On the non-singularity of the thermal conductivity tensor and its consequences

Transcription

1 On the non-sngularty of the thermal conductvty tensor and ts consequences Al R. Hadjesfandar Department of Mechancal and Aerospace Engneerng Unversty at Buffalo, State Unversty of New Yor Buffalo, NY 1426 USA January 28, 215 Abstract In ths paper, the symmetrc character of the thermal conductvty tensor for lnear ansotropc materal s establshed as the result of arguments from tensor analyss for Duhamel s generalzaton of Fourer s heat conducton. The non-sngular nature of the conductvty tensor plays the fundamental role n establshng ths symmetry, as well as ts postve (or negatve) defnteness. Sgnfcantly, the second law of thermodynamcs does not contrbute here n establshng these characterstcs, but does ultmately decde that the conductvty tensor s ndeed postve defnte. 1. Introducton The conductvty tensor characterzes the general lnear heat conducton relaton between temperature gradents and heat flux n heterogeneous ansotropc materal. By usng nonequlbrum statstcal mechancs, Onsager [1] has shown that the conductvty tensor s symmetrc. Snce classcal contnuum thermodynamcs dd not provde any drect reasonng for ths property for a long tme, there had been the general belef that the symmetry condton can only be derved based on addtonal physcal assumptons. For nstance, Day and Gurtn [2] have ntroduced the requrement that the thermal wor functonal has a wea relatve mnmum at equlbrum. 1

2 Recently, Hadjesfandar [] has establshed the symmetrc character of the conductvty tensor by usng arguments from tensor analyss and lnear algebra regardng the generalzaton of Fourer s heat conducton law. Interestngly, the fundamental ground n ths establshment s the non-sngularty of the conductvty tensor. The method of proof s based on the consstency of the system of lnear equatons representng the heat conducton law n dfferent coordnate systems. Ths proof clearly demonstrates that classcal contnuum thermodynamcs can provde the mathematcal reason for the symmetrc character of the conductvty tensor, whch s a necessary condton for havng consstent tensoral relatons n classcal heat conducton theory. As a result, one mght speculate that there are other ways to establsh ths character, although the fundamental step stll remans the non-sngularty of the conductvty tensor. Here we establsh the symmetrc character of the conductvty tensor by focusng on the conductvty tensor tself, wthout usng the lnear heat conducton equaton. As mentoned, the fundamental ground n ths establshment s also the non-sngularty of the conductvty tensor. The method of proof s based on the fact that the conductvty tensor cannot be sew-symmetrc. Sgnfcantly, ths proof shows the subtle character of tensors n three-dmensonal space, whch has not been recognzed prevously. Interestngly, the form of ths proof also shows that the conductvty tensor s ether postve or negatve defnte. It should be emphaszed that the second law of thermodynamcs and Clausus-Duhem nequalty do not have any role here n establshng the symmetry character of the conductvty tensor. They only establsh that the conductvty tensor s postve, rather than negatve, defnte. The paper s organzed as follows. In Secton 2, we provde an overvew of the classcal heat conducton relatons for lnear ansotropc materal. After that n Secton, the symmetrc character of the conductvty tensor s establshed by usng the arguments from tensor analyss. Fnally, Secton 4 contans a summary and some general conclusons. Appendx A presents propertes of the egenvalue problem for second order tensors. 2. Lnear heat conducton theory Consder the three dmensonal orthogonal coordnate system x 1x2x as the reference frame. For lnear ansotropc materal, Duhamel s generalzaton of Fourer s heat conducton law [4] s 2

3 q = T. (1), j Here the tensor s the materal thermal conductvty tensor, whch relates the heat flux vector q to the gradent of the temperature feld T. The mnus sgn n (1) assures that the heat flow occurs from a hgher to a lower temperature when the materal s sotropc wth postve conductvty n Fourer s orgnal law. From a physcal standpont, we postulate that there s a one to one relatonshp between the temperature gradent T, and the heat flux q n (1). Ths condton requres that the conductvty tensor be non-sngular, as wll be demonstrated n detal n the next secton. In terms of components, the second order conductvty tensor n the coordnate system x1 x2x can be wrtten as = (2) Snce we have not establshed the symmetry character of, the nne components of are ndependent of each other at ths stage. Therefore, the conductvty tensor s specfed by nne ndependent components n the general case. As a result, the conductvty tensor can be represented by ponts of an abstract nne-dmensonal space. However, as wll be seen, there are some restrctons on the conductvty tensor, whch confnes the doman of the conductvty tensor n ths abstract space. By decomposng the thermal conductvty tensor nto symmetrc ( ) and sew-symmetrc [ ] parts, we have = +, () ( ) [ ] where 1 ( ) = ( + j ) = ( j), (4) 2

4 1 [ ] = ( j ) = [ j]. (5) 2 Notce that here we have ntroduced parentheses surroundng a par of ndces to denote the symmetrc part of a second order tensor, whereas square bracets are assocated wth the sewsymmetrc part. Snce the general conductvty tensor s specfed by nne ndependent components, the tensors ( ) and [ ] are specfed by sx and three ndependent components, respectvely. In the followng secton, we prove that [ ] vanshes based exclusvely on tensor analyss.. Symmetrc character of the conductvty tensor Now we establsh that the conductvty tensor cannot be sngular. Appendx A demonstrates that sngular tensors have at least one zero egenvalue. Therefore, f we assume that the conductvty tensor s sngular, at least one of ts egenvalues vanshes. Let us arbtrarly choose the thrd egenvalue to be one of these egenvalues, that s, λ =, where ts correspondng real egenvector s ( ) v. For a non-zero temperature gradent T, n the drecton of T, ( ) ( ) v, where = ϑv, (6) wth ϑ as an arbtrary non-zero constant, there would be no heat flux; that s ( ) q = T = ϑ v =. (7), j However, ths physcally contradcts the fact that there s a one to one relatonshp between the temperature gradent T, and the heat flux q n (1). Therefore, ths contradcton requres that the conductvty tensor be non-sngular, that s ( ) Ths shows that the conductvty tensor s nvertble. det = det. (8) Interestngly, we notce that ( ) det = specfes an eght-dmensonal hyper-surface n the abstract nne-dmensonal space of the conductvty tensor. Ths hyper-surface dvdes the nnedmensonal space nto two exclusve subspaces. Snce the doman of the conductvty tensor s 4

5 contnuous n the abstract nne-dmensonal space, the constrant (8) requres that ths doman be n only one of these two subspaces. The more fundamental meanng of ths restrcton wll be elucdated shortly. In Appendx A, we demonstrate the well-nown fact that all three-dmensonal sew-symmetrc tensors are sngular. As a result, the non-sngular conductvty tensor cannot be sewsymmetrc. For now, we concentrate on ths mportant character and gnore the general nonsngularty character of the conductvty tensor. For further nvestgaton, we consder the decomposton = +. (9) ( ) [ ] Let us assume the sew-symmetrc tensor [ ] part s non-zero. Snce, the conductvty tensor cannot become sew-symmetrc, the symmetrc part ( ) s also non-zero. However, the symmetrc part ( ) can become as arbtrarly small as we wsh. Ths means that the tensor can approach to the non-zero lmt value [ ] n many arbtrary ways, but t cannot become equal to [ ]. Mathematcally, ths states that the conductvty tensor s not defned at [ ], although t s defned n ts neghborhood. However, we notce that ths restrcton s n contradcton wth the contnuty of the doman of defnton of the conductvty tensor. contradcton requres that the sew-symmetrc part [ ] vansh, that s [ ] = Ths result states that the conductvty tensor s symmetrc, that s Therefore, ths, = ( ). (1) =. (11) Therefore, the general conductvty tensor s specfed by sx ndependent components. j For more clarfcaton, we demonstrate the above reasonng by usng the symbolc threedmensonal coordnate system n Fg. 1, where the horzontal plane ncludng two coordnate 5

6 axs represents the sx-dmensonal space of ( ), and the vertcal axs represents the threedmensonal space of [ ]. Notce that the orgn corresponds to the zero of ( ) and [ ]. Snce the conductvty tensor cannot be sew-symmetrc, t cannot be on the vertcal lne, although t can be at any pont around t. Therefore, the vertcal lne s the locaton of mpossble values for the conductvty tensor. However, ths s nconsstent wth the contnuty of the conductvty tensor n ts doman. As a result, ths contradcton requres that ponts representng the consstent conductvty tensor must le n the horzontal plane that passes contnuously through the pont [ ] =. Only n ths plane s the conductvty tensor contnuous. Snce the sew symmetrc part s zero everywhere n ths plane, the thermal conductvty tensor must be symmetrc. [ ] ( ) Fg. 1 Symbolc representaton of abstract nne-dmensonal conductvty space Now we notce that the egenvalues of the symmetrc conductvty tensor are all real and ther correspondng real egenvectors are mutually orthogonal for dstnct egenvalues or can be taen mutually orthogonal for repeated egenvalues. Consequently, we can generally dagonalze the conductvty tensor by choosng the new coordnate system x 1x 2 x, such that the coordnate axes x, x and 1 2 x are along the orthogonal egenvectors ( 1) v, ( 2) v and representaton of the conductvty tensor n ths new coordnate system becomes ( ) v. As a result, the 6

7 where we have =, (12) λ 1 = 11, λ 2 = 22, λ =. (1) On the other hand, the non-sngularty of the conductvty tensor also mposes more restrctons on ths symmetrc tensor. In partcular, f there s no restrcton on the sgns of egenvalues λ 1, λ 2 and λ, then they can also become zero. However, ths result s n contradcton to the nonsngularty of the conductvty tensor. Therefore, the strctly non-sngular conductvty tensor s symmetrc and s ether postve or negatve defnte. Interestngly, we notce that ths shows that the hyper-surface ( ) det = dvdes the abstract space of the symmetrc conductvty tensor space nto postve and negatve defnte conductvty tensor subspaces. Notce that to ths stage, there has been no reference to the second law of thermodynamcs. The combnaton of the frst and second law of thermodynamcs [5] results n the Clausus- Duhem nequalty q T. (14), Ths nequalty shows that the heat flux vector cannot have any postve component n the drecton of temperature gradent. By usng the relaton (1) for heat flux, we can wrte the Clausus-Duhem nequalty (14) as T T. (15),, j Snce the conductvty tensor s a non-sngular symmetrc tensor, the Clausus-Duhem nequalty n the form of (15) shows that ths tensor s postve defnte. Ths means that all of the egenvalues (1) are postve. As mentoned before, the second law of thermodynamcs and Clausus-Duhem nequalty do not have any role here n establshng the symmetrc character of the conductvty tensor. The proof has been solely based on the tensoral character of quanttes n Duhamel s generalzaton of 7

8 Fourer s heat conducton law (1). The Clausus-Duhem nequalty (14) mposes only the postve defnte restrcton on the symmetrc conductvty tensor. 4. Conclusons By usng arguments from tensor analyss, we have establshed the symmetrc character of the conductvty tensor for lnear ansotropc materal. Interestngly, the non-sngular character of the conductvty tensor s fundamental n establshng ths statement. The proof here shows that classcal contnuum thermodynamcs can provde the mathematcal reason for the symmetrc character of the conductvty tensor. Remarably, ths method of proof also shows that the conductvty tensor s ether postve or negatve defnte. However, the Clausus-Duhem nequalty requres ths tensor to be postve defnte. The method of proof used here shows the subtle character of tensors and ther nterrelatonshps n three-dmensonal space, whch has not been fully exploted n studyng physcal phenomena from ths mathematcal vew. By usng the character of tensors, we may fnd mportant results, whch could not have been magned prevously n classcal physcs. As shown n ths paper, the tensoral proof refutes the general belef that the symmetry condton for the conductvty tensor can be derved only based on addtonal physcal assumptons, such as the requrement that the thermal wor functonal have a wea relatve mnmum at equlbrum. It should be emphaszed that the sngularty of all three-dmensonal sew-symmetrc tensors has played a vtal role n establshng the symmetrc character of the conductvty tensor. However, we should remember that sew-symmetrc tensors are only sngular n odd dmensonal spaces, such as the three-dmensonal physcal space consdered here. Sew-symmetrc tensors n even dmensonal spaces are not necessarly sngular. Ths means that f our physcal space had been even dmensonal, e.g., two-dmensonal, the conductvty tensor would not have been symmetrc, unless other physcal arguments were mposed. As one mght expect, the symmetrc character of the resstvty tensor n Ohm s law for electrc conducton and the dffuson coeffcent tensor for Fc s law n mass transfer and other dffusve systems can be establshed usng analogous methods. 8

9 Appendx A. Egenvalue problem for second order tensors Consder the general second order tensor C n three-dmensonal space. problem for ths tensor s defned as The egenvalue Cv j = λv, (A1) where the parameter λ s the egenvalue or prncpal value and the vector v s the egenvector or prncpal drecton. The egenvalue problem (A1) can be wrtten as ( λδ ) j = The condton for (A2) to possess a non-trval soluton for v s whch n terms of elements can be wrtten as C v. (A2) ( λδ ) det C =, (A) C11 λ C12 C1 det λ C C C2 =. (A4) C1 C2 C λ Ths gves the cubc characterstc equaton for λ as λ I λ + I λ I =, (A5) where the real coeffcents I 1, I 2 and I are the nvarants of the tensor C expressed as ( ) I1 = trace C = C, (A6) I2 = ( trace C) trace ( C ) = ( C ) CC j, (A7) 2 2 I 1 = det ( C ) = εε pqrcpc jqcr. (A8) 6 The symbol ε n (A8) s the alternatng or Lev-Cvta symbol. It should be notced that snce the vector v s normalzed, we have where v s the complex conjugate of v. v v =1, (A9) 9

10 Let us call the egenvalues λ 1, λ 2 and λ. The cubc equaton (A5) wth real coeffcents has at least one real root. Therefore, n any case, one egenvalue and ts correspondng egenvector are real, whch we denote as the thrd egensoluton λ and egenvalues λ 1 and λ 2, and ther correspondng egenvectors ( ) v. We notce that the other two ( 1) v and ( 2) v are ether real or complex conjugate of each other. However, for the real egenvalue λ wth the correspondng real normalzed egenvector ( ) v, we have ( ) ( ) v v =1. (A1) Interestngly, we notce that the nvarants of the tensor C can be expressed n terms of egenvalues, where I1 = λ1 + λ2 + λ, (A11) I2 = λ1λ 2 + λ2λ + λλ1, (A12) I = λ λ λ. (A1) 1 2 If a tensor s sngular, the relaton (A1) shows that at least one of the egenvalues vanshes. A second order tensor P s symmetrc, f = P t P P, j = P. (A14) It s seen that a general symmetrc second order tensor n three-dmensonal space s specfed by sx ndependent values. The egenvalues of the symmetrc tensor P are all real and ther correspondng real egenvectors are mutually orthogonal for dstnct egenvalues or can be taen mutually orthogonal for repeated egenvalues. Ths means there s a prmed orthogonal coordnate system x 1x 2x, where the representaton of P s dagonal, that s P P = P. (A15) P A second order tensor Q s sew-symmetrc, f 1

11 t Q = Q, Qj Q =. (A16) As a result, a general sew-symmetrc tensor n three-dmensonal space s specfed by three ndependent values. For the determnant of ths tensor, we have det Snce the tensor Q s three-dmensonal, we have Therefore, (A17) becomes ( ) = det ( ) Q Q. (A17) ( Q) = ( ) ( Q) det 1 det det = det ( Q) ( ) = det ( ). (A18) Q Q. (A19) Ths shows that the determnant of the tensor Q vanshes; that s ( ) det Q =. (A2) Ths n turn shows that one of the egenvalues of the sew-symmetrc tensor Q s zero. Therefore, all three-dmensonal sew-symmetrc tensors are sngular and have one zero egenvalue. Interestngly, the other two egenvalues of any sew-symmetrc tensor purely magnary conjugate par. Q form a References 1. Onsager, L.: Recprocal relatons n rreversble processes, I. Phys. Rev. Lett. 7, (191) 2. Day, D. A., Gurtn, M. E.: On the symmetry of the conductvty tensor and other restrctons n the nonlnear theory of heat conducton. Arch. Raton. Mech. Anal.,,26-2 (1969). Hadjesfandar, A.R..: On the symmetrc character of the thermal conductvty tensor. Internatonal Journal of Materals and Structural Integrty, 8 (4), (214) 4. Duhamel, J.-M.-C.: Sur les équatons générales de la propagaton de la chaleur dans les corps soldes dont la conductblté n est pas la même dans tous les sens. J. Ec. Polytech. Pars. 1 (21), (182) 11

12 5. Malvern, L.E.: Introducton to the Mechancs of a Contnuous Medum. Prentce-Hall, Englewood Clffs, New Jersey (1969) 12