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  • TENSOR ALGEBRAContinuum Mechanics Course (MMC) - ETSECCPB - UPC

  • Tensor Algebra

    Introduction to Tensors

    2

  • Introduction

    SCALAR

    VECTOR

    MATRIX

    ?

    , , ...

    , , ...v f

    , , ...

    , ...C

    v

    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.

    Scalar: zero dimension

    Vector: 1 dimension

    2nd order: 2 dimensions

    3rd order: 3 dimensions

    4th order

    a,a a

    ,A A

    , AA, AA

    3.14 1.20.30.8

    vi

    0.1 0 1.30 2.4 0.5

    1.3 0.5 5.8ijE

    5

  • Cartesian Coordinate System

    Given an orthonormal basis formed by three mutually perpendicular unit vectors:

    Where:

    Note that

    1 2 2 3 3 1 , , e e e e e e

    1 2 3 1 , 1 , 1 e e e

    1

    0i j iji ji j

    e e

    if

    if

    6

  • Cylindrical Coordinate System

    1 2

    1 2

    3

    cos sin sin cos

    r

    z

    e e ee e ee e

    1

    2

    3

    cos ( , , ) sin

    x rr z x r

    x z

    x

    1x

    2x

    3x

    7

  • Spherical Coordinate System

    1 2 3

    1 2

    1 2 3

    sin sin sin cos cos cos sin cos sin cos cos sin

    r

    e e e ee e ee e e e

    1

    2

    3

    sin cos, , sin sin

    cos

    x rr x r

    x r

    x

    3x

    1x

    2x

    8

  • Tensor Algebra

    Indicial or (Index) Notation

    9

  • Tensor Bases VECTOR

    A vector can be written as a unique linear combination of the three vector basis for .

    In matrix notation:

    In index notation:

    ie

    1 1 2 2 3 3 v v v v e e ev

    v

    1

    2

    3

    vvv

    v

    vi ii

    v e vii v

    tensor as a physical entity

    component i of the tensor in the given basis

    1v2v

    3v

    1,2,3i

    1,2,3i

    10

  • Tensor Bases 2nd ORDER TENSOR

    A 2nd order tensor can be written as a unique linear combination of the nine dyads for .

    Alternatively, this could have been written as:

    , 1,2,3i j i j i j e e e eA

    1 1 1 2 1 3

    2 1 2 2 2 3

    3 1 3 2 3 3

    11 12 13

    21 22 23

    31 32 33

    A A AA A AA A A

    A e e e e e ee e e e e ee e e e e e

    1 1 1 2 1 3

    2 1 2 2 2 3

    3 1 3 2 3 3

    11 12 13

    21 22 23

    31 32 33

    A A A

    A A A

    A A A

    A e e e e e e

    e e e e e e

    e e e e e e

    11

  • Tensor Bases 2nd ORDER TENSOR

    In matrix notation:

    In index notation:

    11 12 13

    21 22 23

    31 32 33

    A A AA A AA A A

    A

    Aij i jij

    A e e ijij AA

    tensor as a physical entity

    component ij of the tensor in the given basis

    1 1 1 2 1 3

    2 1 2 2 2 3

    3 1 3 2 3 3

    11 12 13

    21 22 23

    31 32 33

    A A A

    A A A

    A A A

    A e e e e e e

    e e e e e e

    e e e e e e

    , 1,2,3i j

    12

  • Tensor Bases 3rd ORDER TENSOR

    A 3rd order tensor can be written as a unique linear combination of the 27 tryads for .

    Alternatively, this could have been written as:

    i j k i j k e e e e e eA

    , , 1,2,3i j k

    1 1 1 1 2 1 1 3 1

    2 1 1 2 2 1 2 3 1

    3 1 1 3 2 1 3 3 1

    1 1 2 1 2 2

    ...

    111 121 131

    211 221 231

    311 321 331

    112 122

    e e e e e e e e e

    e e e e e e e e e

    e e e e e e e e e

    e e e e e e

    A A A A

    A A A

    A A A

    A A

    1 1 1 1 2 1 1 3 1

    2 1 1 2 2 1 2 3 1

    3 1 1 3 2 1 3 3 1

    1 1 2 1 2 2

    ...

    111 121 131

    211 221 231

    311 321 331

    112 122

    e e e e e e e e ee e e e e e e e ee e e e e e e e ee e e e e e

    A A A A

    A A A

    A A A

    A A

    13

  • Tensor Bases 3rd ORDER TENSOR

    In matrix notation:

    1 1 1 1 2 1 1 3 1

    2 1 1 2 2 1 2 3 1

    3 1 1 3 2 1 3 3 1

    1 1 2 1 2 2

    ...

    111 121 131

    211 221 231

    311 321 331

    112 122

    e e e e e e e e e

    e e e e e e e e e

    e e e e e e e e e

    e e e e e e

    A

    A A A

    A A A

    A A A

    A A

    113 123 133

    213 223 233

    313 323 333

    A A A

    A A A

    A A A

    112 122 132

    212 222 232

    312 322 332

    A A A

    A A A

    A A A

    111 121 131

    211 221 231

    311 321 331

    A A A

    A A A

    A A A

    A

    14

  • Tensor Bases 3rd ORDER TENSOR

    In index notation:

    ijk i j kijk

    ijk i j k ijk i j k

    e e e

    e e e e e e

    A A

    A A

    ijkijk A A

    1 1 1 1 2 1 1 3 1

    2 1 1 2 2 1 2 3 1

    3 1 1 3 2 1 3 3 1

    1 1 2 1 2 2

    ...

    111 121 131

    211 221 231

    311 321 331

    112 122

    e e e e e e e e e

    e e e e e e e e e

    e e e e e e e e e

    e e e e e e

    A A A A

    A A A

    A A A

    A A

    tensor as a physical entity

    component ijk of the tensor in the given basis , , 1,2,3i j k

    15

  • A tensor of order n is expressed as:

    The number of components in a tensor of order n is 3n.

    Higher Order Tensors

    1 2 1 2 3, ... ...

    n ni i i i i i iA A e e e e

    1 2, ... 1,2,3ni i i where

    16

  • The Einstein Summation Convention: repeated Roman indices are summed over.

    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).

    3

    1 1 2 2 3 313

    1 1 2 2 3 31

    i i i ii

    ij j ij j i i ij

    a b a b a b a b a b

    A b A b A b A b A b

    REMARKAn index can only appear up to two times in a monomial.

    Repeated-index (or Einsteins) Notation

    i is a mute index

    i is a talking index and j is a

    mute index

    17

  • 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 .

    5. The number of talking indices indicates the order of the tensor result

    Repeated-index (or Einsteins) Notation

    3

    ,1

    i ii i

    ii i

    u uux x

    2 23

    , 21

    i ii jj

    jj j j

    u uux x x

    3

    ,1

    ij ijij j

    jj j

    A AA

    x x

    18

  • Kronecker Delta

    The Kronecker delta ij is defined as:

    Both i and j may take on any value in Only for the three possible cases where i = j is ij non-zero.

    10ij

    i ji j

    if

    if

    11 22 33

    12 13 21

    1 10 ... 0ij

    i ji j

    if

    if

    ij ji REMARKFollowing Einstens notation: Kronecker delta serves as a replacement operator:

    11 22 33 3ii

    ,ij j i ij jk iku u A A

    1,2,3

    19

  • Levi-Civita Epsilon (permutation)

    The Levi-Civita epsilon is defined as:

    3 indices 27 possible combinations.

    01 123, 231 3121 213,132 321

    ijk ijkijk

    if there is a repeated index

    if or

    if or

    e

    REMARKThe Levi-Civita symbol is also named permutation or alternating symbol.

    ijk ikj e e

    ijke

    20

  • Relation between and

    1 2 3

    1 2 3

    1 2 3

    deti i i

    ijk j j j

    k k k

    e detip iq ir

    ijk pqr jp jq jr

    kp kq kr

    e e

    ijk pqk ip jq iq jp e e

    2ijk pjk pie e

    6ijk ijk e e

    21

  • Example

    Prove the following expression is true:

    6ijk ijk e e

    22

  • 211 211 212 212 213 213

    221 221 222 222 223 223

    231 231 232 232 233 233

    e e e e e e

    e e e e e e

    e e e e e e

    2i

    311 311 312 312 313 313

    321 321 322 322 323 323

    331 331 332 332 333 333

    e e e e e e

    e e e e e e

    e e e e e e

    3i

    121 121 122 122 123 123 e e e e e e 2j 131 131 132 132 133 133 e e e e e e 3j

    Example - Solution

    111 111 112 112 113 113ijk ijk e e e e e e e e1

    1

    1

    1

    1

    1

    6

    1i

    1j 1k 2k 3k

    23

  • Tensor Algebra

    Vector Operations

    24

  • Sum and Subtraction. Parallelogram law.

    Scalar multiplication

    Vector Operations

    a b b a ca b d

    1 1 2 2 3 3 a a a a b e e e

    i i i

    i i i

    c a bd a b

    i ib a

    25

  • Scalar or dot product yields a scalar

    In index notation:

    Norm of a vector

    Vector Operations

    cos u v u vwhere is the angle

    between the vectors u and v

    2 i i j j i j ij i iu u u u u uu u e eu