MATRICES AND TENSORS

Transcription

1 APPENDIX MATRICES AND TENSORS A.. INTRODUCTION AND RATIONALE The purpose of this appedix is to preset the otatio ad most of the mathematical techiques that are used i the body of the text. The audiece is assumed to have bee through several years of college-level mathematics, which icluded the differetial ad itegral calculus, differetial equatios, fuctios of several variables, partial derivatives, ad a itroductio to liear algebra. Matrices are reviewed briefly, ad determiats, vectors, ad tesors of order two are described. The applicatio of this liear algebra to material that appears i udergraduate egieerig courses o mechaics is illustrated by discussios of cocepts like the area ad mass momets of iertia, Mohr s circles, ad the vector cross ad triple scalar products. The otatio, as far as possible, will be a matrix otatio that is easily etered ito existig symbolic computatioal programs like Maple, Mathematica, Matlab, ad Mathcad. The desire to represet the compoets of three-dimesioal fourth-order tesors that appear i aisotropic elasticity as the compoets of six-dimesioal secod-order tesors ad thus represet these compoets i matrices of tesor compoets i six dimesios leads to the otraditioal part of this appedix. This is also oe of the otraditioal aspects i the text of the book, but a mior oe. This is described i A., alog with the ratioale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES A r-by-c matrix, M, is a rectagular array of umbers cosistig of r rows ad c colums: M M2... M c M2 M22... M 2c M = (A) Mr.... M rc The typical elemet of the array, M ij, is the ith elemet i the jth colum; i this text elemets M ij will be real umbers or fuctios whose values are real umbers. The traspose of matrix M is deoted by M T ad is obtaied from M by iterchagig the rows ad colums: M M2... M r M2 M22... M T r 2 M = (A2) M c.... M rc 595

2 596 APPENDIX: MATRICES AND TENSORS The operatio of obtaiig M T from M is called traspositio. I this text we are iterested i special cases of r-by-c matrix M. These special cases are those of the square matrix, r = c =, the case of the row matrix, r =, c =, ad the case of the colum matrix, r =, c =. Further, the special subcases of iterest are = 2, =, ad = 6; subcase = reduces all three special cases to the trivial situatio of a sigle umber or scalar. Square matrix A has the form while row ad colum matrices r ad c have the forms A A2... A A2 A22... A 2 A =......, (A) A.... A [ r r r ] r = 2..., c c 2. c =., (A4). c respectively. The traspose of a colum matrix is a row matrix, ad thus [ ] T c = c c2... c. (A5) To save space i books ad papers, the form of c i (A5) is used more frequetly tha the form i the secod of (A4). Wherever possible, square matrices will be deoted by upper-case boldface Lati letters, while row ad colum matrices will be deoted by lower-case boldface Lati letters, as is the case i eqs. (A) ad (A4). A.. THE TYPES AND ALGEBRA OF SQUARE MATRICES The elemets of square matrix A give by (A) for which the row ad colum idices are equal, amely elemets A, A 22,, A, are called diagoal elemets. A matrix with oly diagoal elemets is called a diagoal matrix: A A A = (A6) A The sum of the diagoal elemets of a matrix is a scalar called the trace of the matrix ad, for matrix A, it is deoted by tra: tr = A + A A A. (A7) If the trace of a matrix is zero, the matrix is said to be traceless. Note also that tra = tra T.

3 TISSUE MECHANICS 597 The zero ad the uit matrix, 0 ad, respectively, costitute the ull elemet, the 0, ad the uit elemet, the, i the algebra of square matrices. The zero matrix is a matrix whose every elemet is zero ad the uit matrix is a diagoal matrix whose diagoal elemets are all oe: =......, = (A8) A special symbol, the Kroecker delta, δ ij, is itroduced to represet the compoets of the uit matrix. Whe i = j the value of the Kroecker delta is, δ = δ 22 = = δ =, ad whe i j the value of the Kroecker delta is 0, δ 2 = δ 2 = = δ = δ = 0. Multiplicatio of matrix A by a scalar is defied as multiplicatio of every elemet of matrix A by scalar α; thus, α αa αa... αa αa αa... αa αa.... αa A. (A9) It is the easy to show that A = A, A = A, 0A = 0, ad αo = 0.The additio of square matrices is defied oly for matrices with the same umber of rows (or colums). The sum of two matrices, A ad B, is deoted by A + B, where A + B A2 + B2... A + B A2 + B2 A22 + B22... A2 + B 2 A+ B (A0) A + B.... A + B Matrix additio is commutative ad associative, A+ B= B+ A ad A+ ( B+ C) = ( A+ B) + C, (A) respectively. The followig distributive laws coect matrix additio ad matrix multiplicatio by scalars: α( A+ B) = αa+ αb ad ( α+ β) A= αa+ βa, (A2) where α ad β are scalars. Negative square matrices may be created by employig the defiitio of matrix multiplicatio by scalar (A8) i the special case whe α =. I this case the defiitio of additio of square matrices (A0) ca be exteded to iclude subtractio of square matrices, A B. A matrix for which B = B T is said to be a symmetric matrix, while a matrix for which C = C T is said to be a skew-symmetric or ati-symmetric matrix. The symmetric ad skewsymmetric parts of a matrix, say A, are costructed from A as follows: symmetric part of A ( T A ) 2 + A, ad (A) skew-symmetric part of A ( T AA ). (A4) 2

4 598 APPENDIX: MATRICES AND TENSORS It is easy to verify that the symmetric part of A is a symmetric matrix ad that the skewsymmetric part of A is a skew-symmetric matrix. The sum of the symmetric part of A ad the skew-symmetric part of A is A: ( T ) A = ( T ) 2 A + A + 2 A A. (A5) This result shows that ay square matrix ca be decomposed ito the sum of a symmetric ad a skew-symmetric matrix. Usig the trace operatio itroduced above, represetatio (A5) ca be exteded to three-way decompositio of matrix A: (tr A) (tr ) ( T ) 2 A = + ) ( T + ) 2 A A + 2 A A A. (A6) The last term i this decompositio is still the skew-symmetric part of the matrix. The secod term is the traceless symmetric part of the matrix, ad the first term is simply the trace of the matrix multiplied by the uit matrix. Example A.. Costruct the three-way decompositio of matrix A give by 2 A = Solutio: The symmetric ad skew-symmetric parts of A, as well as the trace of A are calculated: ( T ) 5 7, ( T ) 0, tr 5 2 A+ A = 2 A A = A = ; the, sice =, it follows from (A6) that A = Itroducig the otatio for the deviatoric part of -by- square matrix A, (tr ) deva= A A, (A7) the represetatio for matrix A give by (A6) may be rewritte as A= H+ D+ S, (A8) where H is called the hydrostatic compoet, D is called the deviatoric compoet, ad S is the skew-symmetric compoet, (tr ) H= A, (dev dev T ) 2 D = A + A, = ( T ) 2 S A A. (A9)

5 TISSUE MECHANICS 599 Example A..2 Show that tr(deva) = 0. Solutio: Applyig the trace operatio to both sides of (A7), oe obtais tr(deva) = tra (/)tra tr; the, sice tr =, it follows that tr(deva) = 0. The product of two square matrices, A ad B, with equal umbers of rows (colums) is a square matrix with the same umber of rows (colums). The matrix product is writte as A B where A B is defied by k= ( AB ) = A B ; (A20) ij ik kj k= thus, for example, the elemet i the rth row ad cth colum of product A B is give by ( AB ) = A B + A B A B. rc r c r2 2c r c The dot iside matrix product A B idicates that oe idex from A ad oe idex from B are to be summed over. The positioig of the summatio idex o the two matrices ivolved i a matrix product is critical ad is reflected i the matrix otatio by the traspose. I the three equatios below, (A2), study carefully how the positios of the summatio idices withi the summatio sig chage i relatio to the positio of the traspose o the matrices i the associated matrix product: k= k= k= T T T T ( AB ) = A B, ( A B) = A B, ( A B ) = A B. (A2) ij ik jk ij ki kj ij ki jk k= k= k= A widely used otatioal covetio, called the Eistei summatio covetio, drops the summatio symbol i (A20) ad writes ( AB ) ij = A ik B kj, (A22) where the covetio is the uderstadig that the repeated idex, k, is to be summed over its rage of admissible values from to. For = 6, the rage of admissible values is to 6, icludig 2,, 4, ad 5. The two k idices are the summatio or dummy idices. A summatio idex is defied as a idex that occurs i a summad twice ad oly twice. Note that summads are terms i equatios separated from each other by plus, mius, or equal sigs. The existece of summatio idices i a summad requires that the summad be summed with respect to those idices over the etire rage of admissible values. Note that the summatio idex is oly a meas of statig that a term must be summed, ad the letter used for this idex is immaterial; thus A im B mj has the same meaig as A ik B kj. The other idices i formula (A22), the i ad j idices, are called free idices. A free idex is free to take o ay oe of its rage of admissible values from to. For example, if were, the free idex could be, 2 or. A free idex is formally defied as a idex that occurs oce ad oly oce i every summad of a equatio. A free idex may take o ay or all of its admissible values; the total umber of equatios that may be represeted by a equatio with oe free idex is the rage of admissible values. Thus, equatio (A22) represets 2 separate equatios. For two 2-by-2 matrices A ad B, the product is writte as AB A A 2 B B 2 AB + A2B2 AB2 + A2B 22 = = A2 A 22 B2 B 22 A2B + A22B2 A2B2 + A22B, (A2) 22

6 600 APPENDIX: MATRICES AND TENSORS where, i this case, products (A20) ad (A22) stad for 2 = 2 2 = 4 separate equatios, the righthad sides of which are the four elemets of the last matrix i (A2). A very sigificat feature of matrix multiplicatio is ocommutativity, that is to say, A B B A. Note, for example, that trasposed product B A of the multiplicatio represeted i (A2), BA B B 2 A A 2 BA + B2 A2 BA2 + B2 A 22 = = B2 B 22 A2 A 22 B2A + B22 A2 B2A2 + B22 A, (A24) 22 is a illustratio of the fact that A B B A, i geeral. If A B = B A, matrices A ad B are said to commute. Fially, matrix multiplicatio is associative: ad matrix multiplicatio is distributive with respect to additio: A( B C) = ( AB) C, (A25) A ( B+ C) = A B+ AC ad ( B+ C) A= B A+ C A, (A26) provided the results of these operatios are defied. Example A.. Costruct products A B ad B A of matrices A ad B give by A= 4 5 6, 4 5 B = Solutio: Products A B ad B A are give by Observe that A B B A A B= , B A = The colo or double dot otatio betwee the two secod-order tesors is a extesio of the sigle dot otatio betwee the matrices, A B, ad idicates that oe idex from A ad oe idex from B are to be summed over; the double dot otatio betwee the matrices, A:B, idicates that both idices of A are to be summed with differet idices from B, ad thus = k= A: B A B. i= k= This colo otatio stads for the same operatio as the trace of the product, A:B = tr(a B). Although tr(a B) ad A:B mea the same thig, A:B ivolves fewer characters ad it will be the otatio of choice. Note that A:B = A T :B T ad A T :B= A:B T but that A:B A T :Bi geeral. I the cosideratios of mechaics, matrices are ofte fuctios of coordiate positios x, x 2, x ad time t. I this case the matrix is writte A(x, x 2, x, t), which meas that each elemet of A is a fuctio of x, x 2, x ad t: ik ki

7 TISSUE MECHANICS 60 A( x, x, x, t ) = 2 A( x, x2, x, t) A2( x, x2, x, t)... A ( x, x2, x, t) A2( x, x2, x, t) A22( x, x2, x, t)... A2 ( x, x2, x, t) (A27) A ( x, x, x, t).... A ( x, x, x, t) 2 2 Let operator stad for a total derivative, or a partial derivative with respect to x, x 2, x, or t, or a defiite or idefiite (sigle or multiple) itegral; the the operatio of the operator o the matrix follows the same rule as multiplicatio of a matrix by a scalar (A9); thus, A( x, x, x, t) = 2 A( x, x2, x, t) A2( x, x2, x, t)... A ( x, x2, x, t) A2( x, x2, x, t) A22( x, x2, x, t)... A2 ( x, x2, x, t) A ( x, x, x, t).... A ( x, x, x, t) 2 2 The followig distributive laws coect matrix additio ad operator operatios:. (A28) ( A+ B) = B+ A ad ( + )A= A+ A, (A29) where ad 2 are two differet operators. 2 2 Problems A... Simplify the followig expressio by usig the summatio idex covetio: 0 = rw + r2w2 + rw, ψ = ( uv + uv + uv )( uv + uv + uv ), φ = A x + A xx + A xx+ A xx + A xx A22x2 + A2xx2+ A2x2x+ Ax. A..2. Matrix M has the umbers 4, 5, 5 i its first row,,, i its secod row, ad 7,, i its third row. Fid the traspose of M, the symmetric part of M, ad the skewsymmetric part of M. A... Prove that xi / xj = δij. A..4. Cosider hydrostatic compoet H, deviatoric compoet D, ad skew-symmetric compoet S of square -by- matrix A defied by (A7) ad (A8). Evaluate the followig: trh, trd, trs, tr(h D) =H:D, tr(h S) = H:S, ad tr(s D)=S:D. A..5. For the matrices i Example A. show that tra B = trb A = 666. I geeral, will A:B = B:A, or is this a special case? A..6. Prove that A:B is zero if A is symmetric ad B is skew-symmetric. A..7. Calculate A T B, A B T ad A T B T for matrices A ad B of Example A... A..8. Fid the derivative of matrix A(t) with respect to t: 2 t t siωt A () t = cosht lt 7t. / t / t2 lt2

8 602 APPENDIX: MATRICES AND TENSORS A..9. Show that (A B) T = B T A T. A..0. Show that (A B C) T = C T B T A T. A.4. THE ALGEBRA OF N-TUPLES The algebra of colum matrices is the same as the algebra of row matrices. The colum matrices eed oly be trasposed to be equivalet to row matrices, as illustrated i eqs. (A) ad (A4). A phrase that describes both row ad colum matrices is -tuples. This phrase will be used here because it is descriptive ad iclusive. A zero -tuple is a -tuple whose etries are all zero; it is deoted by 0 = [0, 0,, 0]. The multiplicatio of -tuple r by scalar α is defied as multiplicatio of every elemet of -tuple r by scalar α, ad thus αr = [αr, αr 2,, αr ]. As with square matrices, it is the easy to show for -tuples that r = r, r = r, 0r = 0, ad α0 = 0. Additio of -tuples is oly defied for -tuples with the same. The sum of two -tuples, r ad t, is deoted by r + t, where r + t = [r + t, r 2 + t 2,, r + t ]. Row-matrix additio is commutative, r + t = t + r, ad associative, r + (t + u) = (r + t) + u. The followig distributive laws coect -tuple additio ad -tuple multiplicatio by scalars; thus, α(r + t) =αr + αt ad (α + β)r = αr + βr, where α ad β are scalars. Negative -tuples may be created by employig the defiitio of -tuple multiplicatio by a scalar, αr = [αr, αr 2,, αr ], i the special case whe α =. I this case the defiitio of additio of -tuples, r + t = [r + t, r 2 + t 2,, r + t ], ca be exteded to iclude subtractio of -tuples, r t, ad the differece betwee -tuples, r t. Two -tuples may be employed to create a square matrix. The square matrix formed from r ad t is called the ope product of -tuples r ad t; it is deoted by r t, ad defied by rt rt 2... rt rt rt... rt = rt.... rt r t. (A0) The America physicist J. Willard Gibbs itroduced the cocept of the ope product of vectors, callig the product a dyad. This termiology is still used i some books, ad the otatio is spoke of as the dyadic otatio. The trace of this square matrix, tr{r t} is the scalar product of r ad t: tr{ r t} = r t = rt + rt rt. (A) I the special case of =, the skew-symmetric part of ope product r t, 0 rt 2rt 2 rt rt rt 2 rt 2 0 rt 2 rt 2 2, (A2) rt rt rt 2rt 2 0 provides the compoets of the cross product of r ad t, deoted by r x t, ad writte as r x t = [r 2 t r t 2, r t r t, r t 2 r 2 t ]. These poits cocerig dot product r t ad cross product r x t will be revisited later i this Appedix. Example A.4. Give -tuples a = [, 2, ] ad b = [4, 5, 6], costruct ope product matrix a b, the skew-symmetric part of the ope product matrix, ad trace of the ope product matrix.

9 TISSUE MECHANICS 60 Solutio: ad tr{a b} = a b = , ( T ) a b= 0 aba b = Frequetly, -tuples are cosidered as fuctios of coordiate positios x, x 2, x ad time t. I this case the -tuple is writte r(x, x 2, x, t), which meas that each elemet of r is a fuctio of x, x 2, x, ad t: r ( x, x, x, t) = [ r( x, x, x, t), r ( x, x, x, t),..., r ( x, x, x, t)]. (A) Agai, lettig the operator stad for a total derivative, or a partial derivative with respect to x, x 2, x, or t, or a defiite or idefiite (sigle or multiple) itegral, the the operatio of the operator o the -tuple follows the same rule as the multiplicatio of a -tuple by a scalar (A9), ad thus r( x, x, x, t) = [ r( x, x, x, t), r ( x, x, x, t),..., r ( x, x, x, t)]. (A4) The followig distributive laws coect matrix additio ad operator operatios: ( r+ t) = r+t ad ( + )r= r+ r, (A5) where ad 2 are two differet operators. 2 2 Problems A.4.. Fid the derivative of -tuple r(x, x 2, x, t) = [x x 2 x, 0x x 2, cosh αx ] T with respect to x. A.4.2. Fid the symmetric ad skew-symmetric parts of matrix r s, where r = [, 2,,4] ad s = [5, 6,7,8]. A.5. LINEAR TRANSFORMATIONS A system of liear equatios, r = A t + A t A t, 2 2 r2 = A2t+ A22t A2t, r = A t + A t + + A t, may be cotracted horizotally usig the summatio symbol, ad thus k= r = A t, k k k= r k= = A t, 2 2k k k= k= r = A t. k= k k (A6) (A7)

10 604 APPENDIX: MATRICES AND TENSORS Itroductio of the free idex covetio codeses this system of equatios vertically: i k= r = A t. (A8) k= This result may also be represeted i matrix otatio as a combiatio of -tuples, r ad t, ad square matrix A: ik k r= At, (A9) where the dot betwee A ad t idicates that summatio is with respect to oe idex of A ad oe idex of t, or r t r 2 A A2... A t 2. A2 A22... A 2. = A.... A. r t (A40) if the operatio of matrix A upo colum matrix t is iterpreted as the operatio of the square matrix upo the -tuple defied by (A8). This is a operatio very similar to square matrix multiplicatio. This may be see easily by rewritig the -tuple i (A40) as the first colum of a square matrix whose etries are all otherwise zero; thus, the operatio is oe of multiplicatio of oe square matrix by aother: r r 2 A A2... A t A2 A22... A 2 t = (A4). A.... A t r The operatio of square matrix A o -tuple t is called a liear trasformatio of t ito - tuple r. The liearity property is reflected i the property that A applied to the sum (r + t) follows a distributive law A ( r+ t ) = Ar + At ad that multiplicatio by scalar α follows rule α( Ar ) = A( αr. ) These two properties may be combied ito oe, A ( αr+ βt) = αar + βat, where α ad β are scalars. The compositio of liear trasformatios is agai a liear trasformatio. Cosider liear trasformatio t = B u, u t (meaig u is trasformed ito t), which is combied with liear trasformatio (A9), r = A t, t r, to trasform u r, ad thus r = A B u, ad if we let C A B, the r = C u. The result of the compositio of the two liear trasformatios, r = A t ad t = B u, is the a ew liear trasformatio, r = C u, where square matrix C is give by matrix product A B. To verify that it is, i fact, a matrix multiplicatio, the compositio of trasformatios is doe agai i the idicial otatio. Trasformatio t = B u i the idicial otatio, t m= = B u, (A42) k km m m=

11 TISSUE MECHANICS 605 is substituted ito r = A t i idicial otatio (A8), which may be rewritte as where C is defied by k= m= r A B u =, (A4) i ik km m k= m= C m= i Cimum m= r =, (A44) k= = A B. (A45) im ik km k= Compariso of (A45) with (A20) shows that C is the matrix product of A ad B, C = A B. The calculatio from (A42) to (A45) may be repeated usig the Eistei summatio covetio. The calculatio will be similar to the oe above, with the exceptio that the summatio symbols will ot appear. Example A.5. Determie result r = C u of the compositio of the two liear trasformatios, r = A t ad t = B u, where A ad B are give by A= 4 5 6, 4 5 B = Solutio: Square matrix C represetig the composed liear trasformatio is give by the matrix product A B: A B = It is importat to be able to costruct the iverse of liear trasformatio r = A t, t = A r, if it exists. The iverse trasformatio exists if iverse matrix A ca be costructed from matrix A. The costructio of the iverse of a matrix ivolves the determiat of the matrix ad the matrix of the cofactors. The determiat of A is deoted by DetA. A matrix is said to be sigular if its determiat is zero, o-sigular if it is ot. The cofactor of elemet A ij of A is deoted by coa ij ad is equal to ( ) i+j times the determiat of a matrix costructed from matrix A by deletig the row ad colum i which elemet A ij occurs. A matrix formed of cofactors coa ij is deoted by coa. Example A.5.2 Compute the matrix of cofactors of A: A = a d e d b f. e f c

12 606 APPENDIX: MATRICES AND TENSORS Solutio: The cofactors of the distict elemets of matrix A are coa = (bc f 2 ), cob = (ac e 2 ), coc = (ab d 2 ), cod = (dc fe), coe = (df eb), ad cof = (af de); thus, the matrix of cofactors of A is coa = 2 bcf ( dcfe) ( df eb) 2 ( dc fe) ac e ( af de) ( df eb) ( af de) abd 2. The formula for the iverse of A is writte i terms of coa as T ( coa) A = Det A, (A46) where ( coa ) T is the matrix of cofactors trasposed. The iverse of a matrix is ot defied if the matrix is sigular. For every osigular square matrix A the iverse of A ca be costructed, ad thus AA = A A=. (A47) It follows the that the iverse of liear trasformatio r = A t, t = A r, exists if matrix A is osigular, DetA 0. Example A.5. Show that the determiat of a -by- ope product matrix, a b, is zero. Solutio: ab ab 2 ab a b = ab 2 ab 2 2 ab 2 = ab abab 2 2 abab 2 2 ab ab 2 ab Det{ } Det ( ). ab( abab abab) + ab( abab abab) = Example A.5.4 Fid the iverse of matrix A = Solutio: The matrix of cofactors is give by thus, the iverse of A is the give by coa = ;

13 TISSUE MECHANICS 607 A T coa = = Det A The eigevalue problem for liear trasformatio r = A t addresses the questio of -tuple t beig trasformed by A ito some scalar multiple of itself, λt. Specifically, for what values of t ad λ does λt = A t? If such values of λ ad t exist, they are called eigevalues ad eige -tuples of matrix A, respectively. The eigevalue problem is the to fid solutios to the equatio ( Aλ) t = 0. (A48) This is a system of liear equatios for the elemets of -tuple t. For the case of = it may be writte i the form ( A λ) t + A t + A t = 0, 2 2 A t + ( A λ) t + A t = 0, (A49) A t + A t + ( A λ) t = The stadard approach to the solutio of a system of liear equatios like (A48) is Cramer s rule. For a system of three equatios i three ukows, (A6) with =, r = At+ A2t2 + At, r2 = A2t+ A22t2 + A2t, (A50) r = At+ A2t2 + At. Cramer s rule provides the solutio for -tuple t =[t, t 2, t ]: r A2 A A r A A A2 r r2 A22 A2 A2 r2 A2 A2 A22 r2 r A2 A A r A A A2 r t =, t2 =, t = DetA DetA DetA. (A5) Cosiderig the case where = ad applyig Cramer s rule to system of equatios (A49), we fid that 0 A2 A A λ 0 A A λ A2 0 0 A22 λ A2 A2 0 A2 A2 A22 λ 0 0 A2 A λ A 0 A λ A A2 0 t=, t2 =, t = Det[ Aλ] Det[ Aλ] Det[ Aλ] which shows, due to the colum of zeros i each umerator determiat, that the oly solutio is that t =[0, 0, 0], uless Det[A λ] = 0. If Det[A λ] = 0, the values of t, t 2, ad t are all of the form 0/0 ad therefore udefied. I this case Cramer s rule provides o iformatio. I

14 608 APPENDIX: MATRICES AND TENSORS order to avoid trivial solutio t = [0, 0, 0], the value of λ is selected so that Det[A λ] = 0. While the argumet was specialized to = i order to coserve page space, result Det[ A λ ] = 0 (A52) holds for all. This coditio forces matrix [A λ] to be sigular ad forces system of equatios (A48) to be liearly depedet. The further solutio of (A52) is explored, retaiig the assumptio of = for coveiece, but it should oted that all the maipulatios ca be accomplished for ay, icludig the values of of iterest here 2,, ad 6. I the case of =, (A52) is writte i the form A λ A2 A A2 A22 λ A2 = 0, (A5) A A A λ 2 ad, whe the determiat is expaded, oe obtais a cubic equatio for λ: where 2 λ λ λ IA + IIA IIIA = 0 (A54) k= A = tra = kk = kk = , (A55) k= I A A A A A II A A A A A A = A2 A + 22 A A + A, (A56) A2 A A A A III A A A A A A 2 A = DetA = (A57) 2 This argumet the geerates a set of three λ's that allow determiat (A5) to vaish. We ote agai that the vaishig of the determiat makes set of equatios (A49) liearly depedet. Sice the system is liearly depedet, all of the compoets of t caot be determied from (A49). Thus, for each value of λ that is a solutio to (A54), we ca fid oly two ratios of the elemets of t t, t 2, ad t. It follows that, for each eige -tuple, there will be oe scalar ukow. I this text we will oly be iterested i the eigevalues of symmetric matrices. I A.7 it is show that a ecessary ad sufficiet coditio for all the eigevalues to be real is that the matrix be symmetric. Example A.5.5 Fid the eigevalues ad costruct the ratios of the eige -tuples of matrix A = (A58) 6 0 2

15 TISSUE MECHANICS 609 Solutio: The cubic equatio associated with this matrix is, from (A54), (A55), (A56). ad (A57), 2 λ λ λ = 0, (A59) which has three roots 27, 8, ad 9. The eige -tuples are costructed usig these eigevalues. The first eige -tuple is obtaied by substitutio of (A58) ad λ = 27 ito (A49), ad thus 9t + 6t + 6t = 0, 6t 2t = 0, 6t 6t = 0. (A60) 2 2 Note the liear depedece of this system of equatios; the first equatio is equal to the secod multiplied by ( /2) ad added to the third multiplied by ( ). Sice there are oly two idepedet equatios, the solutio to this system of equatios is t = t ad t =2t 2, leavig a udetermied parameter i eige -tuple t. Similar results are obtaied by takig λ = 8 ad λ = 9. Problems A.5.. Show that the eigevalues of matrix 2 G = are.45, 0.7, ad A.5.2. Costruct the iverse of matrix A, where a b A =. b c A.5.. Show that the iverse of matrix G of Problem A.5. is give by 2 G =. 2 0 A.5.4. Show that the eigevalues of matrix G of Problem A.5. are the iverse of the eigevalues of matrix G of Problem A.5.. A.5.5. Solve matrix equatio A 2 = A A = A for A assumig that A is osigular. A.5.6. Why is it ot possible to costruct the iverse of a ope product matrix, a b? A.5.7. Costruct a compositioal trasformatio based o matrix G of Problem A.5. ad the ope product matrix, a b, where the -tuples are a = [, 2, ] ad b = [4, 5, 6]. A.5.8. If F is a square matrix ad a is a -tuple, show that a T F T = F a. A.6. VECTOR SPACES Loosely, vectors are defied as -tuples that follow the parallelogram law of additio. More precisely, vectors are defied as elemets of a vector space called the arithmetic -space. Let A deote the set of all -tuples, u = [u, u2, u,..., un], v = [v, v2, v,..., vn], etc., iclud-

16 60 APPENDIX: MATRICES AND TENSORS ig the zero -tuple, 0 = [0, 0, 0,..., 0], ad the egative -tuple, u = [ u, u2, u,..., un]. A arithmetic -space cosists of set A together with the additive ad scalar multiplicatio operatios defied by u + v = [u + v, u2 + v2, u + v,..., un + vn] ad αu = [αu, αu2, αu,..., αun], respectively. The additive operatio defied by u + v = [u + v, u2 + v2, u + v,..., un+ vn] is the parallelogram law of additio. The parallelogram law of additio was first itroduced ad proved experimetally for forces. A vector is defied as a elemet of a vector space, i our case a particular vector space called the arithmetic -space. The scalar product of two vectors i dimesios was defied earlier, (A). This defiitio provided a formula for calculatig scalar product u v ad the magitude of vectors u ad v, u = u u ad v = v v. Thus, oe ca cosider the elemetary defiitio of the scalar product below as the defiitio of agle ζ: i= u v= uv = u v cosζ. (A6) i= i i Recallig that there is a geometric iterpretatio of ζ as the agle betwee two vectors u ad v i two or three dimesios, it may seem strage to have cos ζ appear i formula (A6), which is valid i dimesios. However, sice u v divided by u v is always less tha oe, ad thus defiitio (A6) is reasoable ot oly for two ad three dimesios, but for a space of ay fiite dimesio. It is oly i two ad three dimesios that agle ζ may be iterpreted as the agle betwee the two vectors. Example A.6. Show that the magitude of the sum of two uit vectors e = [,0] ad e 2 = [cos α, si α] ca vary i magitude from 0 to 2, depedig o the value of agle α. Solutio: e + e 2 = [ + cos α, si α], ad thus e + e 2 = 2(+ cos α). It follows that e + e 2 = 2 whe α = 0, e + e 2 = 0 whe α = π, ad e + e 2 = 2 whe α = π/2. Thus, the sum of two uit vectors i two dimesios ca poit i ay directio i the two dimesios ad ca have a magitude betwee 0 ad 2. A set of uit vectors ei, i =, 2,...,, is called a orthoormal basis of the vector space if all the base vectors are of uit magitude ad are orthogoal to each other, ei ej = δij for i, j havig rage. From the defiitio of orthogoality oe ca see that, whe i j, uit vectors ei ad ej are orthogoal. I the case where i = j the restrictio reduces to the requiremet that the ei's be uit vectors. The elemets of -tuples v = [v, v 2, v,..., v ] referred to a orthoormal basis are called compoets. A importat questio cocerig vectors is the maer i which their compoets chage as their orthoormal basis is chaged. I order to distiguish betwee the compoets referred to two differet bases of a vector space we itroduce two sets of idices. The first set of idices is composed of lowercase Lati letters i, j, k, m,, p, etc. which have admissible values, 2,,..., as before; the secod set is composed of lowercase Greek letters α, β, γ, δ,..., etc., whose set of admissible values are Roma umerals I, II, III,...,. The Lati basis refers to base vectors e i while the Greek basis refers to base vectors e α. The compoets of vector v referred to a Lati basis are the v i, i =, 2,,...,, while the compoets of the same vector referred to a Greek basis are v α, α = I, II, III,...,. It should be clear that e is ot the same as e Ι, v 2 is ot the same as v ΙΙ, etc., that e, v 2 refer to the Lati basis while e I, v II refer to the Greek basis. The termiology of callig a set of idices "Lati" ad the other "Greek" is arbitrary; we could have itroduced the secod set of idices as i', j', k', m', ',

17 TISSUE MECHANICS 6 p', etc., which would have had admissible values of ', 2', ',...,, ad subsequetly spoke of the uprimed ad primed sets of idices. The rage of the idices i the Greek ad Lati sets must be the same sice both sets of base vectors e i ad e α occupy the same space. It follows the that the two sets, e i ad e α, take together are liearly depedet ad therefore we ca write that e i is a liear combiatio of the e α 's ad vice versa. These relatioships are expressed as liear trasformatios: α= ei = Qiαe α ad α= i= eα = Qiα e i, (A62) i= where Q = [Q iα ] is the matrix characterizig the liear trasformatio. I the case of = the first of these equatios may be expaded ito a system of three equatios: e = Q e + Q e + Q e, I I II II III III e = Q e + Q e + Q e, (A6) 2 2I I 2II II 2III III e = Q e + Q e + Q e. I I II II III III If oe takes the scalar product of eι with each of these equatios ad otes that sice the e α, α = I, II, III, form a orthoormal basis, the e Ι e II = e Ι e III = 0, ad Q Ι = e e Ι = e Ι e, Q 2Ι = e 2 e Ι = e Ι e 2, ad Q Ι = e e Ι = e Ι e. Repeatig the scalar product operatio for e Ι Ι ad e Ι ΙΙ shows that, i geeral, Q iα = e i e α = e α e i. Recallig that the scalar product of two vectors is the product of magitudes of each vector ad the cosie of the agle betwee the two vectors (A6), ad that the base vectors are uit vectors, it follows that Q iα = e i e α = e α e i are just the cosies of agles betwee the base vectors of the two bases ivolved. Thus, the compoets of liear trasformatio Q = [Q iα] are the cosies of the agles betwee the base vectors of the two bases ivolved. Because the defiitio of scalar product (A6) is valid i dimesios, all these results are valid i dimesios eve though the two- ad three- dimesioal geometric iterpretatio of the compoets of liear trasformatio Q as the cosies of the agles betwee coordiate axes is o loger valid. The geometric aalogy is very helpful, so cosideratios i three dimesios are cotiued. Three-dimesioal Greek ad Lati coordiate systems are illustrated o the left-had side of Figure A.. Matrix Q with compoets Q iα = e i e α relates the compoets of vectors ad base vectors associated with the Greek system to those associated with the Lati system: eei eeii ee III Qi α i α 2 I 2 II 2 III [ ] [ ] Q= = e e = e e e e e e e e e e e e I II III. (A64) I the special case whe the e ad e I are coicidet, the relative rotatio betwee the two observers' frames is a rotatio about that particular selected ad fixed axis, ad matrix Q has the special form This situatio is illustrated o the left i Figure A Q = 0 cosθ siθ. (A65) 0 siθ cosθ

18 62 APPENDIX: MATRICES AND TENSORS Figure A.. The relative rotatioal orietatio betwee coordiate systems. Matrix Q = [Qiα] characterizig the chage from Lati orthoormal basis ei i a N- dimesioal vector space to Greek basis eα (or vice versa) is a special type of liear trasformatio called a orthogoal trasformatio. Takig the scalar product of ei with ej, where ei ad ej both have represetatio (A62), α= ei = Qiαe α ad α= β= ej = Qjβe β, (A66) β= it follows that α= β= α= β= α= e e = δ = Q Q e e = Q Q δ = Q Q. (A67) i j ij iα jβ α β iα jβ αβ iα jα α= β= α= β= α= There are a umber of steps i calculatio (A67) that should be cosidered carefully. First, the coditio of orthoormality of the bases has bee used twice, e i e j = δ ij ad e α e β = δ αβ. Secod, the trasitio from the term before the last equal sig to the term after that sig is characterized by a chage from a double sum to a sigle sum over ad the loss of Kroecker delta δ αβ. This occurs because the sum over β i the double sum is always zero except i the special case whe α = β due to the presece of Kroecker delta δ αβ. Third, a compariso of the last term i (A67) with the defiitio of matrix product (A20) suggests that it is a matrix product of Q with itself. However, a careful compariso of the last term i (A67) with the defiitio of matrix product (A20) shows that the summatio is over a differet idex i the secod elemet of the product. I order for the last term i (A67) to represet a matrix product, idex α should appear as the first subscripted idex rather tha the secod. However, this α idex may be relocated i the secod matrix by usig the traspositio operatio. Thus, the last term i eq. (A67) is the matrix product of Q with Q T, as may be see from the first of eqs. (A8). Thus, sice the matrix of Kroecker delta compoets is uit matrix, it has bee show that = Q Q T. (A68) If we repeat the calculatio of the scalar product, this time usig e α ad e β rather tha e i ad e j, the it is foud that = Q T Q ad, combied with the previous result, = Q Q T = Q T Q. (A69)

19 TISSUE MECHANICS 6 Usig the fact that Det =, ad two results that are proved i A.8, Det A B = Det A Det B, ad Det A = Det A T, it follows from = QQ T or = Q T Q that Q is osigular ad Det Q = ±. Comparig matrix equatios = Q Q T ad = Q T Q with the equatios defiig the iverse of Q, = Q Q = Q Q, it follows that Q = Q T, (A70) sice the iverse exists (Det Q is ot sigular) ad is uique. Ay matrix that satisfies eq. (A69) is called a orthogoal matrix. Ay chage of orthoormal bases is characterized by a orthogoal matrix ad is called a orthogoal trasformatio. Fially, sice Q = Q T the represetatios of the trasformatio of bases (A62) may be rewritte as α= ei = Qiαe α ad α= i= eα = Qiαe i. (A7) i= Orthogoal matrices are very iterestig, useful, ad easy to hadle; their determiat is always plus or mius oe ad their iverse is obtaied simply by computig their traspose. Furthermore, the multiplicatio of orthogoal matrices has the closure property. To see that the product of two -by- orthogoal matrices is aother -by- orthogoal matrix, let R ad Q be orthogoal matrices ad cosider their product deoted by W = R Q. The iverse of W is give by W = Q R ad its traspose by W T = Q T R T. Sice R ad Q are orthogoal matrices, Q R = Q T R T, it follows that W = W T, ad therefore W is orthogoal. It follows the that the set of all orthogoal matrices has the closure property as well as the associative property with respect to the multiplicatio operatio, a idetity elemet (the uit matrix is orthogoal), ad a iverse for each member of the set. Here we shall cosider chagig the basis to which a give vector is referred. While vector v itself is ivariat with respect to a chage of basis, the compoets of v will chage whe the basis to which they are referred is chaged. The compoets of vector v referred to a Lati basis are the v i, i =, 2,,...,, while the compoets of the same vector referred to a Greek basis are v α, α = I, II, III,...,. Sice vector v is uique, i= α= i i α α i= α= v = v e = v e. (A72) Substitutig the secod of (A7) ito the secod equality of (A72), oe obtais which may be rewritte as i= α= i= vi i = Qiαvα i i= α= i= e e, (A7) = 0 i= α= vi Qiαvα i i= α= e. (A74) Takig the dot product of (A74) with ej, it follows that the sum over i is oly ozero whe i = j, ad thus j α= v = Q v. (A75) α= jα α

20 64 APPENDIX: MATRICES AND TENSORS If the first, rather tha the secod, of (A7) is substituted ito the secod equality of (A72), ad similar algebraic maipulatios accomplished, oe obtais v β i= = Q v. (A76) i= Results (A75) ad (A76) are writte i matrix otatio usig superscripted (L) ad (G) to distiguish betwee compoets referred to the Lati or Greek bases: iβ i ( L) ( G) v = Qv, ( G ) T ( L ) v = Q v. (A77) Problems A.6.. Is matrix a orthogoal matrix?. A.6.2. Are matrices A, B, C, ad Q, where Q = C B A, ad where cosφ si φ 0 A = si φ cosφ 0, B = 0 cosθ siθ, 0 siθ cosθ cos ψ 0 si ψ C = 0 0 si ψ 0 cos ψ all orthogoal matrices? A.6.. Does a iverse of the compositioal trasformatio costructed i Problem A.5.7 exist? A.6.4. Is it possible for a ope product of vectors to be a orthogoal matrix? A.6.5. Trasform the compoets of vector v (L) = [, 2, ] to a ew (Greek) coordiate system usig trasformatio 2 2 Q =

21 TISSUE MECHANICS 65 A.7. SECOND-ORDER TENSORS Scalars are tesors of order zero; vectors are tesors of order oe. Tesors of order two will be defied usig vectors. For brevity, we shall refer to "tesors of order two" simply as "tesors" throughout most of this sectio. The otio of a tesor, like the otio of a vector, was geerated by physicists for applicatio i physical theories. I classical dyamics the essetial cocepts of force, velocity, ad acceleratio are all vectors; hece, the mathematical laguage of classical dyamics is that of vectors. I the mechaics of deformable media the essetial cocepts of stress, strai, rate of deformatio, etc., are all secod-order tesors; thus, by aalogy, oe ca expect to deal quite frequetly with secod-order tesors i this brach of mechaics. The reaso for this widespread use of tesors is that they ejoy, like vectors, the property of beig ivariat with respect to the basis, or frame of referece, chose. The defiitio of a tesor is motivated by a cosideratio of the ope or dyadic product of vectors r ad t. Recall that the square matrix formed from r ad t is called the ope product of the -tuples r ad t; it is deoted by r t ad defied by (A0) for -tuples. We employ this same formula to defie the ope product of vectors r ad t. Both of these vectors have represetatios relative to all bases i the vector space, i particular the Lati ad the Greek bases, ad thus from (A72) i= α= r = r e = r e, i i α α i= α= j= β= t = t e = t e. (A78) j j β β j= β= The ope product of vectors r ad t, r t, the has represetatio j= i= β= α= rt i j i j rαtβ α β j= i= β= α= r t = e e = e e. (A79) This is a special type of tesor, but it is referred to the geeral secod-order tesor basis, e i e j, or e α e β. A geeral secod-order tesor is quatity T, defied by the formula relative to bases e i e j, e α e β ad, by implicatio, ay basis i the vector space: j= i= β= α= ij i j αβ α β j= i= β= α= T = T e e = T e e. (A80) Formulas (A78) ad (A80) have similar cotet i that vectors r ad t ad tesor T are quatities idepedet of a base or coordiate system while the compoets of r, t, ad T may be expressed relative to ay basis. I formulas (A78) ad (A80), r, t, ad T are expressed as compoets relative to two differet bases. The vectors are expressed as compoets relative to bases e i ad e α, while tesor T is expressed relative to bases e i e j ad e α e β. Tesor bases e i e j ad e α e β are costructed from vector bases e i ad e α. Example A.7. If base vectors e, e 2, ad e are expressed as e = [, 0, 0] T, e 2 = [0,, 0] T, ad e = [0, 0, ] T, the it follows from (A77) that ad we ca express v i this form: i= v= vie i, i=

22 66 APPENDIX: MATRICES AND TENSORS Create a similar represetatio for T give by (A80) for =. Solutio: The represetatio for T give by (A80), 0 0 v = v 0 v 2 v (A8) 0 0 j= i= T= T e e, j= i= ij i j ivolves base vectors e e, e e 2 etc. These base vectors are expressed as matrices of tesor compoets by The represetatio for T, e e = 0 0 0, e e = 0 0 0, e e = 0 0, etc. (A82) 2 2 the ca be writte i aalogy to (A8) as j= i= T= T e e, j= i= ij i j T = T T 0 0 T T T T T T T The compoets of tesor T relative to the Lati basis, T (L) =[T ij ], are related to the compoets relative to the Greek basis, T (G) =[T αβ ], by ( L ) ( G ) T = QT Q T ad ( G ) T ( L ) T = Q T Q. (A8) These formulas relatig the compoets are the tesorial equivalet of vectorial formulas ( L) ( G) ( ) ( ) v = Qv ad v G = Q T v L give by (A77), ad their derivatio is similar. First, substitute the secod of (A66) ito (A80) twice, oce for each base vector: j= i= j= i= β= α= T = T e e = T Q Q e e. (A84) ij i j αβ iα jβ i j j= i= j= i= β= α= The gather together the terms referred to basis e i e j, ad thus β α ( ) j= i= = = T T Q Q e e = 0. (A85) ij αβ iα jβ i j j= i= β= α=

23 TISSUE MECHANICS 67 Next, take the scalar product of (A85), first with respect to e k, ad the with respect to e m. Oe fids that the oly ozero terms that remai are T β= α= = Q T Q. (A86) km kα αβ mβ β= α= A compariso of the last term i (A86) with the defiitio of matrix product (A20) suggests that it is a triple matrix product ivolvig Q twice ad T (G) oce. Careful compariso of the last term i (A86) with the defiitio of matrix product (A20) shows that the summatio is over a differet idex i the third elemet of the product. I order for the last term i (A86) to represet a triple matrix product, the β idex should appear as the first subscripted idex rather tha the secod. However, this β idex may be relocated i the secod matrix by usig the traspositio operatio, as show i the first equatio of (A2). Thus, the last term i eq. (A86) is the matrix product of Q T with Q T. The result is the first equatio of (A8). If the first, rather tha the secod, of (A67) is substituted ito the secod equality of (A80), ad similar algebraic maipulatios accomplished, oe obtais the secod equatio of (A8). The word tesor is used to refer to quatity T defied by (A80), a quatity idepedet of ay basis. It is also used to refer to the matrix of tesor compoets relative to a particular basis, for example, T (L) =[T ij ] or T (G) = [T αβ ]. I both cases tesor should be tesor of order two, but the order of the tesor is geerally clear from the cotext. A tesor of order N i a space of dimesios is defied by k= j= i= γ= β= α= B = B e e e = B e e e. (A87) ij... k i j k αβ... γ α β γ k= j= i= γ= β= α= The umber of base vectors i the basis is the order N of the tesor. It is easy to see that this defiitio specializes to that of secod-order tesor (A80). The defiitio of a vector as a tesor of order oe is easy to see, ad the defiitio of a scalar as a tesor of order 0 is trivial. I the sectio before last, A.5 o Liear Trasformatios, the eigevalue problem for a liear trasformatio, r = At, was cosidered. Here we exted those results by cosiderig r ad t to be vectors ad A to be a symmetric secod-order tesor, A = A T. The problem is actually little chaged util its coclusio. The eigevalues are still give by (A52) or, for =, by (A54). The values of three quatities I A, II A, III A defied by (A55), (A56), ad (A57) are the same except that A 2 = A 2, A = A, ad A 2 = A 2 due to the assumed symmetry of A, A = A T. These quatities may ow be called the ivariats of tesor A sice their value is the same idepedet of the coordiate system chose for their determiatio. As a example of the ivariace with respect to basis, this property will be derived for I A = tr A. Let T = A i (A86), ad the set idices k = m ad sum from to over idex k; thus, k= k= β= α= β= α= k= β= α= α= A = Q A Q = A Q Q = A δ = A. (A88) kk kα αβ kβ αβ kα kβ αβ αβ αα k= k= β= α= β= α= k= β= α= α= The trasitio across the secod equal sig is a simple rearragemet of terms. The trasitio across the third equal sig is based o coditio k= δ αβ Qk α Qk β k= = (A89) which is a alterate form of (A67), a form equivalet to = Q T Q. The trasitio across the fourth equal sig employs the defiitio of the Kroecker delta ad summatio over β. The

24 68 APPENDIX: MATRICES AND TENSORS result is that the trace of the matrix of secod-order tesor compoets relative to ay basis is the same umber: k= α= A = Aαα. (A90) kk k= α= It may also be show that II A ad III A are ivariats of tesor A. Example A.7.2 (extesio of Example A.5.5) Cosider the matrix give by (A58) i Example A.5.5 to be the compoets of a tesor. Costruct the eigevectors of that tesor ad use those eigevectors to costruct a eigebasis: A = (A58) repeated Solutio: The eigevalues were show to be 27, 8, ad 9. It ca be show that the eigevalues must always be real umbers if A is symmetric. Eige -tuples were costructed usig these eigevalues. The first eige -tuple was obtaied by substitutio of (A58) ad λ = 27 ito (A49), ad thus 9t + 6t + 6t = 0,6t 2t = 0,6t 6t = 0. (A60) repeated 2 2 These three coditios, oly two of which are idepedet, gave t = t ad t =2t 2, leavig a udetermied parameter i eige -tuple t. Now that t is a vector, we ca specify the legth of a vector. Aother cosequece of the symmetry of A is that these eigevectors are orthogoal if the eigevalues are distict. Hece, if we set the legth of the eigevectors to be oe to remove the udetermied parameter, we will geerate a orthoormal basis from the set of three eigevectors, sice the eigevalues are distict. If we use ormality coditio t 2 + t 2 + t 2 = 2 ad the results that follow from (A56), t = t ad t =2t 2, we fid that t =± (2 e+ e2 + 2 e ) (A9) which shows that both t ad t are eigevectors. This will be true for ay eigevector because they are really eige-directios. For the secod ad third eigevalues, 8 ad 9, we fid that =± ( ) t =± (2 e 2 e e ), (A92) t e e2 e ad 2 respectively. It is easy to see that these three eigevectors are mutually orthogoal. It was oted above that, sice the eigevectors costitute a set of three mutually perpedicular uit vectors i a three-dimesioal space, they ca be used to form a basis or a coordiate referece frame. Let the three orthogoal eigevectors be base vectors e I, e II, ad e III of a Greek referece frame. From (A9) ad (A92) we form a ew referece basis for the example eigevalue problem, ad thus = ( ) e = ( e 2 e + 2 e ), ei e e2 e, II 2

25 TISSUE MECHANICS 69 eiii = (2 e 2 e2 e ). (A9) It is easy to verify that both the Greek ad Lati base vectors form right-haded orthoormal systems. Orthogoal matrix Q for trasformatio from the Lati to the Greek system is give by (A64) ad (A9) as 2 2 Q = [ Qi α ] = 2 2. (A94) 2 2 Substitutig Q of (A94) ad the A specified by (A58) ito the secod of (A8), with T = A, the followig result is determied: A ( G) ( G ) T ( L ) A = Q A Q, (A95) = =. (A96) Thus, relative to the basis formed of its eigevectors, a symmetric matrix takes o a diagoal form, the diagoal elemets beig its eigevalues. This result, which was demostrated for a particular case, is true i geeral i a space of ay dimesio as log as the matrix is symmetric. There are two poits i the above example that are always true if the matrix is symmetric. The first is that the eigevalues are always real umbers ad the secod is that the eigevectors are always mutually perpedicular. These poits will ow be proved i the order stated. To prove that λ is always real we shall assume that it could be complex; the we show that the imagiary part is zero. This proves that λ is real. If λ is complex, say λ + iμ, the associated eigevector t may also be complex ad we deote it by t = + im. With these otatios (A48) ca be writte Equatig the real ad imagiary parts, we obtai two equatios, ( A { λ+ iμ} ) ( + im ) = 0. (A97) A = λμm, Am = λm+ μ. (A98) The symmetry of matrix A meas that, for ay vectors ad m, T ma = ma = Am, (A99) a result that ca be verified i may ways. Substitutig the two equatios of (A98) ito the first ad last equalities of (A99), we fid that μ = μ, which meas that μ must be zero ad λ real. This result also shows that must be real. We will ow show that ay two eigevectors are orthogoal if the two associated eigevalues are distict. Let λ ad λ 2 be the eigevalues associated with eigevectors ad m, respectively; the

26 620 APPENDIX: MATRICES AND TENSORS A = λ ad Am = λ2m. (A00) Substitutig the two equatios of (A00) ito the first ad last equalities of (A99), we fid that ( λ λ ) m = 0. (A0) 2 Thus, if λ λ 2, the ad m are perpedicular. If the two eigevalues are ot distict, the ay vector i a plae is a eigevector, so that oe ca always costruct a mutually orthogoal set of eigevectors for a symmetric matrix. Geeralizig Example A.7.2 above from to, it may be cocluded that ay -by- matrix A of symmetric tesor compoets has a represetatio i which the eigevalues lie alog the diagoal of the matrix ad the off-diagoal elemets are all zero. The last expressio i (A96) is a particular example of this whe =. If symmetric tesor A has eigevalues λ i, the quadratic form ψ may be formed from A ad vector -tuple x, ad thus i= 2 ψ = xa x = λi xi. (A02) If all the eigevalues of A are positive, this quadratic form is said to be positive defiite ad i= i= i= 2 xa x= λi xi 0 for all x 0. (A0) (If all the eigevalues of A are egative, the quadratic form is said to be egative defiite.) Trasformig tesor A ito a arbitrary coordiate system, eq. (A02) takes the form i= xa x= Axx ij i j 0 for all x 0. (A04) i= Tesor A with property (A04), whe used as the coefficiets of a quadratic form, is said to be positive defiite. I the mechaics of materials there are a umber of tesors that are positive defiite due to the physics they represet. The momet of iertia tesor is a example. Others will be ecoutered as material coefficiets i costitutive equatios i Chapter 5. Problems A.7.. Cosider two three-dimesioal coordiate systems. Oe coordiate system is a right-haded coordiate system. The secod coordiate system is obtaied from the first by reversig the directio of the first ordered base vector ad leavig the other two base vectors to be idetical with those i the first coordiate system. Show that the orthogoal trasformatio relatig these systems is give by 0 0 Q = ad that its determiat is. A.7.2. Costruct the eigevalues ad the eigevectors of matrix T of tesor compoets where

27 TISSUE MECHANICS 62 T = A.7.. Costruct the eigevalues ad the eigevectors of matrix A of tesor compoets where 7 A =. 4 A.7.4. Show that the eigevalues of matrix H, H =, are 7.227, , 0.602, 0.62, ad.294. A.7.5. Cosider the compoets of tesor T give i Problem A.7.2 to be relative to a (Lati) coordiate system ad deote them by T (L). Trasform these compoets to a ew coordiate system (Greek) usig trasformatio 2 2 Q = A.7.6. Show that if a tesor is symmetric (skew-symmetric) i oe coordiate system, the it is symmetric (skew-symmetric) i all coordiate systems. Specifically, show that if A (L) = (A (L) ) T, the A (G) = (A (G) ) T. A.8. THE MOMENT OF INERTIA TENSOR The mass momet of iertia tesor illustrates may features of the previous sectios such as the tesor cocept ad defiitio, the ope product of vectors, the use of uit vectors, ad the sigificace of eigevalues ad eigevectors. The mass momet of iertia is the secod momet of mass with respect to a axis. The first ad zeroth momet of mass with respect to a axis is associated with the cocepts of the ceter of mass of the object ad the mass of the object, respectively. Let dv represet the differetial volume of object O. The volume of that object V is the give by O VO = dv, (A05) 0

28 622 APPENDIX: MATRICES AND TENSORS ad, if ρ(x, x 2, x, t) = ρ(x, t) is the desity of object O, the mass M O of O is give by MO Cetroid x cetroid ad ceter of mass x cm of object O are defied by = ρ( x, tdv ). (A06) 0 xcetroid = dv V x, O 0 xcm = ρ(, tdv ) M x x (A07) O 0 where x is a positio vector locatig the differetial elemet of volume or mass with respect to the origi. The power of x occurrig i the itegrad idicates the order of the momet of mass it is to the zero power i the defiitio of the mass of the object itself ad it is to the first order i the defiitio of the mass ceter. The secod momets of area ad mass with respect to the origi of coordiates are called the area ad mass momets of iertia, respectively. Let e represet the uit vector passig through the origi of coordiates; the x (x e)e is the perpedicular distace from the e axis to the differetial elemet of volume or mass at x (Figure A.2). The secod or mass momet of iertia of object O about axis e, a scalar, is deoted by I ee ad give by I ee = ( x ( x e ))( e x ( x e ))(,) e ρ x t dv. (A08) O This expressio for I ee may be chaged i algebraic form by otig first that 2 x xe e x x e e = xx xe ad ( ( ) ) ( ( ) ) ( ) 2 xx ( xe ) = e{( xx ) ( xx)} e; thus, from A(08), Iee = e {( ) ( )} ρ(, t) dv x x x x x e. (A09) 0 If the otatio for the mass momet of iertia tesor I is itroduced, the (A09) ad (A08) simplify to I= {( xx) ( xx)} ρ( x, tdv ), (A0) 0 I ee = eie. (A) I this sectio mass momet of iertia I has bee referred to as a tesor. A short calculatio will demostrate that the termiology is correct. From (A0) is easy to see that I may be writte relative to the Lati ad Greek coordiate systems as ( ) ( ) ( ) ( ) ( ) ( ) I L = ( x L x L ) ( x L x L ) ρ( x L, tdv ), (A2) ad { } 0 I x x x x x, (A) ( G ) ( ) ( ) ( ) ( ) ( ) = {( G G ) ( G G )} ρ( G, tdv ) 0

29 TISSUE MECHANICS 62 Figure A.2. A diagram for the calculatio of the mass momet of iertia of a object about the axis characterized by the uit vector e. x is the vector from the origi O of coordiates to the elemet of mass dm; x (xe)e is the perpedicular distace from the axis e to the elemet of mass dm. respectively. The trasformatio law for the ope product of x with itself ca be calculated by twice usig the trasformatio law for vectors (A77) applied to x, ad thus ( ) ( ) ( ) ( ) ( ) ( ) x L x L = Qx G Qx G = Q( x G x G ) Q T. (A4) The occurrece of the traspose i the last equality of the last equatio may be more easily perceived by recastig the expressio i idicial otatio: x x = Q x Q x = Q x x Q. (A5) ( L) ( L) ( G) ( G) ( G) ( G) i j iα α jβ β iα α β jβ Now, cotractig the ope product of vectors i (A4) above to the scalar product, it follows T T that sice QQ = Q Q= ( Qi α Qi β = δ αβ ), ( L ) ( L ) ( G ) ( G ) x x = x x. (A6) Combiig results (A4) ad (A6), it follows that the o-scalar portios of the itegrads i (A2) ad (A) are related by ( L) ( L) ( L) ( L) ( G) ( G) ( G) ( G) T {( ) ( )} = {( ) ( )} x x x x Q x x x x Q. Thus, from this result ad (A2) ad (A), the trasformatio law for secod-order tesors is obtaied: ( L ) ( G ) I = QI Q T, (A7) ad it follows that tesor termiology is correct i describig the mass momet of iertia. The matrix of tesor compoets of momet of iertia tesor I i a three-dimesioal space is give by I I2 I I = I 2 I22 I 2, (A8) I I2 I

30 624 APPENDIX: MATRICES AND TENSORS where the compoets are give by I = ( x + x ) ρ( x, t) dv, I = ( x + x ) ρ( x, t) dv, I = ( x + x ) ρ( x, t) dv, I2 ( x x2 ) ρ(, t) dv I ( x x ) ρ(, t) dv 0 = x (A9) = x, I2 = ( x2x) ρ( x, t) dv. 0 0 Example A.8. Determie the mass momet of iertia of a rectagular prism of homogeeous material of desity ρ ad side legths a, b, ad c about oe corer. Select the coordiate system so that its origi is at oe corer ad let a, b, ad c represet the distaces alog the x, x 2, x axes, respectively. Costruct the matrix of tesor compoets referred to this coordiate system. Solutio: Itegratios (A9) yield the followig results: I = ( x + x ) ρ( x, t) dv = ρ ( x + x ) dx dx dx bc, 2 2 ρabc 2 2 ( 2 ) 2 ( ), 0,0 = aρ x + x dx dx = b + c ρabc 2 2 I22 = ( a + c ), ρabc 2 2 I = ( a + b ), ρabc I = ( x x ) ρ( x, t) dv= ρc ( x x ) dx dx = ( ab) 4 ab ,0 ρabc ρabc I = ( ac), I2 = ( bc), 4 4 ad thus 2 2 4( b c ) ab ac ρabc I = ab 4( a c ) bc 2 + ac bc a + b 2 2 4( ). Example A.8.2 I the special case whe the rectagular prism i Example A.8. is a cube, that is to say, a = b = c, fid the eigevalues ad eigevectors of the matrix of tesor compoets referred to the coordiate system of the example. The fid the matrix of tesor compoets referred to the pricipal, or eigevector, coordiate system.

31 TISSUE MECHANICS 625 Solutio: The matrix of tesor compoets referred to this coordiate system is 8 5 ρa I = The eigevalues of I are ρa 5 /6, ρa 5 /2, ad ρa 5 /2. Eigevector (/ )[,, ] is associated with eigevalue ρa 5 /6. Due the multiplicity of the eigevalue ρa 5 /2, ay vector perpedicular to the first eigevector, (/ )[,, ], is a eigevector associated with the multiple eigevalue ρa 5 /2. Thus, ay mutually perpedicular uit vectors i the plae perpedicular to the first eigevector may be selected as the base vectors for the pricipal coordiate system. The choice is arbitrary. I this example the two perpedicular uit vectors, (/ 2)[, 0, ] ad (/ 6)[, 2, ], are the eigevectors associated with multiple eigevalue ρa 5 /2, but ay perpedicular pair of vectors i the plae may be selected. The orthogoal trasformatio that will trasform the matrix of tesor compoets referred to this coordiate system to the matrix of tesor compoets referred to the pricipal, or eigevector, coordiate system is the give by Q. = Applyig this trasformatio produced the matrix of tesor compoets referred to the pricipal, or eigevector, coordiate system: ρ T a QIQ = Formulas for the mass momet of iertia of a thi plate of thickess t ad a homogeeous material of desity ρ are obtaied by specializig these results. Let the plate be thi i the x directio ad cosider the plate to be so thi that terms of order t 2 are egligible relative to the others; the formulas (A9) for the compoets of the mass momet of iertia tesor are give by I = ρt x dx dx, I = ρt x dx dx, I = ρt ( x + x ) dx dx. (A20) I = ρt ( x x ) dx dx, I = 0, I 2 = Whe divided by ρt, these compoets of the mass momet of iertia of a thi plate of thickess t are called the compoets of the area momet of iertia matrix:

32 626 APPENDIX: MATRICES AND TENSORS I I = x dx dx ρt =, Area O I I = x dx dx ρt =, Area O Area I 2 2 I = = ( x + x2 ) dxdx2 ρt, O Area I2 I2 = ( xx2 ) dxdx2 ρt =, O Area I = 0, Area I 2 = 0. (A2) Example A.8. Determie the area momet of iertia of a thi rectagular plate of thickess t, height h, ad width of base b. Specify precisely where the origi of the coordiate system that you are usig is located ad how the base vectors of that coordiate system are located relative to the sides of the rectagular plate. Solutio: The coordiate system that makes this problem easy is oe that passes through the cetroid of the rectagle ad has axes parallel to the sides of the rectagle. If base b is parallel to the x axis ad height h is parallel to the x 2 axis, the itegratios (A2) yield the followig results: Area bh I =, 2 Area hb I 22 =, 2 Area bh 2 2 I = ( b + h ), 2 Area I 2 = 0, Area I = 0, Area I 2 = 0. Example A.8.4 Determie the area momets ad product of iertia of a thi right-triagular plate of thickess t, height h, ad width of base b. Let base b be alog the x axis ad height h be alog the x 2, axis ad the slopig face of the triagle have edpoits at (b, 0) ad (0, h). Determie the area momets ad product of iertia of the right-triagular plate relative to this coordiate system. Costruct the matrix of tesor compoets referred to this coordiate system. Solutio: Itegratios (A2) yield the followig results: h Area 2 x 2 2 = 2 2 ( ) bh = h 2 2 =, 2 O 0 I x dx dx b x dx Area 22 2 I = hb h Area x2 2 = ( 2 ) 2 ( ) ( ) 2 2 ( bh = ) 2 = ; h 24 O 0 I x x dx dx b x dx thus, the matrix of tesor compoets referred to this coordiate system is bh 2h bh = 24 hb 2b 2 Area I. 2 Example A.8.5 I the special case whe the triagle i Example A.8.4 is isosceles, that is to say, b = h, fid the eigevalues ad eigevectors of the matrix of tesor compoets referred to the coordiate

33 TISSUE MECHANICS 627 system of the example. The fid the matrix of tesor compoets referred to the pricipal, or eigevector, coordiate system. Solutio: The matrix of tesor compoets referred to this coordiate system is bh 2h bh = 24 hb 2b 2 Area I. 2 The eigevalues of I are h 4 /8 ad h 4 /24. Eigevector (/ 2)[,,},{,} ] is associated with eigevalue h 4 /8 ad eigevector (/ 2)[, ] is associated with eigevalue h 4 /24. The orthogoal trasformatio that will trasform the matrix of tesor compoets referred to this coordiate system to the matrix of tesor compoets referred to the pricipal, or eigevector, coordiate system is the give by Q =. 2 Applyig this trasformatio produced the matrix of tesor compoets referred to the pricipal, or eigevector, coordiate system: 4 Area 0 QI Q h T = The parallel axis theorem for momet of iertia matrix I is derived by cosiderig the mass momet of iertia of object O about two parallel axes, I ee about e ad I e'e' about e. Ie'e' is give by I = e' I' e ', (A22) where momet of iertia matrix I is give by 0 e'e' I' = {( x' x') ( x' x')} ρ( x ', tdv ) '. (A2) Let d be a vector perpedicular to both e ad e ad equal i magitude to the perpedicular distace betwee e ad e, thus x = x + d, e d = 0, ad e d = 0. Substitutig x = x + d i I, it follows that 0 I' = {( x x+ d d+ 2 xd) } ρ( x, tdv ) 0 {( x x + d d + d x + x d )} ρ( x, tdv ), (A24) or if (A24) is rewritte so that costat vector d is outside the itegral sigs, I' = {( x x)(,) ρ x tdv+ ( d d) ρ(,) x tdv+ (2 d x)(,) ρ x tdv {( xx) ρ( x, tdv ) ( dd) ρ( x, tdv ) 0 0 d ρ(, tdv ) ρ(, tdv ) x x x x d ; 0 0

34 628 APPENDIX: MATRICES AND TENSORS the recallig defiitios (A06) of mass M O of O ad (A07) of ceter of mass x cm of object O, this result simplifies to I' = I+ {( dd) ( d d)} M O + 2 M x ( d) M ( d x + x d ). (A25) O cm O cm cm Thus, whe the origi of coordiates is take at the ceter of the mass, it follows that x = cm 0 ad I' = I + {( dd) ( dd )} M. (A26) cm I the special case of the area momet of iertia this formula becomes I' = I + {( dd) ( dd )}A, (A27) cetroid where I are ow the area momets of iertia ad the mass of the object, M O, has bee replaced by the area of thi plate A. O Example A.8.6 Cosider agai the rectagular prism of Example A.8.. Determie the mass momet of iertia tesor of that prism about its cetroid or ceter of mass. Solutio: The desired result, the mass momet of iertia about the cetroidal axes, is I cm i (A26), ad the momet of iertia about the corer, I, is the result calculated i Example A.8.: ρabc 2 Formula (A26) is the writte i the form b + c ab ac 2 2 4( ) 2 2 I ' = ab 4( a + c ) bc. ac bc a + b 2 2 4( ) I = I' {( dd) ( dd )} M, cm where M o = ρabs. Vector d is a vector from the cetroid to the corer: O d= ( 2 ) 2 a e + b e + c e. Substitutig I ad the formula for d ito the equatio for I above, it follows that the mass momet of iertia of the rectagular prism relative to its cetroid is give by ρabc ( b c ) I = 0 ( a + c ) 0. cm ( a b ) Example A.8.7 Cosider agai the thi right-triagular plate of Example A.8.4. Determie the area momet of iertia tesor of that right-triagular plate about its cetroid.

35 TISSUE MECHANICS 629 Solutio: The desired result, the area momet of iertia about the cetroidal axes is the Area I cetroid i (A27) ad momet of iertia I ' Area about the corer is the result calculated i Example A.8.4: Formula (A26) is the writte i the form 2 ' bh 2h bh I Area = hb 2b Area ' I = I dd dd, cetroid Area {( ) ( )}A where A = bh. Vector d is a vector from the cetroid to the corer: d= ( )( 2) be + he. Substitutig I ad the formula for d ito the equatio for I cetroid above, it follows that the mass momet of iertia of the rectagular prism relative to its cetroid is give by 2 Area bh 2h bh I cetroid == hb 2b Problems A.8.. Fid the ceter of mass of a set of four masses. The form masses ad their locatios are mass (2 kg) at (, ), mass 2 (4 kg) at (4, 4), mass (5 kg) at (-4, 4), ad mass 4 ( kg) at (, ). A.8.2. Uder what coditios does the ceter of mass of a object coicide with the cetroid? A.8.. Fid the cetroid of a cylider of legth L with a semicircular cross-sectio of radius R. A.8.4. Fid the ceter of mass of a cylider of legth L with a semicircular cross-sectio of radius R (R < 2L) if the desity varies accordig to rule ρ = ρ o ( + c(x 2 ) 2 ). The coordiate system for the cylider has bee selected so that x is alog its legth L, x 2 is across its smallest dimesio (0 x 2 R), ad x is alog its itermediate dimesio ( R x R). A.8.5. Show that momet of iertia matrix I is symmetric. A.8.6. Develop the formulas for the mass momet of iertia of a thi plate of thickess t ad a homogeeous material of desity ρ. Illustrate these specialized formulas by determiig the mass momet of iertia of a thi rectagular plate of thickess t, height h, ad width of base b, ad a homogeeous material of desity ρ. Specify precisely where the origi of the coordiate system that you are usig is located ad how the base vectors of that coordiate system are located relative to the sides of the rectagular plate. A.8.7. I Example A.8.2 the occurrece of a multiple eigevalue (ρa 5 /2) made ay vector perpedicular to the first eigevector, e = (/ )[,, ], a eigevector associated with multiple eigevalue ρa 5 /2. I Example A.8.2 the two perpedicular uit vectors, e 2 = (/ 2) [, 0, ] ad e = (/ 6)[, 2, ], were selected as the eigevectors associated with multiple eigevalue ρa 5 /2, but ay two perpedicular vectors i the plae could have bee selected. Select two other eigevectors i the plae ad show that these two eigevectors are give by e II = cos γ e 2 + si γ e ad e III = si γ e 2 + cos γ e. Let R be the orthogoal trasformatio betwee these Lati ad Greek systems:

36 60 APPENDIX: MATRICES AND TENSORS 0 0 R = 0 cosγ siγ. 0 siγ cosγ Show that whe the Greek coordiate system is used rather tha the Lati oe, the coordiate trasformatio that diagoalizes matrix I is Q R rather tha Q. Show that both R Q ad Q trasform matrix I ito the coordiate system i which it is diagoal: ρ T T T a QIQ = RQIQ R = A.9. THE ALTERNATOR AND VECTOR CROSS-PRODUCTS There is a strog emphasis o the idicial otatio i this sectio. It is advised that the defiitios (i A.) of free idices ad summatio idices be reviewed carefully if oe is ot altogether comfortable with idicial otatio. It would also be beeficial to redo some idicial otatio problems. The alterator i three dimesios is a three-idex umerical symbol that ecodes the permutatios that oe is taught to use expadig a determiat. Recall the process of evaluatig the determiat of -by- matrix A: A A2 A A A2 A DetA = Det A2 A22 A2 = A2 A22 A2 = (A28) A A2 A A A2 A AA22 A A A2 A2 A2 A2A + A2 AA2 + AA2A2 A A A22. The permutatios that oe is taught to use expadig a determiat are permutatios of a set of three objects. The alterator is deoted by e ijk ad defied so that it takes o values +, 0, or accordig to rule e ijk + if P is a eve permutatio 0 otherwise, if P is a odd permutatio 2 P, (A29) i j k where P is the permutatio symbol o a set of three objects. The oly + values of e ijk are e 2, e 2, ad e 2. It is easy to verify that 2, 2, ad 2 are eve permutatios of 2. The oly values of e ijk are e 2, e 2, ad e 2. It is easy to verify that 2, 2, ad 2 are odd permutatios of 2. The other 2 compoets of e ijk are all zero because they are either eve or odd permutatios of 2 due to the fact that oe umber (either, 2, or ) occurs more tha oce i the idices (for example, e 22 = 0 sice 22 is ot a permutatio of 2). Oe memoic device for the eve permutatios of 2 is to write 22, the read the first set of three digits, 2, the secod set, 2, ad the third set, 2. The odd permutatios may be read off 22 also by readig from right to left rather tha from left to right; readig from the right (but recordig them the from the left, as usual) the first set of three digits, 2, the secod set, 2, ad the third set, 2.

37 TISSUE MECHANICS 6 The alterator may ow be employed to shorte formula (A28) for calculatig the determiat: k= j= i= k= j= i= ijk im j kp = ijk mi j pk k= j= i= k= j= i= empdeta e A A A e A A A. (A0) = This result may be verified by selectig the values of mp to be 2, 2, 2, 2, 2, or 2, the performig summatio over three idices i, j, ad k over, 2, ad, as idicated o the right-had side of (A0). I each case the result is the right-had side of (A28). It should be oted that (A0) may be used to show DetA = DetA T. The alterator may be used to express the fact that iterchagig two rows or two colums of a determiat chages the sig of the determiat: Am A Ap Am Am2 Am empdet = A2m A2 A2p = A A2 A Am A Ap Ap Ap2 Ap A. (A) Usig the alterator agai may combie these two represetatios: Aim Ai Aip eijkempdeta = Ajm Aj Ajp. (A2) Akm Ak Akp I the special case whe A = (A ij = δ ij ), a importat idetity relatig the alterator to the Kroecker delta is obtaied: eijkemp δim δi δip δjm δj δjp δkm δk δkp =. (A) The followig special cases of (A) provide three more very useful relatios betwee the alterator ad the Kroecker delta: k= k= e e = δ δ δ δ ijk mk im j i jm, k= j= e e = 2δ ijk mjk im, k= j= k= j= i= e e = 6. (A4) ijk ijk k= j= i= The first of these relatios is obtaied by settig idices p ad k equal i (A) ad the expadig the determiat. The secod is obtaied from the first by settig idices ad j equal i the first. The third is obtaied from the secod by settig idices i ad m equal i the secod. Example A.9. Derive the first of (A4) from (A). Solutio: The first of (A4) is obtaied from (A) by settig idices p ad k equal i (A) ad the expadig the determiat: oe fids that δ δ δ im i ik eijke mk = δjm δj δjk ; k= δkm δk

38 62 APPENDIX: MATRICES AND TENSORS k= eijkemk= δimδjδimδjkδk δiδjm+ δiδkmδjk + δikδjmδkδikδkmδj ). ( k= Carryig out the idicated summatio over idex k i the expressio above, k= eijkemk= δ δ δ δ δ δ + δ δ + δ δ δ δ = im j im j i jm i jm jm i im j δ δ This is the desired result, the first of (A4). δ δ. im j i jm Example A.9.2 Prove that Det(A B) = DetADetB. Solutio: Replacig A i (A0) by C ad selectig the values of mp to be 2, the (A0) becomes Now C is replaced by product A B (with ad thus or C m= i A im B m m= k= j= i= DetC e C C C. = k= j= i= = ijk i j2 k =, C j 2 = A j B 2, ad C k = A kp B p, = p= p= k= j= i= m= = p= eijk Aim Bm AjB 2 AkpBp k= j= i= m= = p= DetA B =, m= = p= k= j= i= DetA B = eijk Aim Aj A kp BmB2B p, k= j= i= m= = p= where the order of the terms i the secod sum has bee rearraged from the first. Compariso of the first four rearraged terms from the secod sum with the right-had side of (A0) shows that the first four terms i the sum o the right may be replaced by e mp DetA; thus, applyig the first equatio of this solutio agai with C replaced by B, the desired result is obtaied: k= j= i= DetA B= DetA e B B B = Det A DetB. k= j= i= mp m 2 p

39 TISSUE MECHANICS 6 Cosider ow the questio of the tesorial character of the alterator. Vectors were show to be characterized by symbols with oe subscript, (secod rak) tesors were show to be characterized by symbols with two subscripts. What is the tesorial character of a symbol with three subscripts; is it a third-order tesor? Almost. Tesors are idetified o the basis of their tesor trasformatio law. Recall tesor trasformatio laws (A75) ad (A76) for a vector, (A86) for a secod-order tesor, ad (A87) for a tesor of order. A equatio that cotais a trasformatio law for the alterator is obtaied from (A0) by replacig A by the orthogoal trasformatio Q give by (A64) ad chagig the idices as follows: m α, β, p γ, ad thus k= j= i= e αβγ DetQ eijkqiαqjβqk γ. (A5) = k= j= i= This is a uusual trasformatio law because the determiat of orthogoal trasformatio Q is either + or. The expected trasformatio law, o the basis of tesor trasformatio laws (A75) ad (A76) for a vector, (A86) for a secod-order tesor, ad (A87) for a tesor of order, is that DetQ = +. DetQ = + occurs whe the trasformatio is betwee coordiate systems of the same hadedess (right- to right-haded or left- to left-haded). Recall that a right(left)- had coordiate system or orthoormal basis is oe that obeys the right(left)-had rule, that is to say, if the curl of your figers i your right (left) had fist is i the directio of rotatio from the first ordered positive base vector ito the secod ordered positive base vector, your exteded thumb will poit i the third ordered positive base vector directio. DetQ = occurs whe the trasformatio is betwee coordiate systems of the opposite hadedess (left to right or right to left). Sice hadedess does ot play a role i the trasformatio law for eve order tesors, this depedece o the sig of DetQ ad therefore the relative hadedess of the coordiate systems for the alterator trasformatio law is uexpected. The title to this sectio metioed both the alterator ad the vector cross-product. How are they coected? If you recall the defiitio of vector cross-product a x b i terms of a determiat, the coectio betwee the two is made: e e2 e = a a2 a = b b2 b a b (A6) ( ab 2 ba 2 ) e+ ( ab ba ) e2 + ( ab 2 ba 2) e. I idicial otatio, vector cross-product a x b is writte i terms of a alterator as k= j= i= a b e ab e, (A7) = k= j= i= ijk i j k a result that may be verified by expadig it to show that it coicides with (A6). If c = a x b deotes the result of the vector cross-product, the from (A7), ck j= i= eijkab i j j= i= =. (A8) Is vector cross-product c = a x b a vector or a tesor? It is called a vector, but a secodorder tesorial character is suggested by the fact that the compoets of a x b coicide with the compoets of the skew-symmetric part of 2(a b) (see eq. A2). The aswer is that vector

40 64 APPENDIX: MATRICES AND TENSORS cross-product c = a x b is uusual. Although it is, basically, a secod-order tesor, it ca be treated as a vector as log as the trasformatios are betwee coordiate systems of the same hadedess. I that case eq. (A6) shows that the alterator trasforms as a proper tesor of order three, ad thus there is o ambiguity i represetatio (A7) for a x b. Whe studets first lear about the vector cross-product they are admoished (geerally without explaatio) to always use right-haded coordiate systems. This hadedess problem is the reaso for that admoishmet. The vector that is the result of the cross-product of two vectors has ames like axial vector or pseudo vector to idicate its special character. Typical axial vectors are momets i mechaics ad the vector curl operator ( A.). Example A.9. Prove that a x b = b x a. Solutio: I formula (A7), let i j ad j i; thus, k= j= i= a b e a be. = k= j= i= jik j i k Next, chage e jik to e ijk ad rearrage the order of a j ad b i, the the result is proved: k= j= i= a b= e ba j e = b a. k= j= i= ijk i k The scalar triple product of three vectors is a scalar formed from three vectors, a ( b c ), ad the triple-vector-product is a vector formed from three vectors, (r x (p x q)). A expressio for the scalar-triple-product is obtaied by takig the dot product of vector c with the cross-product i represetatio (A7) for a x b, ad thus k= j= i= c ( a b ) =eijkab i jc k. (A9) k= j= i= From the properties of the alterator it follows that c ( a b) = a ( b c) = b ( c a ) = (A40) a ( c b) =b ( a c) =c ( b a ). If three vectors a, b, ad c coicide with the three o-parallel edges of a parallelepiped, scalar triple product a ( b c ) is equal to the volume of the parallelepiped. I the followig example a useful vector idetity for triple vector product (r x (p x q)) is derived. Example A.9.4 Prove that (r x (p x q)) = (r q)p (r p)q. Solutio: First rewrite (A7) with chage a r, ad agai with chages a p ad b q, where b = (p x q): k= j= i= r b e rb e, = k= j= i= ijk i j k m= = j= b p q e p q e. = = m= = j= mj m j

41 TISSUE MECHANICS 65 Note that the secod of these formulas gives the compoets of b as j m= = mj m m= = b e p q =. This formula for the compoets of b is the substituted ito the expressio for (r x b) = (r x (p x q) above, ad thus k= j= i= m= = r ( p q) = ijk mj i m e k. k= j= i= m= = e e rp q O the right-had side of this expressio for r ( p q ), e ijk is ow chaged to e ikj ad the first of (A4) is the employed: the summig over k ad i, k= i i= m = r ( p q) = ( δ δ δ δ ) rp q e ; im k i km i m k k= m= = m= = rmpmq rpmq m m= = r ( p q) = e e = ( rq) p( rp) q. I the process of calculatig area chages o surfaces of objects udergoig large deformatios, like rubber or soft tissue, certai idetities that ivolve both vectors ad matrices are useful. Two of these idetities are derived i the followig two examples. Example A.9.5 Prove that A a (A b A c) = a (b c)deta, where A is a -by- matrix, ad a, b, ad c are vectors. Solutio: Notig the formula for the scalar triple product as a determiat, a (b c) = a a a 2 b b b 2 c c c 2 ad the represetatio for multiplicatio of A times a, A A A a A a + A a + A a Aa = A A A a = A a + A a + A a, A A2 A a Aa A2a2 Aa + + the A a + A a + A a A a + A a + A a A a + A a + A a Aa (A b A c) = Det A b + A b + A b A b + A b + A b A b + A b + A b A c + A c + A c A c + A c + A c A c + A c + A c

42 66 APPENDIX: MATRICES AND TENSORS Recallig from Example A.9.2 that Det(A B) = DetA DetB, it follows that the previous determiat may be writte as a product of determiats: A A A Det A A A A A A a b c Det a2 b2 c 2 a b c = a (b c) DetA, which is the desired result. I the last step the fact that the determiat of the traspose of a matrix is equal to the determiat of the matrix, DetA = DetA T, was employed. Example A.9.6 Prove vector idetity (A b A c) A = (b c) DetA, where A is a -by- matrix, ad b ad c are vectors. Solutio: Recall the result of Example A.9.5, amely, that A a (A b A c) = a (b c) DetA, ad let a = e, the e 2 ad the e, to obtai the followig three scalar equatios: Aw + A2w2 + Aw = q DetA A2w + A22w2 + A2w = q 2 DetA Aw + A2w2 + Aw = q DetA where w = (A b A c), q = (b c). These three equatios may be recast i matrix otatio: A A2 A w A2 A22 A 2 w 2 = A A A w 2 q q 2 q DetA, or A T w = q DetA, ad sice w = (A b A c), q = (b c), A T (A b A c) = (b c) DetA, or (A b A c) A = (b c) DetA, which is the desired result. I the last step, relatio a F T = F a (Problem 5.8) was employed. Problems A.9.. Fid cross-products a x b ad b x a of two vectors a = [, 2, ] ad b = [4, 5, 6]. What is the relatioship betwee a x b ad b x a? A.9.2. Show that if A is a skew-symmetric -by- matrix, A = A T, the DetA = 0. A.9.. Evaluate Det(a b). A.9.4. Show DetA = DetA T. A.9.5. Show DetQ = ± if Q T Q = Q Q T =. A.9.6. Fid the volume of the parallelepiped if the three o-parallel edges of a parallelepiped coicide with three vectors a, b, ad c, where a = [, 2, ] meters, b = [, 4, 6] meters, ad c = [,, ] meters. A.9.7. If v = a x x ad a is a costat vector, usig idicial otatio, evaluate div v ad curl v.

43 TISSUE MECHANICS 67 A.0. CONNECTION TO MOHR S CIRCLES The material i the sectio before last, amely, trasformatio law (A8) for tesorial compoets ad the eigevalue problem for liear trasformatios, is preseted i stadard textbooks o the mechaics of materials i a more elemetary fashio. I those presetatios the secodorder tesor is take to be the stress tesor ad a geometric aalog calculator is used for trasformatio law (A8) for tesorial compoets i two dimesios, ad for the solutio of the eigevalue problem i two dimesios. The geometric aalog calculator is called a Mohr circle. A discussio of the coectio is icluded to aid i placig the material just preseted i perspective. ( ) ( ) The special case of the first trasformatio law from (A8), T L = QT G Q T, is rewritte i two dimesios ( = 2) i form σ' = Q σ Q T ; thus, T (L) = σ' ad T (G) = σ, where matrix of stress tesor compoets σ, matrix of trasformed stress tesor compoets σ', ad orthogoal trasformatio Q represetig rotatio of the Cartesia axes are give by σx τ xy σ = τxy σ, y σx ' τ xy ' ' σ ' = τxy ' ' σ, y' cosθ si θ Q =. (A4) siθ cosθ Expasio of matrix equatio σ' = Q σ Q T, σx' τ x' y' cosθ siθσx τ xy cosθ siθ σ ' = = τxy ' ' σ y' si θ cosθ τxy σ, (A42) y si θ cosθ ad subsequet use of double-agle trigoometric formulas si 2θ = 2si θ cos θ ad cos 2θ = cos 2 θ si 2 θ yield the followig: σx = (/2)(σx + σy) + (/2)(σx σy) cos 2θ + τxy si 2θ σy = (/2)(σx + σy) (/2)(σx σy) cos 2θ τxy si 2θ, (A4) τx y = (/2)(σx σy) si 2θ + τxy cos 2θ. These are formulas for stresses σx, σy, ad τx y, as fuctios of stresses σx, σy, ad τxy ad agle 2θ. Note that the sum of the first two equatios i (A98) yields the followig expressio, which is defied as 2C: 2C σx + σy = σx + σy. (A44) The fact that σx + σy = σx + σy is a repetitio of result (A90) cocerig the ivariace of the trace of a tesor, the first ivariat of a tesor, uder chage of basis. Next cosider the followig set of equatios i which the first is the first of (A4), icorporatig defiitio (A44) ad trasposig the term ivolvig C to the other side of the equal sig, ad the secod equatio is the third of (A4): σx C = (/2)(σx σy) cos 2θ + τxy si 2θ, τx y = (/2)(σx σy) si 2θ + τxy cos 2θ. If these equatios are ow squared ad added we fid that (σx C ) 2 + (τx y ) 2 = R 2 (A45)

44 68 APPENDIX: MATRICES AND TENSORS where R 2 (/4)(σx σy) 2 + (τxy) 2. (A46) Equatio (A45) is the equatio for a circle of radius R cetered at poit σx = C, τx y = 0. The circle is illustrated i Figure A.. Figure A.. A illustratio of Mohr's circle for a state of stress. The poits o the circle represet all possible values of σx, σy, ad τx y ; they are determied by the values of C ad R, which are, i tur, determied by σx, σy, ad τxy. The eigevalues of matrix σ are the values of ormal stress σx whe the circle crosses the σx axis. These are give by umbers C + R ad C R, as may be see from Figure A.. Thus, Mohr s circle is a graphical aalog calculator for the eigevalues of two-dimesioal secod-order tesor σ, as well as a graphical aalog calculator for equatio σ' = Q σ Q T represetig trasformatio of compoets. The maximum shear stress is simply the radius of circle R, a importat graphical result that is readable from Figure A.. As a graphical calculatio device, Mohr s circles may be exteded to three dimesios, but the graphical calculatio is much more difficult tha doig the calculatio o a computer, so it is o loger doe. A illustratio of three-dimesioal Mohr s circles is show i Figure A.4. The shaded regio represets the set of poits that are possible stress values. The three poits where the circles itersect the axis correspod to the three eigevalues of the threedimesioal stress tesor ad the radius of the largest circle is the magitude of the largest shear stress.

45 TISSUE MECHANICS 69 Figure A.4. Mohr's circles i three dimesios. Problems A.0.. Costruct the two-dimesioal Mohr s circle for matrix A give i Problem A.7.. A.0.2. Costruct the three-dimesioal Mohr s circles for matrix T give i Problem A.7.2. A.. SPECIAL VECTORS AND TENSORS IN SIX DIMENSIONS The fact that the compoets of a secod-order tesor i dimesios ca be represeted as a -by- square matrix allows the powerful algebra of matrices to be used i the aalysis of secod-order tesor compoets. I geeral this use of the powerful algebra of matrices is ot possible for tesors of other orders. For example, i the case of the third-order tesor with compoets A ijk oe could imagie a geeralizatio of a matrix from a array with rows ad colums to oe with rows, colums, ad a depth dimesio to hadle the iformatio of the third idex. This would be like a -by--by- cube sub-partitioed ito cells that would each cotai a etry like the etry at a row/colum positio i a matrix. Moder symbolic algebra programs might be exteded to hadle these -by--by- cubes ad to represet them graphically. By extesio of this idea, fourth-order tesors would require a -by--by--by- hypercube with o possibility of graphical represetatio. Fortuately, for certai fourth-order tesors (a case of special iterest i cotiuum mechaics) there is a way to agai employ the