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Tensor Algebra TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC Introduction to Tensors Tensor Algebra 2 Introduction SCALAR , , ... v VECTOR vf, , ... MATRIX σε,,... ? 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 Scalar: zero dimension 3.14 1.2 v 0.3 a , a Vector: 1 dimension i 0.8 0.1 0 1.3 2nd order: 2 dimensions A, A E 02.40.5 ij rd A , A 3 order: 3 dimensions 1.3 0.5 5.8 A , A 4th order … 5 Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: eeˆˆ12,, ee ˆˆ 23 ee ˆˆ 31 Where: eeeˆˆˆ1231, 1, 1 Note that 1 if ij eeˆˆi j ij 0 if ij 6 Cylindrical Coordinate System x3 xr1 cos x(,rz , ) xr2 sin xz3 eeeˆˆˆr cosθθ 12 sin eeeˆˆˆsinθθ cos x2 12 eeˆˆz 3 x1 7 Spherical Coordinate System x3 xr1 sin cos xrxr, , 2 sin sin xr3 cos ˆˆˆˆ x2 eeeer sinθφ sin 123sin θ cos φ cos θ eeeˆˆˆ cosφφ 12sin x1 eeeeˆˆˆˆφ cosθφ sin 123cos θ cos φ sin θ 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 eˆ for i 1, 2, 3 . i v ve vv11ˆˆˆ 2 e 2 v 33 e v3 In matrix notation: v1 v v 2 v1 v3 v2 In index notation: ˆ tensor as a physical entity ve vii i i v v component of the tensor in the i i given basis i 1, 2, 3 10 Tensor Bases – 2nd ORDER TENSOR A 2nd order tensor A can be written as a unique linear combination of the nine dyads for . eeeeˆˆˆˆijij ij,1,2,3 AeeeeeeAA11ˆˆ11 12 ˆˆ 12 A 13 ˆˆ 13 AA21eeˆˆ21 22 ee ˆˆ 22 A 23 ee ˆˆ 23 AA31eeˆˆ31 32 ee ˆˆ 32 A 33 ee ˆˆ 33 Alternatively, this could have been written as: AeeeeeeAA11ˆˆ11 12 ˆˆ 12 A 13 ˆˆ 13 AA21eeˆˆ21 22 ee ˆˆ 22 A 23 ee ˆˆ 23 AA31eeˆˆ31 32 ee ˆˆ 32 A 33 ee ˆˆ 33 11 Tensor Bases – 2nd ORDER TENSOR Aeeeeee AA11ˆˆ11 12 ˆˆ 12 A 13 ˆˆ 13 AA21eeˆˆ21 22 ee ˆˆ 22 A 23 ee ˆˆ 23 AA31eeˆˆ31 32 ee ˆˆ 32 A 33 ee ˆˆ 33 In matrix notation: A11AA 12 13 A A AA 21 22 23 A31AA 32 33 In index notation: AeeA ˆˆ tensor as a ij i j physical entity ij A A component ij of the tensor ij ij in the given basis ij,1,2,3 12 Tensor Bases – 3rd ORDER TENSOR A 3rd order tensor A can be written as a unique linear combination of the 27 tryads eeˆˆ ijkijk eeee ˆˆˆˆ for ijk ,, 1,2,3 . A AA111eeeˆˆˆ111 121 eee ˆˆˆ 121 A 131 eee ˆˆˆ 131 AAA211eeeˆˆˆ211 221 eee ˆˆˆ 221 231 eee ˆˆˆ 231 AA311eeeˆˆˆ311 321 eee ˆˆˆ 321 A 331 eee ˆˆˆ 331 AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2... Alternatively, this could have been written as: A AA111eeeˆˆˆ111 121 eee ˆˆˆ 121 A 131 eee ˆˆˆ 131 AAA211eeeˆˆˆ211 221 eee ˆˆˆ 221 231 eee ˆˆˆ 231 AA311eeeˆˆˆ311 321 eee ˆˆˆ 321 A 331 eee ˆˆˆ 331 AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2 ... 13 Tensor Bases – 3rd ORDER TENSOR A AA111eeeˆˆˆ111 121 eee ˆˆˆ 121 A 131 eee ˆˆˆ 131 AAA211eeeˆˆˆ211 221 eee ˆˆˆ 221 231 eee ˆˆˆ 231 AA311eeeˆˆˆ311 321 eee ˆˆˆ 321 A 331 eee ˆˆˆ 331 AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2... In matrix notation: AAA 113 123 133 AAA112 122 132 AAA111 121 131 AAA 213 223 233 AAA212 222 232 A AAA211 221 231 AAA313 323 333 AAA312 322 332 AAA311 321 331 14 Tensor Bases – 3rd ORDER TENSOR A AA111eeeˆˆˆ111 121 eee ˆˆˆ 121 A 131 eee ˆˆˆ 131 AAA211eeeˆˆˆ211 221 eee ˆˆˆ 221 231 eee ˆˆˆ 231 AA311eeeˆˆˆ311 321 eee ˆˆˆ 321 A 331 eee ˆˆˆ 331 AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2... In index notation: ˆˆˆ A Aijkeee i j k ijk tensor as a AAijkieeˆˆj e ˆkijki eee ˆˆˆj k physical entity component ijk of the tensor A Aijk ijk in the given basis ijk,, 1,2,3 15 Higher Order Tensors A tensor of order n is expressed as: Aeeee A ˆˆˆ... ˆ ii12, ... inn i 1 i 2 i 3 i where ii12, ... in 1,2,3 The number of components in a tensor of order n is 3n. 16 Repeated-index (or Einstein’s) Notation The Einstein Summation Convention: repeated Roman indices are summed over. 3 i is a mute index abii ab ii ab11 ab 2 2 ab 33 i1 i is a talking 3 index and j is a A bAbAbAbAb mute index ij j ij j i11 i 2 2 i 33 j1 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”). 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 xi . uu3 22uu3 A 3 A u iiu iiA ij ij ii, ijj, 2 ij, j xiii1 x x jjxxj1 j x jjj1 x 5. The number of talking indices indicates the order of the tensor result 18 Kronecker Delta δ The Kronecker delta δij is defined as: 1 if ij ij 0 if ij Both i and j may take on any value in 1, 2, 3 Only for the three possible cases where i = j is δij non-zero. 11if ij11 22 33 ij 0if ij 12 13 21 ... 0 REMARK Following Einsten’s notation: 3 ij ji ii 11 22 33 Kronecker delta serves as a replacement operator: ijuu j i, ij A jk A ik 19 Levi-Civita Epsilon (permutation) ϵ The Levi-Civita epsilon eijk is defined as: 0 if there is a repeated index eijk 1ifijk 123, 231 or 312 1ifijk 213,132 or 321 3 indices 27 possible combinations. eeijkik j REMARK The Levi-Civita symbol is also named permutation or alternating symbol. 20 Relation between δ and ε ii123 i ip iq ir e det eeijk pqr det jp jq jr ijk j123 j j kk123 k kp kq kr eeijk pqk ip jq iq jp eeijk pjk 2 pi eeijk ijk 6 21 Example Prove the following expression is true: eeijk ijk 6 22 Example - Solution k 1 k 2 k 3 eeijk ijk e111 e 111 e 112 e 112 e 113 e 113 j 1 1 ee121 121 ee 122 122 ee 123 123 j 2 i 1 1 ee131 131 ee 132 132 ee 133 133 j 3 1 ee211 211 ee 212 212 ee 213 213 i 2 ee221 221 ee 222 222 ee 223 223 1 ee231 231 ee 232 232 ee 233 233 1 ee311 311 ee 312 312 ee 313 313 1 i 3 ee321 321 ee 322 322 ee 323 323 ee331 331 ee 332 332 ee 333 333 6 23 Vector Operations Tensor Algebra 24 Vector Operations Sum and Subtraction. Parallelogram law. ab bac cabiii ab d dabiii Scalar multiplication abaa11 eˆˆˆ 2 e 2 a 33 e baii 25 Vector Operations Scalar or dot product yields a scalar where is the angle uv uvcos between the vectors u and v In index notation: i3 T ˆˆ ˆˆ uvuuuuuii e vv jj e i ji e e j i vv jij ii ii v u v i1 u v 0(i j ) ij 1(j i ) Norm of a vector 12 12 u uu uuii 2 u uuuii eˆˆ u j e j uu i jij uu ii 26 Vector Operations Some properties of the scalar or dot product uv vu u00 uvw uvuw Linear operator uu0 u 0 uu0 u 0 uv 0, u 0 , v 0 u v 27 Vector Operations Vector product (or cross product ) yields another vector cab ba cab sin where is the angle between the vectors a and b 0 In index notation: cecabcabiˆˆ e {1,2,3} iiiee ijk j k i ijk j k i 1 i 2 i 3 eeeˆˆˆ symb 123 ceeab abˆˆ ab ab ab ab e ˆ det aaa 23 32 1 31 13 2 12 21 3 123 eeab ab eeab ab bbb ee123ab23 132 ab 32 231 3 1 213 1 3 312 1 2 321 2 1 123 11 1111 28 Vector Operations Some properties of the vector or cross product uv vu uv 0,, u 0v0 uv || uvwuvuwab a b 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 2nd order tensor A.
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