TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Introduction to Tensors Tensor Algebra 2
Introduction SCALAR VECTOR,,... v, f,... v MATRIX σ, ε,...? C,... 3
Concept of Tensor A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix. Many physical quantities are mathematically represented as tensors. Tensors are independent of any reference system but, by need, are commonly represented in one by means of their component matrices. The components of a tensor will depend on the reference system chosen and will vary with it. 4
Order of a Tensor The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor. a a, a A, A A, A A, A Scalar: zero dimension Vector: 1 dimension 2 nd order: 2 dimensions 3 rd order: 3 dimensions 4 th order 3.14 1.2 v i 0.3 0.8 0.1 0 1.3 E ij 0 2.4 0.5 1.3 0.5 5.8 5
Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: eˆ eˆ, eˆ eˆ, eˆ eˆ Where: 1 2 2 3 3 1 eˆ 1, eˆ 1, eˆ 1 1 2 3 Note that eˆ 1 if i j eˆ 0 if i j i j ij 6
Cylindrical Coordinate System x 3 x1 r cos x(, r, z) x2 r sin x3 z x 1 x 2 eˆ cos θ eˆ sin θ eˆ 3 1 2 eˆ sin θ eˆ cos θ eˆ eˆ r z eˆ 1 2 7
Spherical Coordinate System x 3 x x1 rsin cos r,, x2 r sin sin x3 r cos x 1 x 2 eˆ sin θsin φ eˆ sin θcos φ eˆ cos θ eˆ r eˆ cos φ eˆ sin φ eˆ 1 2 3 1 2 eˆ cosθsin φ eˆ cosθcos φ eˆ sin θ eˆ φ 1 2 3 8
Indicial or (Index) Notation Tensor Algebra 9
Tensor Bases VECTOR A vector v can be written as a unique linear combination of the three vector basis eˆi for i 1, 2, 3. v In matrix notation: In index notation: v veˆ v eˆ v eˆ 1 1 2 2 3 3 v v v v 1 2 3 v 3 v 1 v 2 v i v i v i v i eˆ i tensor as a physical entity component i of the tensor in the given basis i 1, 2,3 10
Tensor Bases 2 nd ORDER TENSOR A 2 nd order tensor A can be written as a unique linear combination of the nine dyads for. eˆ eˆ ee ˆ ˆ i j i j Alternatively, this could have been written as: A A ee ˆˆ A ee ˆˆ A ee ˆˆ 11 1 1 12 1 2 13 1 3 A ee ˆˆ A ee ˆˆ A ee ˆˆ 21 2 1 22 2 2 23 2 3 A ee ˆˆ A ee ˆˆ A ee ˆˆ 31 3 1 32 3 2 33 3 3 i, j 1,2,3 ˆ ˆ ˆ ˆ ˆ ˆ 1 1 1 2 1 3 eˆ2 eˆ1 eˆ2 eˆ2 eˆ2 eˆ3 eˆ eˆ eˆ eˆ eˆ eˆ A A e e A e e A e e 11 12 13 A A A 21 22 23 A A A 31 3 1 32 3 2 33 3 3 11
Tensor Bases 2 nd ORDER TENSOR ˆ ˆ ˆ ˆ ˆ ˆ 1 1 1 2 1 3 eˆ2 eˆ1 eˆ2 eˆ2 eˆ2 eˆ3 eˆ eˆ eˆ eˆ eˆ eˆ A A e e A e e A e e 11 12 13 A A A 21 22 23 A A A 31 3 1 32 3 2 33 3 3 In matrix notation: In index notation: A A A A A A A A A A 11 12 13 21 22 23 31 32 33 A eˆ eˆ ij A Aij ij A ij i j tensor as a physical entity component ij of the tensor in the given basis i, j1,2,3 12
Tensor Bases 3 rd ORDER TENSOR A 3 rd order tensor A can be written as a unique linear combination of the 27 tryads for. eˆ eˆ eˆ ee ˆ ˆ eˆ i j k i j k A ˆ ˆ ˆ ˆ ˆ ˆ 112 e1e1e2 A122 e1e2 e2... Alternatively, this could have been written as: i, j, k 1,2,3 eˆ1 eˆ1 eˆ1 eˆ1 eˆ2 eˆ1 eˆ1 eˆ3 eˆ1 eˆ2 eˆ1 eˆ1 eˆ2 eˆ2 eˆ1 eˆ2 eˆ3 eˆ1 eˆ ˆ ˆ 3 e1 e1 eˆ ˆ ˆ 3 e2 e1 eˆ ˆ ˆ 3 e3 e1 A A A A 111 121 131 A A A 211 221 231 A A A 311 321 331 A A eee ˆˆˆ A eee ˆˆˆ A eee ˆˆˆ 111 1 1 1 121 1 2 1 131 1 3 1 A eee ˆˆˆA eee ˆˆˆA eee ˆˆˆ 211 2 1 1 221 2 2 1 231 2 3 1 A eee ˆˆˆA eee ˆˆˆ A eee ˆˆˆ 311 3 1 1 321 3 2 1 331 3 3 1 A eee ˆˆˆ A eee ˆˆˆ... 112 1 1 2 122 1 2 2 13
Tensor Bases 3 rd ORDER TENSOR eˆ1 eˆ1 eˆ1 eˆ1 eˆ2 eˆ1 eˆ1 eˆ3 eˆ1 eˆ2 eˆ1 eˆ1 eˆ2 eˆ2 eˆ1 eˆ2 eˆ3 eˆ1 eˆ ˆ ˆ 3 e1 e1 eˆ ˆ ˆ 3 e2 e1 eˆ ˆ ˆ 3 e3 e1 eˆ eˆ eˆ A eˆ eˆ eˆ... A A A A 111 121 131 A A A 211 221 231 A A A 311 321 331 A 112 1 1 2 122 1 2 2 In matrix notation: A A A A A A A A A A A A A A A A A A A A A A A A A A A A 113 123 133 112 111 122 121 132 131 213 223 233 212 222 232 211 221 231 313 323 333 312 322 332 311 321 331 14
Tensor Bases 3 rd ORDER TENSOR eˆ1 eˆ1 eˆ1 eˆ1 eˆ2 eˆ1 eˆ1 eˆ3 eˆ1 eˆ2 eˆ1 eˆ1 eˆ2 eˆ2 eˆ1 eˆ2 eˆ3 eˆ1 eˆ ˆ ˆ 3 e1 e1 eˆ ˆ ˆ 3 e2 e1 eˆ ˆ ˆ 3 e3 e1 eˆ eˆ eˆ A eˆ eˆ eˆ... A A A A 111 121 131 A A A 211 221 231 A A A 311 321 331 A 112 1 1 2 122 1 2 2 In index notation: A A ˆ ˆ ˆ ijk ei ej ek ijk A eˆ eˆ eˆ A ee ˆ ˆ eˆ ijk i j k ijk i j k tensor as a physical entity A A ijk ijk component ijk of the tensor i, j, k 1,2,3 in the given basis 15
Higher Order Tensors A tensor of order n is expressed as: A ˆ ˆ ˆ ˆ,... e e e... e A i i i i i i i 1 2 n 1 2 3 n where 1 2 i, i... i 1,2,3 n The number of components in a tensor of order n is 3 n. 16
Repeated-index (or Einstein s) Notation The Einstein Summation Convention: repeated Roman indices are summed over. 3 ab ab ab a b ab i is a mute index i is a talking index and j is a mute index A MUTE (or DUMMY) INDEX is an index that does not appear in a monomial after the summation is carried out (it can be arbitrarily changed of name ). A TALKING INDEX is an index that is not repeated in the same monomial and is transmitted outside of it (it cannot be arbitrarily changed of name ). i i i i i1 3 1 1 2 2 3 3 A b A b A b A b A b ij j ij j i1 1 i2 2 i3 3 j1 REMARK An index can only appear up to two times in a monomial. 17
Repeated-index (or Einstein s) Notation Rules of this notation: 1. Sum over all repeated indices. 2. Increment all unique indices fully at least once, covering all combinations. 3. Increment repeated indices first. 4. A comma indicates differentiation, with respect to coordinate x i. 3 2 ui u 3 2 i ui u A 3 i ij uii, ui, jj 2 x i1 x Aij, j x x x x i i j j j1 j j j1 x 5. The number of talking indices indicates the order of the tensor result A ij j 18
Kronecker Delta δ The Kronecker delta δ ij is defined as: ij 1 if i j 0 if i j Both i and j may take on any value in Only for the three possible cases where i = j is δ ij non-zero. ij 1 if i j 11 22 33 1 0 if i j 12 13 21... 0 1,2,3 ij ji REMARK Following Einsten s notation: ii Kronecker delta serves as a replacement operator: u u, A A ij j i ij jk ik 11 22 33 3 19
Levi-Civita Epsilon (permutation) ϵ The Levi-Civita epsilon e ijk is defined as: e ijk 0 if there is a repeated index 1 if ijk 123, 231or 312 1 if ijk 213, 132 or 321 3 indices e ijk e ikj 27 possible combinations. REMARK The Levi-Civita symbol is also named permutation or alternating symbol. 20
Relation between δ and ε e det i1 i2 i3 ijk j1 j2 j3 k1 k2 k3 e e ip iq ir det kp kq kr ijk pqr jp jq jr e e ijk pqk ip jq iq jp e e 2 ijk pjk pi e e 6 ijk ijk 21
Example Prove the following expression is true: e e 6 ijk ijk 22
Example - Solution e e e k e 1 e k e 2 e k e 3 ijk ijk 111 111 112 112 113 113 j 1 1 e121e121 e122e122 e123e123 i 1 j 2 1 e131e131 e132e132 e133e133 j 3 1 e211e211 e212e212 e213e213 i 2 e221e221 e222e222 e223e223 1 e231e231 e232e232 e233e233 1 e311e311 e312e312 e313e313 1 i 3 e e e e e e 321 321 322 322 323 323 e e e e e e 331 331 332 332 333 333 6 23
Vector Operations Tensor Algebra 24
Vector Operations Sum and Subtraction. Parallelogram law. a b b a c a b d c a b i i i d a b i i i Scalar multiplication abaeˆ a eˆ a eˆ 1 1 2 2 3 3 b i a i 25
Vector Operations Scalar or dot product yields a scalar uv u v cos where is the angle between the vectors u and v In index notation: i3 uvu ˆ v ˆ v ˆ ˆ iei je j ui jei e j uiv jij uivi uivi u v i1 u v ij 0( i j) 1( j i) Norm of a vector u 2 uu uu u 12 12 i i uu ueˆ u eˆ uu uu i i j j i j ij i i T 26
Vector Operations Some properties of the scalar or dot product u v vu u0 0 u vw uv uw uu 0 uu 0 u0 u0 u v0, u0, v0 u v Linear operator 27
Vector Operations Vector product (or cross product ) yields another vector c abba c a b sin In index notation: where is the angle between the vectors a and b cceˆ e a b eˆ c e a b i{1,2,3} 0 i i ijk j k i i ijk j k ab ab ab ab ab ab c eˆ eˆ eˆ e ab i 1 i 2 i 3 2 3 3 2 1 3 1 1 3 2 1 2 2 1 3 e ab 123 2 3 132 3 2 1 1 e ab e ab 231 3 1 213 1 3 1 1 e ab e ab 312 1 2 321 2 1 1 1 symb det eˆ eˆ eˆ a a a b b b 1 2 3 1 2 3 1 2 3 28
Vector Operations Some properties of the vector or cross product uv vu uv0, u0, v0 u v u avbw auvbuw Linear operator 29
Vector Operations Tensor product (or open or dyadic product) of two vectors: Auvuv Also known as the dyad of the vectors u and v, which results in a 2 nd order tensor A. Deriving the tensor product along an orthonormal basis {ê i }: uˆ v ˆ uv ˆ ˆ A ˆ ˆ A uv e e e e e e v, A A uv u i j{1,2,3} ij ij ij i j In matrix notation: u1 2 1 2 3 u 3 T u v uv u v v v i i j j i j i j ij i j u1v1 u1v2 u1v3 A11 A12 A13 u2v1 u2v2 u2v 3 A21 A22 A 23 u3v1 u3v2 u3v 3 A31 A32 A 33 30
Vector Operations Some properties of the open product: u v vu uvw uvw uvw vwu u vwx uxvw u v w uv uw uvw uv w uvw wuv Linear operator 31
Example Prove the following property of the tensor product is true: uvw uv w 32
Example - Solution u vw uv w vector 2 nd order tensor (matrix) 1 st order tensor (vector) scalar 1 st order tensor (vector) vector c vw c u vw u vw u u vw k k i ik i i k i i k k-component of vector c u v w v w c u u vw k k i i k i i k k-component of vector c cu vweˆ u vw eˆ uv w eˆ i j k k k k k k 33
Example Solution uvw uv w u ˆ v ˆ wˆ ˆ vwˆ ˆ vw ˆ ˆ ˆ i i j j k k ui i j k j k ui j k i j k u vw e e e e e e e e e uv w uˆ v ˆ w ˆ v ˆ ˆ w ˆ v w ˆ ˆ ˆ iei jej kek ui jeiej kek ui j k eiej ek 34
Vector Operations Triple scalar or box product V a bc c ab b ca a cos bc a cos b c sin In index notation: V height base area abc eijkab i jck a b c a a a V det b b b c c c 1 2 3 1 2 3 1 2 3 35
Vector Operations Triple vector product uvw uw vuv w In index notation: u e vw e u e vw u vw eˆ eˆ eˆ j j klm l m k ijk j klm l m i e e u vw eˆ u vw eˆ u ijk lmk j l m i il jm im jl j l m i vw eˆ u vw eˆ m i m i l l i i REMARK e e ijk pqk ip jq iq jp 36
Summary Scalar or dot product yields a scalar T uv u v u v cos u v u i v i Vector or cross product yields another vector c abba ab c i i eijkajbk Triple scalar or box product yields a scalar V cos sin a b c a b c a bc eijkabc i j k Triple vector product yields another vector uvwuw vuvw uvw u wvu vw i k k i k k i 37
Tensor Operations Tensor Algebra 38
Tensor Operations Summation (only for equal order tensors) ABBAC C A B ij ij ij Scalar multiplication (scalar times tensor) A C C ij A ij 39
Tensor Operations Dot product (.) or single index contraction product A b c 2 nd order A 3 rd order 1 st order b 1 st order 1 st order C 2 nd order A B C 2 nd order 2 nd order 2 nd order C C c A b i ij j A b ij ijk k A B ik ij jk AB BA Index j disappears (index contraction) Index k disappears (index contraction) Index j disappears (index contraction) REMARK AA A 2 40
Tensor Operations Some properties: A bc AbAc Linear operator 2 nd order unit (or identity) tensor 1u u1 u 1 e e e e [1] ij ij ij j i i i 1 1 0 0 0 1 0 0 0 1 41
2 nd Order Tensor Operations Some properties: 1A AA1 A BC ABAC A BC AB CABC AB BA 42
Example When does the relation nt Tn hold true? 43
Example - Solution n TT n vector 2 nd order tensor c vector c c nt Tn c nt k i ik c T n nt k ki i i ki c k T T c k if ik ki cc eˆ nt eˆ nt eˆ k k k k i ik k c c eˆ Tn eˆ nt eˆ k k k k i ki k 44
Example - Solution nt Tn COMPACT NOTATION nt i iktkini k 1, 2, 3 INDEX NOTATION T T n T T n c T c c MATRIX NOTATION 45
Example - Solution T T n T T n MATRIX NOTATION c T c c TT11 T12 T13 T12 T13 T11 T12 T13 n1 1 n1, n2, n 3 T21 T22 T 23 23 n2, n 3 T 21 T22 T 23 T21 T22 T 23 n 2 2 TT31 T32 T 33 33 T32 T 33 T31 T32 T 33 n3 3 T c 1c 1 T c c 1 c2 c3 2c 2 c 3c 3 46
2 nd Order Tensor Operations Transpose A11 A12 A13 A11 A21 A31 A A A A A A A T A 21 22 23 12 22 32 A31 A32 A 33 A13 A23 A 33 A T ij A ji Trace yields a scalar A T T A T T T AB B A T u v vu T T T AB A B A ii 11 22 33 Tr A ( A A A ) Some properties: Tr Tr T A A Tr A TrA Tr i j i i Tr a b Trab ab a b AB Tr BA Tr A B TrA TrB 47
2 nd Order Tensor Operations Double index contraction or double (vertical) dot product (:) A 2 nd order A 3 rd order A 4 th order : B c 2 nd order : B 2 nd order : B 2 nd order zero order (scalar) c 1 st order C 2 nd order c C c A B ij ij i Ai jkbjk B ij ijkl kl Indices i,j disappear (double index contraction) Indices j,k disappear (double index contraction) Indices k,l disappear (double index contraction) Indices contiguous to the double-dot (:) operator get vertically repeated (contraction) and they disappear in the resulting tensor (4 order reduction of the sum of orders). 48
2 nd Order Tensor Operations Some properties T T T T A: BTr A B Tr B A Tr AB Tr BA B: A 1: ATrAA: 1 T T A: B C B A : C A C : B A: uvuav uv: wxuwvx REMARK AB : CB : AC 49
2 nd Order Tensor Operations Double index contraction or double (horizontal) dot product ( ) A 2 nd order A 3 rd order A 4 th order 2 nd order B B 2 nd order B 2 nd order c zero order (scalar) c 1 st order C 2 nd order c C c A B ij ji i Ai jkbkj B ij ijkl lk Indices i,j disappear (double index contraction) Indices j,k disappear (double index contraction) Indices k,l disappear (double index contraction) Indices contiguous to the double-dot ( ) operator get horizontally repeated (contraction) and they disappear in the resulting tensor (4 orders reduction of the sum of orders). 50
Tensor Operations Norm of a tensor is a non-negative real number defined by A 12 A A AA 12 : 0 Tr ABTr AB BA BA ij ij 1ATrA A1 REMARK A: B AB Unless one of the two tensors is symmetric. 51
Example Prove that: T A: BTr A B AB Tr AB 52
Example - Solution T T T A B A B A B A B A B c Tr ik kk ki ik ik ij ij k j c Tr AB AB A B A B A B kk ki ik ki ik ij ji i j k i 53
2 nd Order Tensor Operations Determinant yields a scalar A A A det det det A Some properties: A A det A det Inverse 11 12 13 1 A A A21 A22 A 23 eijk A1 i A2 j A3 k eijkepqr Api Aqj Ark det AB det Adet B T 3 A det A det A 31 32 33 There exists a unique inverse A -1 of A when A is nonsingular, which satisfies the reciprocal relation: 1 1 A A 1 A A 1 1 AA ik kj Aik Akj ij ijk,, {1, 2,3} REMARK The tensor A is SINGULAR if and only if det A = 0. A is NONSINGULAR if det A 0. 6 54
2 nd Order Tensor Operations If A and B are invertible, the following properties apply: 1 A 1 1 1 1 AB B A A 1 1 1 T T det A 1 A 2 1 1 1 A A A A A A 1 A det A 1 T 1 det A 55
Example Prove that det A e ijk A. 1 i A2 j A3 k 56
Example - Solution det A det A A A A A A A A A 11 12 13 21 22 23 31 32 33 A A A A A A A A A A A A A A A A A A 11 22 33 21 32 13 31 12 23 13 22 31 23 32 11 33 12 21 57
Example - Solution e AA A ijk 1i 2 j 3k i 1 i 2 i 3 k 1 k 2 k 3 e111a11a21a31 e112 A11A21A32 e113a11 A21A33 1 e121a11a22 A31 e122 A11A22 A32 e123a11a22 A33 1 e131a11a23a31 e132 A11A23A32 e133a11 A23A33 1 A A A e A A A e A A A 211 12 21 31 212 12 21 32 213 12 21 33 e221a12 A22 A31 e222 A12 A22 A32 e223a12 A22 A33 1 e231a12 A23A31 e232 A12 A23A32 e233a12 A23A33 1 e311a13a21a31 e312 A13A21A32 e313a13 A21 A33 1 e A A A e A A A e A A A 321 13 22 31 322 13 22 32 323 13 22 33 e A A A e A A A e A A A 331 13 23 31 332 13 23 32 333 13 23 33 j 1 j 2 j 3 A11A22A33 A12A23 A31 A13A21 A32 A13 A22 A31 A12 A21A33 A11 A23 A32 58
Summary - Tensor Operations Dot product contraction of one index: c uvuivi C AB A B C cj u A d A u i ij ij ik kj ij ua c i j i ij j A u d ij j i D BA B A D ij ij ik kj ij C AB A B C ijkl ijkl ijm mkl ijkl D BA B A D ijkl ijkl ijm mkl ijkl C A B A B C ijk ijk im mjk ijk D BA B A D ijk ijk ijm mk ijk 59
Summary - Tensor Operations Double dot product contraction of two indices: ca B : AijBij C D ij AB : A B C ij ikm kmj ij BA : B A D ij ij ikm kmj ij ck A: B d B : A i A B c i k ij ijk k B A d ijk jk i C A: B A B C ijkl ijkl ijmp mpkl ijkl D B: A B A D ijkl ijkl ijmp mpij ijkl C A : B A B C ijk ijk ijlm lmk ijk D B: A B A D ijk ijk ilm lmjk ijk 60
Summary - Tensor Operations Transposed double dot product contraction of two indexes: c AB A B ij ji C D ij AB A B ij C ikm mkj ij BA B A D ij ij ikm mkj ij ck A B d B A i A B c i k ij jik k B A d ijk kj i C AB A B C ijkl ijkl ijmp pmkl ijkl D BA B A D ijkl ijkl ijmp pmij ijkl C A B A B C ijk ijk ijlm mlk ijk D BA B A D ijk ijk ilm mljk ijk 61
Summary - Tensor Operations Open product expansion of indexes: A u v ij ij C A B ijkl uv ijkl A i j ij AB C ij kl ijkl B v u ij ij D B A ijkl u B v i j ij BA D ijkl ij kl ijkl C ijk u A D A u ijk ua C ijk ijk i jk ijk Au D ij k ijk C A B A B C ijklm ijklm ij klm ijklm D BA B A D ijklm ijklm ijk lm ijklm 62
Differential Operators Tensor Algebra 63
Differential Operators A differential operator is a mapping that transforms a field vx, Ax... into another field by means of partial derivatives. The mapping is typically understood to be linear. Examples: Nabla operator Gradient Divergence Rotation 64
Nabla Operator The Nabla operator is a differential operator symbolically defined as: symbolic symb. ˆ x e In Cartesian coordinates, it can be used as a (symbolic) vector on its own: x1 symb. x2 x3 x i i 65
Gradient 66 The gradient (or open product of Nabla) is a differential operator defined as: Gradient of a scalar field Φ(x): Yields a vector symb. {1, 2,3} i i i i xi xi eˆ ˆ i i ei xi Gradient of a vector field v(x): x i Yields a 2 nd order tensor symb. v j v v v j i, j{1,2,3} ij i j xi xi v j v v e j ˆ i e ˆ v v v eˆ ˆ ˆ ˆ ij i ej ei ej x x i i e ˆi j
Gradient Gradient of a 2 nd order tensor field A(x): Yields a 3 rd order tensor symb. A jk A A A A jk i, j, k{1, 2,3} ijk ijk i jk xi xi A jk A A A eˆ ˆ ˆ ˆ ˆ ˆ ijk i ej ek ei ej ek xi A jk A e ˆ e ˆ e ˆ x i i j k 67
Divergence The divergence (or dot product of Nabla) is a differential operator defined as : Divergence of a vector field v(x): Yields a scalar v Divergence of a 2 nd order tensor A(x): Yields a vector i symb. vi v i i xi xi v symb. Aij ij xi xi v Aj A A j {1, 2,3} i ij A ij A A eˆ ˆ j j ej xi A v i x i Aij e ˆ x i j 68
Divergence The divergence can only be performed on tensors of order 1 or higher. If v 0, the vector field v x is said to be solenoid (or divergence-free). 69
Rotation 70 The rotation or curl (or vector product of Nabla) is a differential operator defined as: Rotation of a vector field v(x): Yields a vector symb. symb. v k v i eijk v v {1, 2,3} j k eijk k eijk i xj xj v v v k eˆ eˆ v i i eijk i x j Rotation of a 2 nd order tensor A(x): Yields a 2 nd order tensor e ijk vk e ˆ symb kl A il e A ijk kl eijk i, j, k {1, 2, 3} xj xj Akl A Ae e ˆ e ˆ A A kl il eˆ eˆ eˆ eˆ i l eijk i l x j. A x j ijk i l x j i
Rotation The rotation can only be performed on tensors of order 1 or higher. If v 0, the vector field v x is said to be irrotational (or curl-free). 71
Differential Operators - Summary GRADIENT scalar field Φ(x) vector field v(x) v i v i x i ij ij v x i j 2 nd order tensor A(x) A A ijk ijk A x i jk DIVERGENCE v A v i x i j A ij x i k ROTATION v i eijk A v x j il e ijk A x j kl 72
Example Given the vector determine vv x x xxeˆ xxeˆ xeˆ v, v, v. 1 2 3 1 1 2 2 1 3 73
Example - Solution vv x x xxeˆ xxeˆ xeˆ 1 2 3 1 1 2 2 1 3 Divergence: v xxx xx 1 2 x1 1 2 3 v v i x i v v v v i 1 2 3 v xx 2 3x1 xi x1 x2 x3 74
Example - Solution Divergence: v v i x i In matrix notation: T symb symb 1 2 3 symb T v,, xx 1 2 x1 x2 x 3 x1 symb xxx v 13 31 xxx xx x x xx xx x xx x x x x x x x 1 2 3 1 2 1 1 2 3 1 2 1 2 3 1 1 2 3 1 2 3 11 v xxx xx 1 2 x1 1 2 3 75
Example - Solution Rotation: v i e ijk v x j k v xxx xx 1 2 x1 1 2 3 In index notation: v v k i eijk x j v v v v v v e e e e e e 2 3 1 3 1 2 i12 i13 i21 i23 i31 i32 x1 x1 x2 x2 x3 x3 76
Example - Solution Rotation: In matrix notation v3 v2 e123 e132 x2 x 3 1 1 0 v 3 v1 v 213 231 xx 1 2 1 e e x1 x 3 1 1 x2 xx 1 3 v2 v 1 e312 e321 x1 x2 1 In compact notation: v 2 v3 v1 v i ei 12 ei 13 ei21 x x x 1 1 1 2 v v v e e e 3 1 2 i23 i31 i32 x2 x3 x3 xx 1 ˆ x xx v e eˆ 1 2 2 2 1 3 3 v xxx xx 1 2 x1 1 2 3 77
Example - Solution Rotation: Calculated directly in matrix notation: x ˆ ˆ ˆ 1 e1 e2 e3 v1 symb v v 2 det x 2 x1 x2 x 3 v3 v1 v2 v3 x3 v3 v 2 v1 v 3 v2 v 1 eˆ ˆ ˆ 1 e2 e3 x2 x3 x3 x1 x1 x2 eˆ xx 1eˆ x xx 1 2 2 2 1 3 3 v v v 1 1 2 3 2 1 2 3 1 xxx xx x 78
Example - Solution Gradient: v j ij ij x i v v In matrix notation x 1 xx 2 3 x2 1 symb symb symb xxx 1 2 3, xx 1 2, x 1 xx 1 3 x1 0 x 2 xx 1 2 0 0 T v v v x 31 In compact notation: 3 13 33 v v v 1 1 2 3 2 1 2 3 1 xxx vxxeˆ eˆ xeˆ eˆ eˆ eˆ xx eˆ eˆ xeˆ eˆ xxeˆ eˆ 2 3 1 1 2 1 2 1 3 1 3 2 1 1 2 2 1 2 3 1 xx x 79
Integral Theorems Tensor Algebra 80
Divergence or Gauss Theorem A Given a field in a volume V with closed boundary surface V and unit outward normal to the boundary n, the Divergence (or Gauss) Theorem states: V A dv na ds V Where: A A dv An ds V V represents either a vector field ( v(x) ) or a tensor field ( A(x) ). 81
Generalized Divergence Theorem A Given a field in a volume V with closed boundary surface V and unit outward normal to the boundary n, the Generalized Divergence Theorem states: V A dv na ds V Where: A dv An ds V V represents either the dot product ( ), the cross product ( ) or the tensor product ( ). A represents either a scalar field ( ϕ(x) ), a vector field ( v(x) ) or a tensor field ( A(x) ). 82
Curl or Stokes Theorem Given a vector field u in a surface S with closed boundary surface S and unit outward normal to the boundary n, the Curl (or Stokes) Theorem states: S u nds udr S where the curve of the line integral must have positive orientation, such that dr points counter-clockwise when the unit normal points to the viewer, following the right-hand rule. 83
Example Use the Generalized Divergence Theorem to show that x i i j ij S x nds V n j where is the position vector of. n j V An ds A V dv 84
Example - Solution x nds S i j x S n ds i j ij S x nds V Applying the Generalized Divergence Theorem: V xnds x V dv Applying the definition of gradient of a vector: j x xi ij x x i x ij x j 85
Example - Solution The Generalized Divergence Theorem in index notation: Then, S x nds i j V x x i j dv i j ij S x nds V x x nds dv dv V i S i j V V ij ij x j 86
References Tensor Algebra 87
References José Mª Goicolea, Mecánica de Medios Continuos: Resumen de Álgebra y Cálculo Tensorial, UPM. Eduardo W. V. Chaves, Mecánica del Medio Continuo,Vol. 1 Conceptos básicos, Capítulo 1: Tensores de Mecánica del Medio Continuo, CIMNE, 2007. L. E. Malvern. Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Clis, NJ, 1969. G. A. Holzapfel. Nonlinear solid mechanics: a continuum approach for engineering. 2000. 88