Open access peer-reviewed chapter

# Some Thermodynamic Problems in Continuum Mechanics

By Zhen-Bang Kuang

Submitted: November 21st 2010Reviewed: April 7th 2011Published: September 22nd 2011

DOI: 10.5772/22610

## 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 crespectively. 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,μare used as independent variables and variables p,σ,D,B,S,care 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 Uand 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

dU(ε,D,B,s,c)=σ:dε+EdD+HdB+Tds+μdc;σ:dε=σijdεijdgeμ(ε,E,H,T,μ)=d(UEDHBTsμc)=σ:dεDdEBdHsdTcdμE1

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=geis the electromagnetic Gibbs free energy. For the case without electromagnetic field geμ=gμis the Gibbs free energy with chemical potential. For the case without chemical potential and electromagnetic field geμ=gis 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 Tis varied, the extra heat or entropy should be input from the environment. When cis varied, the extra chemical potential μis 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 DdEBdHcdμis 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.

Ts˙+Ts˙(a)=r˙qi,i=r˙(Tη˙i),i,s˙(a)=ρsT¨,s(a)=ρsT˙=ρs0(C/T0)T˙s˙=s˙(r)+s˙(i);s˙(r)+s˙(a)=r˙/Tη˙i,i;η˙=q/TTs˙(i)=Ts˙Ts˙(r)=Ts˙+Ts˙(a)r˙+Tη˙i,i=η˙iT,i0;s˙(i)=η˙iT,i/TE2

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

h˙=dh˙/dt=Ts˙(i)=T,iη˙i,h˙=T,iη˙i+(T,iηi)·=ηiT˙,iE3

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

η˙i=η˙i(T,j),orη˙i=λijT1T,j,Tη˙i=qi=λijT,jT,j=λ^ijTη˙i=λ^ijqi,λ^ij=λij1E4

where λis 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

qi+τ0q˙i=λT,i,Ts˙=r˙qi,iE5

where τ0is 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:

λT,ii=C(T˙+τ0T¨)+[2G(1+ν)/(12ν)]αT0(ε˙jj+τ0ε¨jj)[G/(12ν)]uj,ij+Gui.jj[2G(1+ν)/(12ν)]αT,i=ρu¨iE6

where C is the specific heat, αis the thermal expansion coefficient, G and νare 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 sis 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

Vs˙dVV(r/ϕ)dV+a(qi/ϕ)nida0,ϕ=ϕ(T,T˙),T=ϕ(T,0)g=Uϕs,g=g(T,T˙,εij)E7

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

λT,ii=C(T˙+τ0T¨)+γT0ε˙jj,σji,j+ρfi=ρu¨iσij=[2Gν/(12ν)]εkkδij+2Gεijγ(θ+τ1θ˙)E8

whereτ0, τ1and γ are material constants.

Now we discuss the inertial entropy theory (Kuang, 2009b). The Helmholtz free energy gand the complement dissipative energy hassumed in the form

g(εkl,ϑ)=(1/2)Cijklεjiεlkαijεijϑ(1/2T0)Cϑ2a)δh=[0t(λij/T)ϑ,idτ)]δϑ,j,ϑ=TT0Cijkl=Cjikl=Cijlk=Cklij,αij=αji,λij=λjib)g=(1/2)Cijklεjiεlk+g(T),g(T)=(1/2)sϑ(1/2)αijεijϑE9

where T0is the reference (or the environment) temperature, Cijkl,αijare material constants. In Eq. (9a) it is assumed that s=0when T=T0orϑ=0. It is obvious thatT,j=ϑ,j,T˙=ϑ˙.

The constitutive (or state) and evolution equations are

σij=g/εij=Cijklεklαijϑ,s=g/ϑ=αijεij+Cϑ/T0ηi=h/ϑ,i=0t(λij/T)ϑ,jdτ,Tη˙i=qi=λijϑ,jE10

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

T(αijεij+Cϑ/T0)·+ρsTϑ¨=r˙+(λijϑ,j),iE11

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

a)ρsTϑ¨+CTϑ˙/T0λijϑ,ji=r˙αijTε˙ijb)(C/T0)(ρs0ϑ¨+ϑ˙)(λ/T)ϑ,ii=r˙/Tαε˙iiorλϑ,ii=C(ρs0ϑ¨+ϑ˙)+αT0ε˙iir˙E12

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¨jjis 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

σij,j+fi=ρu¨iE13

where fis the body force per volume, ρis the density. Substituting the stress σin Eq. (10) into (13) we get

(Cijklεklαijϑ),j+fi=ρu¨i,orρu¨i=Cijkluk,ljαijϑ,j+fiE14

Comparing the elastic wave equation Eq. (14) with the Green－Lindsay theory (Eq. (8)) it is found that in Eq. (14) a term γτ1ϑ˙,iis 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>ρs0the amplitude of the elastic wave will be increased with time. For BaTio3ρs0is 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

g=(1/2)Cijklεjiεlkαijεijϑ(1/2T0)Cϑ2σij(r)=g/εij=Cijklεklαijϑ,s=g/ϑ=αijεij+Cϑ/T0δh=βijklε˙jiδε˙lk+ηjδϑ,j=βijklε˙jiδε˙lk[0t(λij/T)ϑ,idτ)]δϑ,j,σij(i)=h/ε˙ij=βijklε˙kl,ηi=h/ϑ,i=0t(λij/T)ϑ,jdτ,Tη˙i=qi=λijϑ,jσij=σij(r)+σij(i)=Cijklεkl+βijklε˙klαijϑE15

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˙lkis added to the rate of the complement dissipative energy in Eq. (15). Substituting the entropy sand Tη˙iin Eq. (15) and s(a)in (2) into Ts˙+Ts˙(a)=r˙(Tη˙i),iin Eq. (2) we still get the same equation (12).

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

(Cijklεkl+βijklε˙klαijϑ),j+fi=ρu¨i,orρu¨i=Cijkluk,lj+βijklu˙k,ljαijϑ,j+fiE16

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

σ=Yε+βε˙αϑ,s=αε+Cϑ/T0E17

where Yis 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

C(ρs0ϑ¨+ϑ˙)λϑ+αT0u˙=0ρu¨Yuβu˙+αϑ=0E18

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

u=Uexp[i(kxωt)],θ=Θexp[i(kxωt)]E19

where U,Θare the amplitudes of uand ϑrespectively, kis the wave number and ωis the circular frequency. Substituting Eq. (19) into (18) we obtain

[(Yiβω)k2ρω2]U+iαkΘ=0αT0kωU+[λk2C(ρs0ω2+iω)]Θ=0E20

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

a)|(Yiβω)k2ρω2iαkαT0kωλk2C(ρs0ω2+iω)|=|ak2ρω2iαkαT0kωλk2Cb|=0whereb)a=Yiβω=rYeiθY,rY=aa¯=Y2+β2ω2,sinθY=βω/rYb=ρs0ω2+iω=rTeiθT,rT=bb¯=ωρs02ω2+1,sinθT=ω/rTE21

From Eq. (21) we get

λak4(Cab+λρω2+iα2T0ω)k2+ρω2Cb=0k={12λrY[(CrYrTeiθT+λρω2eiθY+iα2T0ωeiθY)±(CrYrTeiθTλρω2eiθY+iα2T0ωeiθY)2+4iα2T0ωrYrTei(θTθY)]}1/21/2E22

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

k=(2λrY)1{(CrTrYeiθT+λρω2eiθY)±(CrTrYeiθTλρω2eiθY)}kY=ωρ/rYeiθY/2,kT=CrT/λeiθT/2E23

Because θY>0due to β>0and θT>0due 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 iα2T0ωk2is appeared. It is known that |Im(CrYrTeiθT+λρω2eiθY+iα2T0ωeiθY)|>|Im(CrYrTeiθTλρω2eiθY+iα2T0ωeiθY)|It means that ImkT>0or the temperature wave is always an attenuated wave. If

Im[(CrYrTeiθTλρω2eiθY+iα2T0ωeiθY)2+4iα2T0ωrYrTei(θTθY)]<Im(CrYrTeiθT+λρω2eiθY+iα2T0ωeiθY)2E24

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

ge(εkl,Ek,Hk,ϑ)=(1/2)Cijklεjiεlk-ekijeEkεij(1/2)ijEiEjτieEiϑ-ekijmHkεij(1/2)μijHiHjτimHiϑαijεijϑ(1/2T0)Cϑ2δhe=(0t(λij/T)ϑ,idτ)δϑ,j(=ηjδϑ,j),ϑ=TT0Cijkl=Cjikl=Cijlk=Cklij,ekije=ekjie,kl=lk,ekijm=ekjim,μkl=μlk,αij=αjiE25

where ekije,kl,τie,ekijm,μkl,τimare material constants. The constitutive equations are

σij=CijklεklekijeEkekijmHkαijϑ,Di=ijEj+eijkeεjk+τieϑBi=μijHj+eijkmεjk+τimϑ,s=αijεij+τieEi+τimHi+Cϑ/T0ηi=hv/ϑ,i=0t(λij/T)ϑ,jdτ,Tη˙i=qi=λijϑ,jE26

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

T(αijεij+τieEi+τimHi+Cϑ/T0)·+ρsTT¨=r˙+(λijϑ,j),iE27
(CijklεklekijeEkekijmHkαijϑ),j+fi=ρu¨i,(ijEj+ekijeεkl+τieϑ),i=ρe,(μijHj+ekijmεkl+τimϑ),i=0E28

where ρeis 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:

Ts˙=r˙qi,i+μdi,i,Ts˙+μc˙=r˙qi,i=r˙(Tη˙i),i,di,i=c˙U˙=σ:ε˙+Ts˙+μc˙,g˙μ=σ:ε˙sT˙cμ˙E29

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:

s˙=s˙(r)+s˙(i);Ts˙(r)=r˙T(qi/Tμdi/T),iTs˙(i)=Ts˙Ts˙(r)=T,iη˙iμ,iξ˙i=0,μ,i=(μ/T),i,ξ˙i=TdiE30

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

a)Tη˙i=λij(T)T,iLij(T)Tμ,i,T1ξ˙i=Dij(T)Tμ,iLij(T)T,ib)Ts˙(i)T,iη˙iμ,iξ˙i0;ξ˙i=diTη˙i=λij(T)T,iLij(T)μ,i,ξ˙i=Dij(T)μ,iLij(T)T,iT,i=λ^ij(T)Tη˙iL^ij(T)ξ˙i,μ,i=D^ij(T)ξ˙iL^ij(T)Tη˙iE31

where Dijis the diffusing coefficients and Lijis 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^ijin 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)

T(s˙+s˙(a))+μ(c˙+c˙(a))=r˙qi,i=r˙(Tη˙i),i;c˙+c˙(a)=di,is(a)=0ts˙(a)dτ,s˙(a)=ρsT¨;c(a)=0tc˙(a)dτ,c˙(a)=ρcμ¨E32

where ρcis the inertial concentration coefficient. Applying the irreversible thermodynamics we can get the Gibbs free energy gμand the complement dissipative energy hμas

gμ(εkl,ϑ,μ)=12Cijklεjiεlkαijεijϑ12T0Cϑ212bμ2bijεijμaμϑa)δhμ=T,iδηiμ,iδξi+δ(T,iηi)+δ(μ,iξi)=ηjδϑ,j+ξjδμ,j=δϑ,j0t(λijT1ϑ,i+LijT1μ,i)dτδμ,j0t(Lijϑ,i+Dijμ,i)dτb)gμ(εkl,ϑ,μ)=(1/2)Cijklεjiεlk+gμ(T),gμ(T)=(1/2)(sϑ+cμ+αijεijϑ+bijεijμ)E33

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

σij=gμ/εij=Cijklεklαijϑbijμ,s=gμ/ϑ=αijεij+Cϑ/T0+aμ,c=gμ/μ=bμ+bijεij+aϑηi=hμ/ϑ,i=0t(λijT1ϑ,j+LijT1μ,j)dτ,ξi=hμ/μ,i=0t(Lijϑ,j+Dijμ,j)dτE34

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

T(αijεij+Cϑ/T0+aμ)·+Tρsϑ¨+μ(bμ+bijεij+aϑ)·+μρsϑ¨=r˙+(λijϑ,j+Lijμ,j),i(bμ+bijεij+aϑ)·+ρcμ¨=Ljiϑ,ij+Djiμ,ij;InmediumE35

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

T(αiju˙i,j+Cϑ˙/T0+aμ˙+ρsϑ¨)=λijϑ,jbμ˙+biju˙i,j+aϑ˙+ρcμ¨=Dijμ,ji;InmediumE36

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

(Cijkluk,lαijϑbijμ),j+fi=ρu¨iE37

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:

ClassicalEnergyconservation:VdUdVdWdQ=0Classicalphysicalvariationalprinciple:δΠ=VδUdVδWδQ=0E38

where Uis 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 dQcan be substituted byVTdsdV. In practice we prefer to use the free energyg:

g=UTs,dg=dUsdTTdsEnergyPrinciple:VdgdVdWVsdTdV=0PhysicalVariationalPrinciple:δΠ=VδgdVδWVsδTdV=0E39

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 ϕdifferent with displacementu, should be divided into local variation and migratory variation, i.e. the variationδϕ=δϕϕ+δuϕ, where the local variation δϕϕof ϕis the variation duo to the change of ϕitself and the migratory variation δuϕof ϕis the variation of change of ϕdue to virtual displacements. In Eqs. (38) and (39) the new force produced by the migratory variation δuϕwill enter the virtual work δWor δWas 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 gand 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 ϑis divided into local variation δϑϑdue to the variation of ϑitself and the migratory variation δuϑdue toδu:

δϑ=δϑϑ+δuϑ,δuϑ=ϑ,iδuiE40

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

δΠT=Vδ(g+h)dV+Vg(T)δuk,kdVδQδW=0δQ=V0t(r˙/T)δϑdτdVV0ts˙(i)δϑdτdV+aq0tη˙δϑdτda+V0tρsϑ¨δϑdτdVδW=V(fkρu¨k)δukdV+aσTkδukdaE41

where fk,Tkand η˙=η˙iniare 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,idτis the complement dissipative heat rate per volume corresponding to the inner complement dissipation energy rateδh. The entropy sincludes the contribution of0ts˙(i)δϑdτ. The fact that the complement dissipation energy rate VδhdVin δΠTand the internal irreversible complement heat rate V0ts˙(i)δϑdτdVin δQare 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

VδgdV=V(Cijklεklαijϑ)δui,jdVV(αijεij+Cϑ/T0)δϑdV=aσijnjδuidaVσij,jδuidVVsδϑdVVg(T)δuk,kdV=(1/2)V(s+αijεij)ϑδuk,kdV=(1/2)a(s+αijεij)ϑnkδukdV+(1/2)V[(s+αijεij)ϑ],kδukdVVδhdV=a(0tλijT1ϑ,injdτ)δϑda+V[0t(λijT1ϑ,i),jdτ]δϑdVE42

Finishing the variational calculation, we have

δΠ=aσ(σ˜ijnjTi)δuidaV(σ˜ij,j+fiρu¨i)δuidVaq[0tT1λijϑ,injdτ+η]δϑdaV{s+0t[T1r˙(λijT1ϑ,i),js˙(i)]dτ}δϑdVV0tρsϑ¨dτdV}δϑdV=0σ˜ij=σij+σijT,σijT=(1/2)(s+αijεij)ϑδij(1/2)sϑδijE43

where σTis 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 σTcan be neglected for the case of the small strain and small change of temperature. In Eq. (43) it is seen that δϑ=δϑϑ+δuϑis 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 δuiandδϑ, from Eq. (43) we get

σ˜ij,j+fi=ρu¨i;T(s˙+ρsϑ¨)=r˙qi,iinmediumσ˜klnl=Tk,onaσ;η˙i=T1λijϑ,i,η˙ini=η˙i,orqn=qn,onaqE44

Here σTis 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

u=u,onau;ϑ=ϑ(orT=T),onaTE45

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

Under assumptions that the mechanical displacementu, the temperature ϑand the chemical potential μsatisfy their own boundary conditionsu=u, ϑ=ϑand μ=μonau,aTand 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˙iniand ξ˙=ξ˙iniare given values. In Eq. (46)δQis 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 δQandV0tT1μ,iξ˙iδϑdτdVa0tT1μξ˙iniδϑdτda=V0tT1μξ˙i,iδϑdτdV.

It is noted that we have the following relations

VδgμdV=aσijnjδuidaVσij,jδuidVVsδϑdVVcδμdVVgμ(T)δuk,kdV=(1/2)V(sϑ+cμ+αijεijϑ+bijεijμ)δuk,kdV=(1/2)a(sϑ+cμ+αijεijϑ+bijεijμ)nkδukdV+(1/2)V(sϑ+cμ+αijεijϑ+bijεijμ),kδukdVδhμ=V{δϑ,i0t(λijT1T,i+LijT1μ,i)dτ+δμ,i0t(LijT,i+Dijμ,i)dτ}dVE47

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

where

σ˜ij=σij+σijTμ,σijTμ=(1/2)(sϑ+cμ+αijεijϑ+bijεijμ)δij(1/2)(sϑ+cμ)E49

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

σ˜kl,l+fk=ρu¨k,inmedium;σ˜klnl=Tk,onaσE50

and

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

T(s˙+ρsϑ¨)=r˙qj,j+μξ˙j,j;c˙+ρcμ¨=ξ˙j,jT(s˙+ρsϑ¨)+μ(c˙+ρcμ¨)=r˙qj,j;InmediumE52

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

The above variational principle requests prior that the u,ϑand μsatisfy their own boundary conditions, so in governing equations the following equations should also be added

u=u,onau;ϑ=ϑ,onaT;μ=μ,onaμE53

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,iis adopted, then we easily get

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

T(αiju˙i,j+Cϑ˙/T0+aμ˙+ρsϑ¨)=λijϑ,jbμ˙+biju˙i,j+aϑ˙+ρcμ¨=Dijμ,j;InmediumE55

## 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 gemis taken as

gem(εkl,Ek,Hk)=(1/2)Cijklεjiεlk-(ekijeEk+ekijmHk)εij(1/2)(ijEiEj+μijHiHj)a)(1/2)(lijkleEiEj+lijklmHiHj)εkl(kmEmEl+μkmHmHl)εkl(Cijkl,lijkle,lijklm)=(Cjikl,ljikle,ljiklm)=(Cijlk,lijlke,lijlkm)=(Cklij,lklije,lklijm),ekije=ekjie,ekijm=ekjimb)gem=(1/2)Cijklεjiεlk+gem,gem=(1/2)(DkEk+BkHk+Δklεlk)(1/2)(DkEk+BkHk)Δkl=emkleEm+emklmHm=emkleφ,memklmψ,mE56

where lijkleand lijklmare the electrostrictive and megnetostrictive constants respectively; and μmay be asymmetric. The corresponding constitutive equations are

σkl=gem/εkl=CijklεijejkleEjejklmHj(1/2)lijkleEiEj(1/2)lijklmHiHjkmEmElμkmHmHlDk=gem/Ek=[kl+lijkleεij+(mlεmk+mkεml)]El+ekijeεijklElBk=gem/Hk=[μkl+lijklmεij+(αmlmεmk+αmkmεml)]Hl+ekijmεijμklHlE57
Let σsand σabe the symmetric and asymmetric parts of σrespectively, we have
σkls=(1/2)(σkl+σlk)=CijklεjiejkleEj(1/2)lijkleEiEjejklmHj(1/2)lijklmHiHj(1/2)(kmEl+lmEk)Em(1/2)(μkmHl+μlmHk)Hmσkla=(1/2)(σklσlk)=(1/2)(kmEllmEk)Em(1/2)(μkmHlμlmHk)HmDkElDlEk=PkElPlEk(kmEllmEk)EmBkHlBlHk=μ0(MkHlMlHk)(μkmHlμlmHk)HmE58

where D=0E+P,B=μ0(H+M)have been used, Pand Mare the polarization density and magnetization density, 0and μ0are the dielectric constant and magnetic permeability in vacuum respectively. The terms containing εin Dand Bin 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)=0or 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:εis 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,ψenvsatisfy their

boundary conditions on their own boundaries au,aφ,aψ,auenv,aφenv,aψenvand the continuity conditions on the interfaceaint. The Physical variational principle in the nonlinear electro-magneto-elastic analysis is

δΠ=δΠ1+δΠ2δWint=0δΠ1=VδgemdV+Vgemδui,idVδWδΠ2=VenvδgemenvdV+Venvgemenvδui,ienvdVδWenvδW=V(fkρu¨k)δukdVVρeδφdV+aσTkδukdaaqσδφda+aμBiniδψdaδWint=aintTkintδukdaaintσintδφda+aμintBiintniδψenvdagemenv,gemenv,δWenvaresimilarandomitedhereE59

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,Bnintare 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,Hwill be distinguished into local and migratory variations, i.e.

δ(φ,ψ,Ei,Hi)=δ(φ,ψ,Ei,Hi)(φ,ψ,Ei,Hi)+δu(φ,ψ,Ei,Hi)δu(φ,ψ)=(φ,ψ),pδup=(Ep,Hp)δupδu(Ei,Hi)=δu(φ,ψ),i=(φ,ψ),ipδup=(Ep,i,Hp,i)δup=(Ei,p,Hi,p)Ei,pδupE60

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,iδupdVSoδΠ1in Eq. (59) is reduced to

where δΠ1is 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 δΠ1in Eq. (61) we get

where σMis the Maxwell stress:

σikM=DiEk+BiHk(1/2)(DnEn+BnHn)δikE64

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 σand σMare asymmetric. Due to the arbitrariness of δui,δφandδψ, from Eq. (65) we get

For the environment we have the similar formula:

Using ni=nienv,ui=uienv,φ=φenv,ψ=ψenvand δΠ1+δΠ2=δWinton the boundary surface we get

(σ˜ijσ˜ijenv)ni=Tjint,(DiDienv)ni=-σint,(BiBienv)ni=Bi*intni,onaintE68

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

ui=ui,onau;φ=φ,onaφ;ψ=ψ,onaμuienv=uienv,onauenv;φenv=φenv;onaφenv;ψenv=ψenv,onaμenvui=uienv,φ=φenv,ψ=ψenv;onaintE69

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

σkl=CijklεijejkleEjejklmHj(1/2)lijkleEiEj(1/2)lijklmHiHjkmEmElμkmHmHlβkmHmElβkmEmHlDk=[kl+lijkleεij+(mlεmk+mkεml)]El+ekijeεij+βklHlBk=[μkl+lijklmεij+2(αmlmεmk+αmkmεml)]Hl+ekijmεij+βklElE70

where βij=βjiis 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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Zhen-Bang Kuang (September 22nd 2011). Some Thermodynamic Problems in Continuum Mechanics, Thermodynamics - Kinetics of Dynamic Systems, Juan Carlos Moreno Piraján, IntechOpen, DOI: 10.5772/22610. Available from:

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