Brief Review of Tensors

Appendix A Brief Review of Tensors A1 Introductory Remarks In the study of particle mechanics and the mechanics of solid rigid bodies vector notation provides a convenient means for describing many physical quantities and laws In studying the mechanics of solid deformable media, physical quantities of a more complex nature, such as stress and strain, assume importance 1 Mathematically such physical quantities are represented by matrices In the analysis of general problems in continuum mechanics, the physical quantities encountered can be somewhat more complex than vectors and matrices Like vectors and matrices these physical quantities are independent of any particular coordinate system that may be used to describe them At the same time, these physical quantities are very often specified most conveniently by referring to an appropriate system of coordinates Tensors, which are a generalization of vectors and matrices, offer a suitable way of mathematically representing these quantities As an abstract mathematical entity, tensors have an existence independent of any coordinate system or frame of reference, yet are most conveniently described by specifying their components in an appropriate system of coordinates Specifying the components of a tensor in one coordinate system determines the components in any other system Indeed, the law of transformation of tensor components is often used as a means for defining the tensor The objective of this appendix is to present a brief overview of tensors Further details pertaining to this subject can be found in standard books on the subject such as [2, 4], or in books dealing with Continuum Mechanics such as [1, 3] 1 We recall that in describing stresses and strains one must specify not only the magnitude of the quantity, but also the orientation of the face upon which this quantity acts 1

2 A Brief Review of Tensors A2 General Characteristics The following general characteristics apply to all tensors: Tensor Rank Tensors may be classified by rank or order according to the particular form of transformation law they obey This classification is also reflected in the number of components a given tensor possesses in an N-dimensional space Thus, a tensor of order p has N p components For example, in a threedimensional Euclidean space, the number of components of a tensor is 3 p It follows therefore, that in three-dimensional space: A tensor of order zero has one component and is called a scalar Physical quantities possessing magnitude only are represented by scalars A tensor of order one has three components and is called a vector; quantities possessing both magnitude and direction are represented by vectors Geometrically, vectors are represented by directed line segments that obey the Parallelogram Law of addition A tensor of order two has nine components and is typically represented by a matrix Notation The following symbols are used herein: Scalars are represented by lowercase Greek letters For example, α Vectors are represented by lowercase Latin letters For example, a or { a } Matrices and tensors are represented by uppercase Latin letters For example, A or { A } Cartesian Tensors When only transformations from one homogeneous coordinate system (eg, a Cartesian coordinate system) to another are considered, the tensors involved are referred to as Cartesian tensors The Cartesian coordinate system can be rectangular (x 1, x 2, x 3 ) or curvilinear, such as cylindrical (R, θ, z) or spherical (r, θ, φ)

A3 Indicial Notation 3 A3 Indicial Notation A tensor of any order, its components, or both may be represented clearly and concisely by the use of indicial notation This convention was believed to have been introduced by Einstein In this notation, letter indices, either subscripts or superscripts, are appended to the generic or kernel letter representing the tensor quantity of interest; eg A ij, B ijk, δ ij, a kl, etc Some benefits of using indicial notation include: (1) economy in writing; and, (2) compatibility with computer languages (eg easy correlation with do loops ) Some rules for using indicial notation follow Index rule In a given term, a letter index may occur no more than twice Range Convention When an index occurs unrepeated in a term, that index is understood to take on the values 1, 2,, N where N is a specified integer that, depending on the space considered, determines the range of the index Summation Convention When an index appears twice in a term, that index is understood to take on all the values of its range, and the resulting terms are summed For example, A kk = a 11 + a 22 + + a NN Free Indices By virtue of the range convention, unrepeated indices are free to take the values over the range, that is, 1, 2,, N These indices are thus termed free The following items apply to free indices: Any equation must have the same free indices in each term The tensorial rank of a given term is equal to the number of free indices N (no of free indices) = number of components represented by the symbol Dummy Indices In the summation convention, repeated indices are often referred to as dummy indices, since their replacement by any other letter not appearing as a free index does not change the meaning of the term in which they occur In the following equations, the repeated indices are thus dummy indices: A kk = A mm and a ik b kl = a in b nl In the equation E ij = e im e mj i and j represent free indices and m is a dummy index Assuming N = 3 and using the range convention, it follows that E ij = e i1 e 1j +e i2 e 2j +e i3 e 3j Care must be taken to avoid breaking grammatical rules in the indicial language For example, the expression a b = (a k ê k ) (b k ê k ) is erroneous since the summation on the dummy indices is ambiguous To avoid such ambiguity, a dummy index can only be paired with one other dummy index in an expression A good rule to follow is use separate dummy indices for each implied summation in an expression

4 A Brief Review of Tensors Contraction of Indices Contraction refers to the process of summing over a pair of repeated indices This reduces the order of a tensor by two For example: Contracting the indices of A ij (a second-order tensor) leads to A kk (a zeroth-order tensor or scalar) Contracting the indices of B ijk (a third-order tensor) leads to B ikk (a first-order tensor) Contracting the indices of C ijkl (a fourth-order tensor) leads to C ijmm (a second-order tensor) Comma Subscript Convention A subscript comma followed by a subscript index i indicates partial differentiation with respect to each coordinate x i Thus, φ,m φ x m ; a i,j a i x j ; C ij,kl C ij x k x l ; etc (A1) If i remains a free index, differentiation of a tensor with respect to i produces a tensor of order one higher For example A j,i = A j (A2) If i is a dummy index, differentiation of a tensor with respect to i produces a tensor of order one lower For example V m,m = V m x m = V 1 x 1 + V 2 x 2 + + V N x N (A3)

A4 Coordinate Systems 5 A4 Coordinate Systems The definition of geometric shapes of bodies is facilitated by the use of a coordinate system With respect to a particular coordinate system, a vector may be defined by specifying the scalar components of that vector in that system A rectangular Cartesian coordinate (RCC) system is represented by three mutually perpendicular axes in the manner shown in Figure A1 x 3 e ^ 3 e ^ 1 x 1 ^ e 2 x 2 Figure A1: Rectangular Cartesian Coordinate System Any vector in the RCC system may be expressed as a linear combination of three arbitrary, nonzero, non-coplanar vectors called the base vectors Base vectors are, by hypothesis, linearly independent A set of base vectors in a given coordinate system is said to constitute a basis for that system The most frequent choice of base vectors for the RCC system is the set of unit vectors ê 1, ê 2, ê 3, directed parallel to the x 1, x 2 and x 3 coordinate axes, respectively Remark 1 The summation convention is very often employed in connection with the representation of vectors and tensors by indexed base vectors written in symbolic notation In Euclidean space any vector is completely specified by its three components The range on indices is thus 3 (ie, N = 3) A point with coordinates (q 1, q 2, q 3 ) is thus located by a position vector x, where In abbreviated form this is written as x = q 1 ê 1 + q 2 ê 2 + q 3 ê 3 (A4) x = q i ê i (A5) where i is a summed index (ie, the summation convention applies even though there is no repeated index on the same kernal) The base vectors constitute a right-handed unit vector triad or right orthogonal triad that satisfy the following relations: and ê i ê j = δ ij (A6)

6 A Brief Review of Tensors ê i ê j = ɛ ijk ê k (A7) The set of base vectors satisfying the above conditions are often called an orthonormal basis In equation (A6), δ ij denotes the Kronecker delta (a second-order tensor typically denoted by I), defined by δ ij = { 1 if i = j 0 if i j (A8) In equation (A7), ɛ ijk is the permutation symbol or alternating tensor (a third-order tensor), that is defined in the following manner: 1 if i, j, k are an even permutation of 1,2,3 ɛ ijk = 1 if i, j, k are an odd permutation of 1,2,3 0 if i, j, k are not a permutation of 1,2,3 (A9) An even permutation of 1,2,3 is 1, 2, 3, 1, 2, 3, ; an odd permutation of 1,2,3 is 3, 2, 1, 3, 2, 1, The indices fail to be a permutation if two or more of them have the same value Remarks 1 The Kronecker delta is sometimes called the substitution operator since, for example, δ ij b j = δ i1 b 1 + δ i2 b 2 + δ i3 b 3 = b i (A10) δ ij C jk = C ik (A11) and so on From its definition we note that δ ii = 3 2 In light of the above discussion, the scalar or dot product of two vectors a and b is written as a b = (a i ê i ) (b j ê j ) = a i b j δ ij = a i b i (A12) In the special case when a = b, a a = a k a k = (a 1 ) 2 + (a 2 ) 2 + (a 3 ) 2 (A13) The magnitude of a vector is thus computed as a = (a a) 1 2 = (a k a k ) 1 2 (A14) 3 The vector or cross product of two vectors a and b is written as a b = (a i ê i ) (b j ê j ) = a i b j ɛ ijk ê k (A15) 4 The determinant of a square matrix A is

A4 Coordinate Systems 7 A 11 A 12 A 13 det A = A = A 21 A 22 A 23 A 31 A 32 A 33 = A 11 A 22 A 33 + A 12 A 23 A 31 + A 13 A 21 A 32 A 31 A 22 A 13 A 32 A 23 A 11 A 33 A 21 A 12 = ɛ ijk A 1i A 2j A 3k (A16) 5 It can also be shown that A i1 A i2 A i3 ɛ ijk det A = A j1 A j2 A j3 A k1 A k2 A k3 = A 1i A 1j A 1k A 2i A 2j A 2k A 3i A 3j A 3k A ir A is A it ɛ ijk ɛ rst det A = A jr A js A jt A kr A ks A kt δ ir δ is δ it ɛ ijk ɛ rst = δ jr δ js δ jt δ kr δ ks δ kt This leads to the following relations: ɛ ijk ɛ ist = δ js δ kt δ jt δ ks ɛ ijk ɛ ijr = 2δ kr ɛ ijk ɛ ijk = 6

8 A Brief Review of Tensors A5 Transformation Laws for Cartesian Tensors Define a point P in space referred to two rectangular Cartesian coordinate systems The base vectors for one coordinate system are unprimed, while for the second one they are primed The origins of both coordinate systems are assumed to coincide The position vector to this point is given by x = x i ê i = x jê j (A17) To obtain a relation between the two coordinate systems, form the scalar product of the above equation with either set of base vectors; viz, Upon expansion, ê k (x i ê i ) = ê k ( x jê ) j x i (ê k ê i ) = x jδ kj = x k Since ê i and ê k are unit vectors, it follows from the definition of the scalar product that ê k ê i = (1)(1) cos (ê k, ê i ) R ki (A18) (A19) (A20) The R ki are computed by taking (pairwise) the cosines of angles between the x k and x i axes For a prescribed pair of coordinate axes, the elements of R ki are thus constants that can easily be computed From equation (A19) it follows that the coordinate transformation for first-order tensors (vectors) is thus x k = R ki x i where the free index is the first one appearing on R We next seek the inverse transformation Beginning again with equation (A17), we write Thus, ê k (x i ê i ) = ê k ( x jê ) j (A21) (A22) or x i δ ki = x jê k ê j (A23) x k = R jk x j The free index is now the second one appearing on R (A24) Remark 1 In both above transformations, the second index on R is associated with the unprimed system In order to gain insight into the direction cosines R ij, we differentiate equation (A21) with respect to x i giving (with due change of dummy indices), x m = R mj x j = R mj δ ji = R mi (A25)

A5 Transformation Laws for Cartesian Tensors 9 We next differentiate equation (A24) with respect to x m, giving x k x m Using the chain rule, it follows that x k In direct notation this is written as x j = R jk x = R jk δ jm = R mk (A26) m = δ ki = x k x m x = R mk R mi (A27) m I = R T R (A28) implying that the R are orthogonal tensors (ie, R 1 = R T ) Linear transformations such as those given by equations (A21) and (A24), whose direction cosines satisfy the above equation, are thus called orthogonal transformations The transformation rules for second-order Cartesian tensors are derived in the following manner Let S be a second-order Cartesian tensor, and let u = Sv in the unprimed coordinates Similarly, in primed coordinates let u = S v Next we desire to relate S to S Using equation (A21), substitute for u and v to give (A29) (A30) But from equation (A29) implying that Ru = S Rv Ru = RSv (A31) (A32) RSv = S Rv Since v is an arbitrary vector, and since R is an orthogonal tensor, it follows that (A33) In a similar manner, S = RSR T or S ij = R ik R jl S kl (A34) S = R T S R or S ij = R mi R nj S mn (A35) The transformation rules for higher-order tensors are obtained in a similar manner For example, for tensors of rank three, and A ijk = R il R jm R kn A lmn A ijk = R li R mj R nk A lmn (A36) (A37)

10 A Brief Review of Tensors Finally, the fourth-order Cartesian tensor C transforms according to the following relations: and C ijkl = R ip R jq R kr R ls C pqrs C ijkl = R pi R qj R rk R sl C pqrs (A38) (A39) A6 Principal Values and Principal Directions In the present discussion, only symmetric second-order tensors with real components are considered For every symmetric tensor A, defined at some point in space, there is associated with each direction (specified by the unit normal n) at the point, a vector given by the inner product This is shown schematically in Figure A2 n v = An v = An (A40) Figure A2: Normal Direction Associated with the Vector v Remark 1 A may be viewed as a linear vector operator that produces the vector v conjugate to the direction n If v is parallel to n, the above inner product may be expressed as a scalar multiple of n; viz, v = An = λn or A ij n j = λn i (A41) The direction n i is called a principal direction, principal axis or eigenvector of A Substituting the relationship n i = δ ij n j into equation (A41) leads to the following characteristic equation of A: (A λi) = 0 or (A ij λδ ij ) n j = 0 (A42) For a non-trivial solution, the determinant of the coefficients must be zero; viz, det (A λi) = 0 or A ij λδ ij = 0 (A43)

A6 Principal Values and Principal Directions 11 This is called the characteristic equation of A In light of the symmetry of A, the expansion of equation (A43) gives (A 11 λ) A 12 A 13 A 12 (A 22 λ) A 23 A 13 A 23 (A 33 λ) = 0 (A44) The evaluation of this determinant leads to a cubic polynomial in λ, known as the characteristic polynomial of A; viz, where λ 3 Ī1λ 2 + Ī2λ Ī3 = 0 Ī 1 = tr (A) = A 11 + A 22 + A 33 = A kk (A45) (A46) Ī 2 = 1 2 (A iia jj A ij A ij ) (A47) Ī 3 = det (A) (A48) The scalar coefficients Ī1, Ī2 and Ī3 are called the first, second and third invariants, respectively, derived from the characteristic equation of A The three roots ( λ (i) ; i = 1, 2, 3 ) of the characteristic polynomial are called the principal values or eigenvalues of A Associated with each eigenvalue is an eigenvector n (i) For a symmetric tensor with real components, the principal values are real If the three principal values are distinct, the three principal directions are mutually orthogonal When referred to principal axes, A assumes a diagonal form; viz, λ (1) 0 A = 0 λ (2) 0 0 0 λ (3) (A49) Remark 1 Eigenvalues and eigenvectors have a useful geometric interpretation in two- and threedimensional space If λ is an eigenvalue of A corresponding to v, then Av = λv, so that depending on the value of λ, multiplication by A dilates v (if λ > 1), contracts v (if 0 < λ < 1), or reverses the direction of v (if λ < 0) Example 1: Invariants of First-Order Tensors Consider a vector v If the coordinate axes are rotated, the components of v will change However, the length (magnitude) of v remains unchanged As such, the length is said to be invariant In fact a vector (first-order tensor) has only one invariant, its length Example 2: Invariants of Second-Order Tensors

12 A Brief Review of Tensors A second order tensor possesses three invariants Denoting the tensor by A, its invariants are (these differ from the ones derived from the characteristic equation of A) I 1 = tr (A) = A 11 + A 22 + A 33 = A kk (A50) I 2 = 1 2 tr [( A 2)] = 1 2 A ika ki (A51) I 3 = 1 3 tr [( A 3)] = 1 3 A ika kj A ji (A52) Any function of the invariants is also an invariant To verify that the first invariant is unchanged under coordinate transformation, recall that A ij = R ik R jl A kl (A53) Thus, A mm = R mk R ml A kl = δ kl A kl = A kk (A54) For the second invariant, Finally, for the third invariant, A ika ki = (R il R km A lm ) (R kn R ip A np ) = R il R ip A lm R km R kn A np = δ lp A lm δ mn A np = A pm A mp (A55) A ika kma mi = (R il R kp A lp ) (R kn R mq A nq ) (R ms R it A st ) = R il R it A lp R kp R kn A nq R mq R ms A st = δ lt A lp δ pn A nq δ qs A st = A tp A pq A qt (A56)

A7 Tensor Calculus 13 A7 Tensor Calculus Several important differential operators are summarized below Gradient Operator The linear differential operator is called the gradient or del operator = ê 1 + ê 2 + ê 3 = ê i x 1 x 2 x 3 (A57) Gradient of a Scalar Field Let φ(x 1, x 2, x 3 ) be a scalar field The gradient of φ is the vector φ with components φ = grad φ = ê i φ = ê i φ,i If n = n i ê i is a unit vector, the scalar operator n = n i is called the directional derivative operator in the direction n (A58) (A59) Divergence of a Vector Field Let v(x 1, x 2, x 3 ) be a vector field The scalar quantity is called the divergence of v Curl of a Vector Field v = div v = v i = v i,i (A60) Let u(x 1, x 2, x 3 ) be a vector field The vector quantity is called the curl of u u = curl u = ε ijk u k x j ê i = ε ijk u k,j ê i (A61) Remark 1 When using u k,j for u k / x j, the indices are reversed in order as compared to the definition of the vector (cross) product; that is, whereas u v = ε kij u i v j ê k v = ε ijk v k,j ê i (A62)

14 A Brief Review of Tensors The Laplacian Operator The Laplacian operator is defined by 2 ( ) = div grad ( ) = ( ) = 2 ( ) = ( ),ii Let φ(x 1, x 2, x 3 ) be a scalar field The Laplacian of φ is then 2 φ = ( ) ê i (φ,j ê j ) = 2 φ x j (ê i ê j ) = φ,ji δ ij = φ,ii (A63) (A64) Let v(x 1, x 2, x 3 ) be a vector field The Laplacian of v is the following vector quantity: 2 v = 2 u k ê k = u k,ii ê k Remark 1 An alternate statement of the Laplacian of a vector is 2 v = ( v) ( v) (A65) (A66)

References [1] Fung, Y C, A First Course in Continuum Mechanics, Second Edition Englewood Cliffs, NJ: Prentice Hall (1977) [2] Joshi, A W, Matrices and Tensors in Physics, 2nd Edition A Halsted Press Book, New York: J Wiley and Sons (1984) [3] Mase, G E, Continuum Mechanics, Schaums Outline Series New York: McGraw-Hill Book Co (1970) [4] Sokolnikoff, I S, Tensor Analysis, Theory and Applications New York: J Wiley and Sons (1958) 15