On the symmetric character of the thermal conductivity tensor

Transcription

1 On the symmetrc character of the thermal conductvty tensor Al R. Hadjesfandar Department of Mechancal and Aerospace Engneerng Unversty at Buffalo, State Unversty of New York Buffalo, NY 146 USA September 19, 13 Abstract In ths paper, the symmetrc character of the conductvty tensor for lnear heterogeneous ansotropc materal s establshed as the result of arguments from tensor analyss and lnear algebra for Fourer s heat conducton. The non-sngular nature of the conductvty tensor plays the fundamental role n establshng ths statement. 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. However, classcal contnuum thermodynamcs has not been able to provde any drect reasonng for ths property, nor can t explan why we have to appeal to non-equlbrum statstcal mechancs. Here we establsh the symmetrc character of the conductvty tensor by usng arguments from tensor analyss and lnear algebra regardng the Fourer s heat conducton law. Ths ncludes results from the famlar egenvalue problem concept and the theory of equatons. Interestngly, the fundamental step n ths establshment s based on the nvertblty of the conductvty tensor. 1

2 In the followng secton, we provde an overvew of some mportant aspects ofelementary tensor analyss. Ths ncludes the defntons of tensors, ther nvarants and the character of the egenvalues of second order tensors based on the theory of equatons. In Secton 3, we ntroduce the classcal heat conducton relatons for lnear heterogeneous ansotropc materal. After that n Secton 4, the symmetrc character of the conductvty tensor s establshed by usng the arguments from tensor analyss and lnear algebra. Fnally, Secton 5 contans a summary and some general conclusons.. Prelmnares Consder the three dmensonal orthogonal coordnate system x 1xx3 as the reference frame, where e 1, e and e 3 are unt base vectors. Ths s the man coordnate system we use to represent the components of fundamental tensors and tensor equatons. We also consder the prmed orthogonal coordnate system x 1 xx 3 for further nvestgaton, havng the same orgn, but wth e 1, e and e 3 as unt base vectors. The general orthogonal transformaton between these systems s represented by the 3 3 transformaton matrx a, where a a a a m n m n mn Here the symbol s the Kronecker delta. We notce that the components a are the drecton cosnes among the axes x and x j ; that s a (1) cos x, x () j It should be notced that the orthogonalty relatons (1) represents a set of sx ndependent relatons among the nne quanttes a. Ths shows that the orthogonal transformaton matrx a s generally specfed at most by 3 ndependent values. The orthogonal transformatons a among the dfferent coordnate systems are essental n defnng vectors and tensors based on the transformatons of ther components []. For example,

3 the scalar A, vector B and second order tensor C transform such that ther components n the prmed coordnate system are A A (3) B ab j (4) C apajqc pq (5) It should be notced that under orthogonal transformatons some scalar values related to the components of B and C do not change, whch are called nvarants of these quanttes. For the vector B there s only one nvarant, whch s the length L B of ths vector. components, ths nvarant s In terms of L B BB (6) For the second order tensor C, ts three egenvalues are the three ndependent nvarants. The egenvalue problem s defned as Cv j v (7) where the parameter s the egenvalue or prncpal value and the vector v s the egenvector or prncpal drecton. The egenvalue problem (7) can be wrtten as j The condton for (8) to possess non-trval soluton for v s whch n terms of elements can be wrtten as Ths gves the cubc characterstc equaton for as where the real coeffcents I C, II C and C v (8) det C (9) C11 C1 C13 det 1 C C C3 C31 C3 C33 3 C C C (1) I II III (11) III C are I C tracec C (1) 3

4 1 1 II trace trace C C C C CCj (13) III 1 det C 6 C C C (14) C k pqr p jq kr The symbol k n (14) s the alternatng or Lev-Cvta symbol. Let us call the egenvalues 1, and 3. The cubc equaton (1) 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 3 and v 3. We notce that the other two egenvalues 1 and, and ther correspondng egenvectors v 1 and v complex conjugate of each other. As a result, we can see that 1 3 are ether real or I (15) C II (16) C III (17) C 1 3 It should be mentoned that the vector v s usually normalzed such that t becomes a unt vector, that s where v s the complex conjugate of v. vv 1 (18) Instead of the egenvalues we may use ther combnatons I C, II C and III C as the new nvarants, whch can be expressed drectly n terms of the elements of the tensor. Therefore, the real values I C, II C and III C are called the fundamental nvarants of the tensor C. A second order tensor P s symmetrc, f P P (19) j The egenvalues of the symmetrc tensor P are all real and ther correspondng egenvectors are mutually orthogonal for dstnct egenvalues or can be taken mutually orthogonal for repeated 4

5 egenvalues. Ths means there s a prmed orthogonal coordnate system x 1, xx 3 where the representaton of P s dagonal, that s P 11 P P P 33 () A second order tensor Q s skew-symmetrc, f It can be easly shown that the determnant of ths tensor vanshes; that s Q Q (1) j III det Q () Q Ths n turn shows that at least one of the egenvalues of the skew-symmetrc tensor Q s zero. We notce that the symmetry and skew-symmetry character of tensors are preserved n orthogonal transformatons. Interestngly, the general second-order tensor decomposed nto the unque sum of ts symmetrc where C C can be and skew-symmetrc C parts, such that C C C (3) C 1 C Cj C j (4) C 1 C Cj C j (5) Notce that here we have ntroduced parentheses surroundng a par of ndces to denote the symmetrc part of a second order tensor, whereas square brackets are assocated wth the skewsymmetrc part. 3. Fundamental heat conducton theory Consder the heat conducton n a heterogeneous ansotropc sold materal contnuum at rest. In contnuum mechancs, t s postulated that the amount of heat energy flow through a surface element ds wth outward drected unt normal vector n s qds, n where q n s the heat flux or 5

6 thermal flux. Let us denote q 1, q and q 3 as the heat fluxes through surfaces wth unt normal n the drecton of coordnate axes x 1, x and x 3, respectvely. It can be shown that these quanttes defne a heat flux vector q qe (see for example Carslaw and Jaeger [3]). As a result of ths, we have the relaton q qn qn (6) n for the heat flux q n. The combnaton of the frst and second law of thermodynamcs [] results n the Clausus- Duhem nequalty qt (7), Ths nequalty shows that the heat flux vector cannot have any postve component n the drecton of temperature gradent Lnear heat conducton theory For lnear heterogeneous ansotropc materal, Duhamel s generalzaton of Fourer s heat conducton law [4] s q k T (8), j Here the tensor k s the materal thermal conductvty tensor, whch can vary from pont to pont. The mnus sgn n (5) assures that the heat flow occurs from a hgher to a lower temperature. In terms of components, the conductvty tensor n the orgnal coordnate system x1xx 3 can be wrtten as k k k k k1 k k3 k 31 k3 k33 Because t s requred that the lnear relaton (8) be nvertble, the conductvty tensor needs to be non-sngular, that s (9) det k det k (3) 6

7 Snce we have not establshed the symmetry character of k, the nne components of k are ndependent of each other at ths stage. Therefore, the conductvty tensor k s specfed by nne ndependent components n the general case. By decomposng the thermal conductvty tensor k nto symmetrc k and skew-symmetrc k parts, we have k k k (31) where 1 k k k j k j (3) 1 k k k j kj (33) In the general case, the tensors k and k are specfed by sx and three ndependent components, respectvely. By usng the relaton (8) for heat flux, we can wrte the Clausus-Duhem nequalty (7) as ktt (34),, j Snce k s skew-symmetrc, we have k TT (35),, j and ktt k TT (36),, j,, j Therefore, the Clausus-Duhem nequalty can be wrtten as k T T (37),, j whch requres that the tensor kbe postve defnte. However, the Clausus-Duhem nequalty does not mpose any restrcton on the tensor k. In the followng secton, we prove that k 7

8 vanshes based exclusvely on tensor analyss. It should be emphaszed that ths proof s ndependent of the second law of thermodynamcs and Clausus-Duhem nequalty (7), whch mpose only the postve defnte condton restrcton on the symmetrc part of the tensor k. 4. Symmetrc character of the conductvty tensor Consder the heat conducton law n the orgnal coordnate system x 1xx3 q k T (38), j Let us look for a drecton of temperature gradent T,, whch s parallel to the heat flux vector, that s q T (39), Therefore, by usng (39) n (38), we obtan the egenvalue problem kt T (4), j, By consderng the normalzed unt vector v n the drecton of the prncpal drecton T,, we obtan the egenvalue problem as kv j v (41) whch can be wrtten as j k v (4) Therefore, the condton for (4) to have a non-trval soluton for v s det k (43) Ths s the characterstc equaton for the tensor k, whch can also be wrtten as k11 k1 k13 det 1 k k k3 k31 k3 k33 As a result, the characterstc equaton s the cubc equaton (44) 8

9 where 3 k k k I II III (45) I k tracek k (46) 1 1 II trace trace k k k k kkj (47) III 1 det k 6 k k k (48) k k pqr p jq kr Snce III k s non-zero, all egenvalues are non-zero. Therefore, the characterstc equaton (45) has at least one real non-zero egenvalue 3 wth the correspondng real normalzed egenvector 3 v, where v v 1 (49) 3 3 It should be mentoned that the relaton (49) shows that the normalzed egenvector v 3 s specfed by two ndependent values n the orgnal coordnate system x 1xx3. Now we choose the orthogonal coordnate system x 1xx 3 such that the axs x 3 concdes wth the drecton of ths real unt egenvector 3 v. Therefore, we have 3 (5) 1 Let us denote the plane normal to ths drecton as. In ths plane, we choose the orthogonal axes x and 1 x arbtrarly. The Fourer s heat conducton law n ths specal prmed coordnate system x 1xx 3 becomes q kt (51), j As a result, for the egenvalue problem (41) n ths specal prmed coordnate system x 1xx 3, we have 9

10 kv v (5) j where the conductvty tensor s represented n the form k 11 k 1 k 13 k k1 k k3 k 31 k3 k33 By examnng the egenvector (5) n the egenvalue problem (5), we obtan k 13 k As we can see, ths relaton requres that 3 33 k 3 (53) (54) k 13, k 3, k 33 3 (55) Therefore, the representaton of the conductvty tensor n the specal prmed orthogonal coordnate system x 1xx 3 reduces to k k k k k k 31 k3 k33 (56) Snce the determnant of the conductvty s nvarant, we have from (3) Ths obvously requres and k k 11 1 det k k33 det k 1 k k 33 3 (57) k (58) k 11 1 det k11k k1k1 k 1 k (59) Now let us consder the temperature feld, such that 1 T T T x, x (6) As a result, the relaton (51) for the heat flux becomes 1

11 q ktkt 1 11,1 1, q ktk T (61) 1,1, q ktk T 3 31,1 3, We notce that one should be able to obtan,1 T and, T for gven heat flux q q q q. 1 3 However, the system (61) for,1 T and, T s over-determned. Therefore, there must be a lnear dependency among these equatons. By scrutnzng the relaton (59), we realze that the frst two equatons q ktkt (6) 1 11,1 1, q ktk T 1,1, are the requred set of equatons to obtan T and T, for gven heat flux components 1,1 q and q. As a result, the components of temperature gradent are explctly expressed as T 1 k q k q,1 1 1 k11k k1k1 T 1 k q k q, k11k k1k1 (63) From these relatons, t s clearly seen that,1 T and, T are ndependent of the component 3 q. Therefore, the last equaton n (61) has to be trvally satsfed for any arbtrary gven 1 q and q. Ths condton requres q 3, k 31, k 3 (64) As a result of ths, the conductvty tensor s specfed by fve ndependent elements n the specal prmed coordnate system x1; xx 3 that s k k k k k 11 1 k 33 1 (65) 11

12 The components of the conductvty tensor n the orgnal unprmed coordnate system x 1xx3 are obtaned by usng the transformaton k a a k (66) m nj mn where 3 3 a v (67) or explctly a v, 3 3 a v, a v (68) Because of the normalzng condton (49), the relatons n (68) enforce only two ndependent constrant values n (66). As a result, the conductvty tensor k s specfed by seven ndependent elements n the orgnal coordnate system x 1xx3. Ths result s n contradcton wth our orgnal statement that the conductvty tensor k s specfed by nne ndependent components. To resolve ths nconsstency, we consder the symmetrc and skew-symmetrc parts of the tensor k. It s seen that the symmetrc tensor k and skew-symmetrc tensor k cannot be smultaneously specfed by sx and three ndependent components any more. For further nvestgaton, we consder the three followng possble cases: Case (). k and k are specfed by four and three ndependent values, respectvely. However, k s a general symmetrc tensor wth sx ndependent values. Ths contradcton requres k, k k (7) However, we notce that for ths case det k det k (71) whch volates the non-sngularty condton for k. Ths case s obvously not acceptable. 1

13 Case (). k and k are specfed by fve and two ndependent values, respectvely. Ths contradcts wth the generalty of the conductvty tensor. As a result, both tensor parts vansh; that s Therefore, ths case also s not acceptable. k k k (69) Case (). k and k are specfed by sx and one ndependent values, respectvely. However, k s a general skew-symmetrc tensor wth three ndependent values. Ths contradcton requres k (7) Consequently, t s seen that ths case s the only acceptable case, whch states that the conductvty tensor s symmetrc k k (73) Ths smply means k k (74) j Therefore, the general conductvty tensor s specfed by sx ndependent components. Because of ths symmetry character, the egenvalues of ths tensor are real and ther correspondng egenvectors are orthogonal. Ths shows that n our prmed coordnate system x 1 xx 3, the conductvty tensor gven by (65) s specfed by four ndependent elements such that k k (75) 1 1 As a result, we can choose the axes x and 1 x along the other orthogonal egenvectors such that the conductvty tensor becomes dagonal, that s k 11 k k k 33 (76) 13

14 where, we have k 11 1, k, k 33 3 (77) It s seen that the Clausus-Duhem nequalty (37) can be wrtten as ktt (78),, j Snce k s non-sngular and symmetrc, ths nequalty whch requres that the tensor k be postve defnte. Ths means that all egenvalues (77) are postve. As we mentoned before, the second law of thermodynamcs and Clausus-Duhem nequalty do not have any role here n establshng the symmetry character of the conductvty tensor. Our proof has been solely based on the tensoral character of quanttes n Duhamel s generalzaton of Fourer s heat conducton law (8) by usng some fundamentals of algebra. 5. Conclusons By usng arguments from tensor analyss and lnear algebra, the symmetrc character of the conductvty tensor for lnear heterogeneous ansotropc materal has been establshed. Ths shows that classcal contnuum mechancs can provde the mathematcal reason for the symmetrc character of the conductvty tensor, whch s a necessary condton for havng the consstent tensoral relatons n classcal heat conducton theory. The method of proof here shows the subtle character of the tensors and ther nterrelatonshps, whch has not been fully utlzed n studyng physcal phenomena from ths mathematcal vew. By usng the character of tensor relatons, we may fnd mportant results, whch could not have been magned prevously n classcal contnuum mechancs. Interestngly, the symmetrc character of the resstvty tensor n Ohm s law for electrc conducton and the dffuson coeffcent tensor for Fck s law n mass transfer and other dffusve systems can be establshed usng analogous methods. 14

15 References [1] L. Onsager, Recprocal relatons n rreversble processes. I, Phys. Rev. Lett. 37 (1931) [] L. E. Malvern, Introducton to the Mechancs of a Contnuous Medum, Prentce-Hall, Englewood Clffs, New Jersey, [3] H. S. Carslaw, J. C. Jaeger, Conducton of Heat n Solds, nd ed., Clarendon Press, Oxford, [4] J.-M.-C. Duhamel, 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, 13 (1) (183)