TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC - PowerPoint PPT Presentation

TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Introduction to Tensors Tensor Algebra 2

Introduction SCALAR   , , ... v VECTOR v f , , ... σ ε , , ... MATRIX ? , ... 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   a a ,   Vector : 1 dimension v i 0.3     0.8      2 nd order : 2 dimensions 0.1 0 1.3 A A ,    E 0 2.4 0.5   ij  3 rd order : 3 dimensions , A   A 1.3 0.5 5.8    4 th order … , A A 5

Cartesian Coordinate System  Given an orthonormal basis formed by three mutually perpendicular unit vectors:    ˆ ˆ ˆ ˆ ˆ ˆ e e , e e , e e 1 2 2 3 3 1 Where:    ˆ ˆ ˆ e 1 , e 1 , e 1 1 2 3  Note that    1 i j if     ˆ ˆ e e   i j ij  0 if i j   6

Cylindrical Coordinate System x 3    x r cos 1      x ( , , ) r z x r sin  2   x z  3   ˆ ˆ ˆ e cos θ e sin θ e r 1 2    ˆ ˆ ˆ e sin θ e cos θ e x  1 2 2  ˆ ˆ e e x z 3 1 7

Spherical Coordinate System x 3     x r sin cos 1          x r , , x r sin sin  2    x r cos  3    ˆ ˆ ˆ ˆ e sin sin θ φ e sin θ cos φ e cos θ e x r 1 2 3 2   ˆ ˆ ˆ e cos φ e sin φ e  1 2 x 1    ˆ ˆ ˆ ˆ e cos sin θ φ e cos cos θ φ e sin θ e φ 1 2 3 8

Indicial or (Index) Notation Tensor Algebra 9

Tensor Bases – VECTOR v  A vector can be written as a unique linear combination of the   ˆ i e i  three vector basis for . 1,2,3 v    ˆ ˆ ˆ v v e v e v e 1 1 2 2 3 3 v 3  In matrix notation:   v 1       v v  2 v   1 v   v 3 2  In index notation:   ˆ v v i e tensor as a physical entity i i   component i of the tensor in the i  v v i   i  given basis 1,2,3 10

Tensor Bases – 2 nd ORDER TENSOR  A 2 nd order tensor can be written as a unique linear combination A     ˆ ˆ ˆ ˆ i j  e e e e of the nine dyads for . , 1,2,3 i j i j              ˆ ˆ ˆ ˆ ˆ ˆ A A e e A e e A e e 11 1 1 12 1 2 13 1 3              ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 21 2 1 22 2 2 23 2 3             ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 31 3 1 32 3 2 33 3 3 Alternatively, this could have been written as:     ˆ ˆ ˆ ˆ ˆ ˆ A A e e A e e A e e 11 1 1 12 1 2 13 1 3     ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 21 2 1 22 2 2 23 2 3    ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 31 3 1 32 3 2 33 3 3 11

Tensor Bases – 2 nd ORDER TENSOR              ˆ ˆ ˆ ˆ ˆ ˆ A A e e A e e A e e 11 1 1 12 1 2 13 1 3              ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 21 2 1 22 2 2 23 2 3             ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 31 3 1 32 3 2 33 3 3  In matrix notation:   A A A 11 12 13       A A A A  21 22 23   A A A   31 32 33  In index notation:    tensor as a   ˆ ˆ A A ij e e i j physical entity ij   component ij of the tensor  A A   i j  , 1,2,3 ij in the given basis ij 12

Tensor Bases – 3 rd ORDER TENSOR  A 3 rd order tensor can be written as a unique linear combination A      i j k  ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e , , 1,2,3 of the 27 tryads for . i j k i j k                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A A 111 1 1 1 121 1 2 1 131 1 3 1                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 211 2 1 1 221 2 2 1 231 2 3 1                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 311 3 1 1 321 3 2 1 331 3 3 1            ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e ... A A 112 1 1 2 122 1 2 2 Alternatively, this could have been written as:     ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A A 111 1 1 1 121 1 2 1 131 1 3 1     ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 211 2 1 1 221 2 2 1 231 2 3 1     ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 311 3 1 1 321 3 2 1 331 3 3 1    ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e ... A A 112 1 1 2 122 1 2 2 13

Tensor Bases – 3 rd ORDER TENSOR                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A A 111 1 1 1 121 1 2 1 131 1 3 1                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 211 2 1 1 221 2 2 1 231 2 3 1                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 311 3 1 1 321 3 2 1 331 3 3 1            ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e ... A A 112 1 1 2 122 1 2 2  In matrix notation:         A A A   113 123 133 A A A      112 122 132  A A A 111 121 131     A A A   213 223 233   A A A        212 222 232 A A A A   211 221 231    A A A  313 323 333   A A A       312 322 332 A A A 311 321 331       14

Tensor Bases – 3 rd ORDER TENSOR                ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A A 111 1 1 1 121 1 2 1 131 1 3 1                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 211 2 1 1 221 2 2 1 231 2 3 1                 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e e e e A A A 311 3 1 1 321 3 2 1 331 3 3 1            ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e ... A A 112 1 1 2 122 1 2 2  In index notation:        ˆ ˆ ˆ e e e A A ijk i j k ijk   tensor as a     ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e A A physical entity ijk i j k ijk i j k   ijk  component ijk of the tensor A A ijk   i j k  , , 1,2,3 in the given basis 15

Higher Order Tensors  A tensor of order n is expressed as:      , ... ˆ ˆ ˆ ˆ A A e e e ... e i i i i i i i 1 2 n 1 2 3 n   i  where i i , ... 1,2,3 1 2 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      i is a mute a b a b a b a b a b index i i i i 1 1 2 2 3 3  i 1 3 i is a talking      A b A b A b A b A b index and j is a ij j ij j i 1 1 i 2 2 i 3 3 mute index  j 1  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