, we have

where D=0E+P,B=μ0(H+M)have been used, Pand M are the polarization density and magnetization density, 0and μ0 are the dielectric constant and magnetic permeability in vacuum respectively. The terms containing ε in D and B in Eq. (58) have been neglected. In the usual electromagnetic theory the electromagnetic body couple isP×E+μ0M×H. From Eq. (58) it is seen that 2σkla+(DkElDlEk)+(BkHlBlHk)=0 or the electromagnetic body couple is balanced by the moment produced by the asymmetric stresses.

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

Because the value of the term Δ:ε 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 u,φ,ψ,uenv,φenv,ψenv satisfy their

Figure 1.

Electromagnetic medium and its environment

boundary conditions on their own boundaries au,aφ,aψ,auenv,aφenv,aψenv and the continuity conditions on the interfaceaint. 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, fk,Tk,σ*,Bn*=Bi*ni;Tkenv,σenv,Bnenv;Tkint,σint,Bnint 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).

As shown in previous paper (Kuang, 2011a, 2011b) and in section 3.1. the variations of φ,ψ,E,H will be distinguished into local and migratory variations, i.e.


Noting that in Eq. (59) we have VδgemdV+Vgemδuk,kdV=VσijδεijdVVDiδEidVVBiδHidV(1/2)V(DkEk+BkHk)δuj,jdV=a[σij(1/2)(DkEk+BkHk)δij]njδuidaV[σij(1/2)(DkEk+BkHk)δij],jδuidV+aDiniδφφdaVDi,iδφφdVVDiEp,iδupdV+aBiniδψψdaVBi,iδψψdVVBiHp,iδupdV SoδΠ1 in Eq. (59) is reduced to


where δΠ1 is the part of δΠ1due to the local variations ofu,φ,ψ; δΠ1is the part of δΠ1due to the migratory variations ofφ,ψ. Substituting the following identity


into δΠ1 in Eq. (61) we get


where σM is the Maxwell stress:


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


where σ˜ is the pseudo total stress (Jiang and Kuang, 2004), which is not the true stress in electromagnetic media. From the expression of σ˜ it is known that σ˜ is symmetric though σ and σM are asymmetric. Due to the arbitrariness of δui,δφ andδψ, from Eq. (65) we get


For the environment we have the similar formula:


Using ni=nienv,ui=uienv,φ=φenv,ψ=ψenv and δΠ1+δΠ2=δWinton 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


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


where βij=βji is the static magnetoelectric coupling coefficient. The electromagnetic body couple is still balanced by the asymmetric stress, i.e. DkElDlEk+BkHlBlHk=[(kmEllmEk)Em+(μkmHlμlmHk)Hm]+[(βkmElβlmEk)Hm+(βkmHlβlmHk)Em]=2σklaIn this case though the constitutive equations are changed, but the electromagnetic Gibbs free energy gein Eq. (56b), governing equations (66)(69) and the Maxwell stress (64) are still tenable.


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.


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Written By

Zhen-Bang Kuang

Submitted: 21 November 2010 Published: 22 September 2011